tag:blogger.com,1999:blog-43834308918652592892024-03-14T10:35:28.426+06:00Continuous Signal and Linear SystemA continuous signal is a mathematical function of an independent variableUnknownnoreply@blogger.comBlogger11125tag:blogger.com,1999:blog-4383430891865259289.post-84385441608093641512010-06-05T00:13:00.000+06:002017-11-02T15:45:59.224+06:00Classification of Signals<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="color: rgb(51 , 102 , 255); font-size: 100%;">Along with the classification of signals below, it is also important to understand the Classification of Systems.</span>
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<span style="font-size: 100%; font-weight: bold;">Continuous-Time vs. Discrete-Time:</span>
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<span style="font-size: 100%;">As the names suggest, this classification is determined by whether or not the time axis (x-axis) is discrete (countable) or continuous (Figure 1). A continuous-time signal will contain a value for all real numbers along the time axis. In contrast to this, a discrete-time signal is often created by using the sampling theorem to sample a continuous signal, so it will only have values at equally spaced intervals along the time axis.</span>
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<tr><td class="tr-caption" style="text-align: center;">Figure 1</td></tr>
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<span style="font-size: 100%;"><span style="font-weight: bold;">Analog vs. Digital:</span>
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<span style="font-size: 100%;">The difference between analog and digital is similar to the difference between continuous-time and discrete-time. In this case, however, the difference is with respect to the value of the function (y-axis) (Figure 2). Analog corresponds to a continuous y-axis, while digital corresponds to a discrete y-axis. An easy example of a digital signal is a binary sequence, where the values of the function can only be one or zero.</span>
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<tr><td class="tr-caption" style="text-align: center;">Figure 2</td></tr>
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<span style="font-weight: bold;">Periodic vs. Aperiodic:</span>
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Periodic signals repeat with some period T, while aperiodic, or nonperiodic, signals do not (Figure 3). We can define a periodic function through the following mathematical expression, where t can be any number and T is a positive constant:
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f(t) = f(T + t).........(1)
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The fundamental period of our function, f(t) is the smallest value of T that the still allows Equation (1) to be true.
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<tr><td class="tr-caption" style="text-align: center;">(a) A periodic signal with period To</td></tr>
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(b) An aperiodic signal</div>
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Figure 3</div>
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<span style="font-weight: bold;">Causal vs. Anticausal vs. Noncausal:</span>
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Causal signals are signals that are zero for all negative time, while anticausal are signals that are zero for all positive time. Noncausal signals are signals that have nonzero values in both positive and negative time (Figure 4).
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<tr><td class="tr-caption" style="text-align: center;">(a) A causal signal</td></tr>
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<tr><td class="tr-caption" style="text-align: center;">(b) An anticausal signal</td></tr>
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(c) A noncausal signal</div>
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Figure 4</div>
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<span style="font-weight: bold;">Even vs. Odd:</span>
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An even signal is any signal f such that f(t)=f(-t). Even signals can be easily spotted as they are symmetric around the vertical axis. An odd signal, on the other hand, is a signal f such that f(t)=-(f(-t)).(Figure 5).
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(b) An odd signal</div>
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Figure 5</div>
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Using the definitions of even and odd signals, we can show that any signal can be written as a combination of an even and odd signal. That is, every signal has an odd-even decomposition. To demonstrate this, we have to look no further than a single equation.
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f(t)=1/2(f(t)+f(-t))+1/2(f(t)−f(-t)).............(2)
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By multiplying and adding this expression out, it can be shown to be true. Also, it can be shown that f(t)+f(-t) fulfills the requirement of an even function, while f(t)−f(-t) fulfills the requirement of an odd function (Figure 6).
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<tr><td class="tr-caption" style="text-align: center;">(a) The signal we will decompose using odd-even decomposition</td></tr>
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<tr><td class="tr-caption" style="text-align: center;">(b) Even part: e(t)=1/2(f(t)+f(-t))</td></tr>
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(c) Odd part: o(t)=1/2(f(t)−f(-t))</div>
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Figure 6</div>
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<span style="font-weight: bold;">Deterministic vs. Random:</span>
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A deterministic signal is a signal in which each value of the signal is fixed and can be determined by a mathematical expression, rule, or table. Because of this the future values of the signal can be calculated from past values with complete confidence. On the other hand, a random signal has a lot of uncertainty about its behavior. The future values of a random signal cannot be accurately predicted and can usually only be guessed based on the averages of sets of signals (Figure 7).
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<tr><td class="tr-caption" style="text-align: center;">(a) Deterministic Signal</td></tr>
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(b) Random Signal</div>
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Figure 7</div>
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<span style="font-weight: bold;">Right-Handed vs. Left-Hand Signal:</span>
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A right-handed signal and left-handed signal are those signals whose value is zero between a given variable and positive or negative infinity. Mathematically speaking, a right-handed signal is defined as any signal where f(t)=0 for<link href="file:///C:%5CDOCUME%7E1%5CRashad%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml" rel="File-List"></link><!--[if gte mso 9]><xml> <w:worddocument> <w:view>Normal</w:View> <w:zoom>0</w:Zoom> <w:compatibility> <w:breakwrappedtables/> <w:snaptogridincell/> <w:wraptextwithpunct/> <w:useasianbreakrules/> </w:Compatibility> <w:browserlevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style> <!-- /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman";} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;} </style> </div>
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--><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman";} </style> <![endif]--> t <>1 < α<o:p></o:p> , and a left-handed signal is defined as any signal where f(t)=0 for <link href="file:///C:%5CDOCUME%7E1%5CRashad%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml" rel="File-List"></link><!--[if gte mso 9]><xml> <w:worddocument> <w:view>Normal</w:View> <w:zoom>0</w:Zoom> <w:compatibility> <w:breakwrappedtables/> <w:snaptogridincell/> <w:wraptextwithpunct/> <w:useasianbreakrules/> </w:Compatibility> <w:browserlevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style> <!-- /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman";} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;} </style> --><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman";} </style> <![endif]--><span style="font-family: "; font-size: 12pt;">t > t<sub>1</sub> > - α</span>. See (Figure 8) for an example. Both figures "begin" at <span style="font-family: "; font-size: 12pt;">t<sub>1</sub></span> and then extends to positive or negative infinity with mainly nonzero values.
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(b) Left-handed signal</div>
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Figure 8<br />
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<span style="font-weight: bold;">Finite vs. Infinite Length:</span>
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As the name applies, signals can be characterized as to whether they have a finite or infinite length set of values. Most finite length signals are used when dealing with discrete-time signals or a given sequence of values. Mathematically speaking, f(t) is a finite-length signal if it is nonzero over a finite interval <link href="file:///C:%5CDOCUME%7E1%5CRashad%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml" rel="File-List"></link><!--[if gte mso 9]><xml> <w:worddocument> <w:view>Normal</w:View> <w:zoom>0</w:Zoom> <w:compatibility> <w:breakwrappedtables/> <w:snaptogridincell/> <w:wraptextwithpunct/> <w:useasianbreakrules/> </w:Compatibility> <w:browserlevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style> <!-- /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman";} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;} </style> </div>
--><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman";} </style> <![endif]--><span style="font-family: "; font-size: 12pt;">t<sub>1</sub> <>2</span> Where t<sub>1</sub> > - α and t<sub>2</sub> < α .An example can be seen in Figure 9. Similarly, an infinite-length signal f(t) is defined as nonzero over all real numbers: α ≤ f(t) ≤ - α <o:p></o:p> <o:p></o:p>
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<tr><td class="tr-caption" style="text-align: center;">Figure 9: Finite-Length Signal.</td></tr>
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Note that it only has nonzero values on a set, finite interval.
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4383430891865259289.post-47473265637791306282010-06-05T00:05:00.000+06:002017-11-02T15:34:25.494+06:00Signal Operations<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="font-family: Arial; font-size: 100%;">This module will look at two signal operations, time shifting and time scaling. Signal operations are operations on the time variable of the signal. These operations are very common components to real-world systems and, as such, should be understood thoroughly when learning about signals and systems.<o:p></o:p></span><br />
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<span style="font-size: 100%;"><b><span style="font-family: "arial";">Time Shifting:<o:p></o:p></span></b></span></div>
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<span style="font-family: Arial; font-size: 100%;">Time shifting is, as the name suggests, the shifting of a signal in time. This is done by adding or subtracting the amount of the shift to the time variable in the function. Subtracting a fixed amount from the time variable will shift the signal to the right (delay) that amount, while adding to the time variable will shift the signal to the left (advance).<o:p></o:p></span></div>
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<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: 16px; text-align: justify;">Figure 1: </span><span style="font-size: 16px; text-align: justify;">f </span><span style="font-size: 16px; text-align: justify;">(t – T) moves</span><span style="font-size: 16px; text-align: justify;"> (delays) </span><span style="font-size: 16px; text-align: justify;">f </span><span style="font-size: 16px; text-align: justify;">to the right by </span><span style="font-size: 16px; text-align: justify;">T</span><span style="font-size: 16px; text-align: justify;">.<br /></span></td></tr>
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<span style="font-size: 100%;"><b><span style="font-family: "arial";">Time Scaling:<o:p></o:p></span></b></span></div>
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<span style="font-family: Arial; font-size: 100%;">Time scaling compresses and dilates a signal by multiplying the time variable by some amount. If that amount is greater than one, the signal becomes narrower and the operation is called compression, while if the amount is less than one, the signal becomes wider and is called dilation. It often takes people quite a while to get comfortable with these operations, as people's intuition is often for the multiplication by an amount greater than one to dilate and less than one to compress.<o:p></o:p></span></div>
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<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: 16px; text-align: justify;">Figure 2: </span><span style="font-size: 16px; text-align: justify;">f </span><span style="font-size: 16px; text-align: justify;">(</span><span style="font-size: 16px; text-align: justify;">at</span><span style="font-size: 16px; text-align: justify;">) </span><span style="font-size: 16px; text-align: justify;">compresses </span><span style="font-size: 16px; text-align: justify;">f </span><span style="font-size: 16px; text-align: justify;">by </span><span style="font-size: 16px; text-align: justify;">a</span><span style="font-size: 16px; text-align: justify;">.</span></td></tr>
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<span style="font-size: 100%;"><b><span style="font-family: "arial";">Time Reversal:<o:p></o:p></span></b></span></div>
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<span style="font-family: Arial; font-size: 100%;">A natural question to consider when learning about time scaling is: What happens when the time variable is multiplied by a negative number? The answer to this is time reversal. This operation is the reversal of the time axis, or flipping the signal over the y-axis.<o:p></o:p></span></div>
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<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: 16px; text-align: justify;">Figure 3: Reverse the time axis</span><span style="font-size: 16px; text-align: justify;">.</span></td></tr>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4383430891865259289.post-79354436988709804102010-06-04T17:42:00.000+06:002017-11-02T15:48:49.715+06:00Useful Signals<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="font-family: Arial; font-size: 100%;">Before looking at this module, hopefully you have some basic idea of what a signal is and what basic classifications and properties a signal can have. To review, a signal is merely a function defined with respect to an independent variable. This variable is often time but could represent an index of a sequence or any number of things in any number of dimensions. Most, if not all, signals that you will encounter in your studies and the real world will be able to be created from the basic signals we discuss below. Because of this, these elementary signals are often referred to as the <b>building blocks</b> for all other signals.<o:p></o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-size: 100%;"><b><span style="font-family: "arial";">Sinusoids</span></b><b><span style="font-family: "arial";">:<o:p></o:p></span></b></span></div>
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<span style="font-family: Arial; font-size: 100%;">Probably the most important elemental signal that you will deal with is the real-valued sinusoid. In its continuous-time form, we write the general form as<o:p></o:p></span></div>
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<span style="font-family: CMMI10; font-size: 100%;">x </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">) = </span><span style="font-family: CMMI10; font-size: 100%;">A</span><span style="font-family: CMR10; font-size: 100%;">cos (ωt + Ф)……(1)<o:p></o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;">Where </span><span style="font-family: CMMI10; font-size: 100%;">A </span><span style="font-family: Arial; font-size: 100%;">is the amplitude; </span><span style="font-family: CMR10; font-size: 100%;">ω </span><span style="font-family: Arial; font-size: 100%;">is the frequency, and </span><span style="font-family: CMR10; font-size: 100%;">Ф</span><span style="font-family: CMMI10; font-size: 100%;"> </span><span style="font-family: Arial; font-size: 100%;">represents the phase. Note that it is common to see </span><span style="font-family: CMR10; font-size: 100%;">ω</span><span style="font-family: CMMI10; font-size: 100%;">t </span><span style="font-family: Arial; font-size: 100%;">replaced with </span><span style="font-family: CMR10; font-size: 100%;">2π</span><span style="font-family: CMMI10; font-size: 100%;">ft</span><span style="font-family: Arial; font-size: 100%;">. Since sinusoidal signals are periodic, we can express the period of these, or any periodic signal, as<o:p></o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;">T = 2</span><span style="font-family: CMR10; font-size: 100%;">π/ω<o:p></o:p></span></div>
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<tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/_Jt8jI5P6sEU/S37r-Ytc1rI/AAAAAAAADx0/-jCJGlzLkck/s1600-h/1.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="margin-left: auto; margin-right: auto;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5440044856927639218" src="https://2.bp.blogspot.com/_Jt8jI5P6sEU/S37r-Ytc1rI/AAAAAAAADx0/-jCJGlzLkck/s400/1.JPG" style="float: left; height: 145px; margin: 0pt 10px 10px 0pt; width: 400px;" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: 16px;">Figure 1: Sinusoid with </span><span style="font-size: 16px;">A </span><span style="font-size: 16px;">= 2</span><span style="font-size: 16px;">,</span><span style="font-size: 16px;"> ω</span><span style="font-size: 16px;"> </span><span style="font-size: 16px;">= 2</span><span style="font-size: 16px;">, and </span><span style="font-size: 16px;">Ф</span><span style="font-size: 16px;"> </span><span style="font-size: 16px;">= 0</span><span style="font-size: 16px;">.</span></td></tr>
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<span style="font-family: Arial; font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-size: 100%;"><b><span style="font-family: "arial";">Complex Exponential Function:<o:p></o:p></span></b></span></div>
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<span style="font-family: Arial; font-size: 100%;">Maybe as important as the general sinusoid, the <b>complex exponential</b> function will become a critical part of your study of signals and systems. Its general form is written as<o:p></o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;">f(t) = B e<sup>st</sup>……….(2)<o:p></o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;">Where </span><span style="font-family: CMMI10; font-size: 100%;">s</span><span style="font-family: Arial; font-size: 100%;">, shown below, is a complex number in terms of σ, the phase constant, and </span><span style="font-family: CMMI10; font-size: 100%;">ω </span><span style="font-family: Arial; font-size: 100%;">the frequency:<o:p></o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;">s = σ + jω<o:p></o:p></span></div>
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<span style="font-size: 100%;"><b><span style="font-family: "arial";">Real Exponentials</span></b><b><span style="font-family: "arial";">:<o:p></o:p></span></b></span></div>
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<span style="font-family: Arial; font-size: 100%;">Just as the name sounds, real exponentials contain no imaginary numbers and are expressed simply as<o:p></o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;"> f(t) = B e<sup>αt</sup>………(3)<o:p></o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;">Where both </span><span style="font-family: CMMI10; font-size: 100%;">B </span><span style="font-family: Arial; font-size: 100%;">and α</span><span style="font-family: CMMI10; font-size: 100%;"> </span><span style="font-family: Arial; font-size: 100%;">are real parameters. Unlike the complex exponential that oscillates, the real exponential either decays or grows depending on the value of </span><span style="font-family: CMMI10; font-size: 100%;">α</span><span style="font-family: Arial; font-size: 100%;">.<o:p></o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: Symbol; font-size: 100%;">·<span style="font-family: ";"> </span></span><!--[endif]--><span style="font-size: 100%;"><b><span style="font-family: "arial";">Decaying Exponential</span></b></span><span style="font-family: Arial; font-size: 100%;">, when </span><span style="font-family: CMMI10; font-size: 100%;">α < </span><span style="font-family: CMR10; font-size: 100%;">0<o:p></o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: Symbol; font-size: 100%;">·<span style="font-family: ";"> </span></span><!--[endif]--><span style="font-size: 100%;"><b><span style="font-family: "arial";">Growing Exponential</span></b></span><span style="font-family: Arial; font-size: 100%;">, when </span><span style="font-family: CMMI10; font-size: 100%;">α > </span><span style="font-family: CMR10; font-size: 100%;">0</span><span style="font-family: Arial; font-size: 100%;"><o:p></o:p></span></div>
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<span style="font-family: CMR10; font-size: 100%;"><o:p> </o:p></span></div>
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<tr><td style="text-align: center;"><a href="http://3.bp.blogspot.com/_Jt8jI5P6sEU/S37r-gF4TyI/AAAAAAAADx8/QduY2vdc7Os/s1600-h/2.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="margin-left: auto; margin-right: auto;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5440044858909151010" src="https://3.bp.blogspot.com/_Jt8jI5P6sEU/S37r-gF4TyI/AAAAAAAADx8/QduY2vdc7Os/s400/2.JPG" style="float: left; height: 170px; margin: 0pt 10px 10px 0pt; width: 400px;" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Figure 2: Examples of Real Exponentials (a) Decaying Exponential (b) Growing Exponential</td></tr>
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<span style="font-size: 100%;"><b><span style="font-family: "arial";">Unit Impulse Function</span></b><b><span style="font-family: "arial";">:<o:p></o:p></span></b></span></div>
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<span style="font-family: Arial; font-size: 100%;">The <b>unit impulse</b> "function" (or <b>Dirac delta</b> function) is a signal that has infinite height and infinitesimal width. However, because of the way it is defined, it actually integrates to one. While in the engineering world, this signal is quite nice and aids in the understanding of many concepts, some mathematicians have a problem with it being called a function, since it is not defined at </span><span style="font-family: CMMI10; font-size: 100%;">t </span><span style="font-family: CMR10; font-size: 100%;">= 0. </span><span style="font-family: Arial; font-size: 100%;">Engineers reconcile this problem by keeping it around integrals, in order to keep it more nicely defined. The unit impulse is most commonly denoted as<o:p></o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;">δ (t)<o:p></o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;">The most important property of the unit-impulse is shown in the following integral:<o:p></o:p></span></div>
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<span style="font-size: 100%;"><a href="http://4.bp.blogspot.com/_Jt8jI5P6sEU/S37r_NqPg1I/AAAAAAAADyE/BHXDMBiDZZ8/s1600-h/3.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5440044871141262162" src="https://4.bp.blogspot.com/_Jt8jI5P6sEU/S37r_NqPg1I/AAAAAAAADyE/BHXDMBiDZZ8/s400/3.JPG" style="float: left; height: 49px; margin: 0pt 10px 10px 0pt; width: 147px;" /></a></span></div>
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<span style="font-size: 100%;"><b><span style="font-family: "arial";">Unit-Step Function</span></b><b><span style="font-family: "arial";">:<o:p></o:p></span></b></span></div>
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<span style="font-family: Arial; font-size: 100%;">Another very basic signal is the <b>unit-step function </b>that is defined as<o:p></o:p></span></div>
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<span style="font-size: 100%;"><a href="http://3.bp.blogspot.com/_Jt8jI5P6sEU/S37r_BuiUdI/AAAAAAAADyM/C2YhdTd8DzI/s1600-h/4.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5440044867938046418" src="https://3.bp.blogspot.com/_Jt8jI5P6sEU/S37r_BuiUdI/AAAAAAAADyM/C2YhdTd8DzI/s400/4.JPG" style="float: left; height: 49px; margin: 0pt 10px 10px 0pt; width: 255px;" /></a></span></div>
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<tr><td class="tr-caption" style="text-align: center;">Figure 5: Basic Step Functions (a) Continuous-Time Unit-Step Function (b) Discrete-Time Unit- Step Function.</td></tr>
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<span style="font-family: Arial; font-size: 100%;">Note that the step function is discontinuous at the origin; however, it does not need to be defined here as it does not matter in signal theory. The step function is a useful tool for testing and for defining other signals. For example, when different shifted versions of the step function are multiplied by other signals, one can select a certain portion of the signal and zero out the rest.<o:p></o:p></span></div>
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<span style="font-size: 100%;"><b><span style="font-family: "arial";">Ramp Function</span></b><b><span style="font-family: "arial";">:<o:p></o:p></span></b></span></div>
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<span style="font-family: Arial; font-size: 100%;">The ramp function is closely related to the unit-step discussed above. Where the unit-step goes from zero to one instantaneously, the ramp function better resembles a real-world signal, where there is some time needed for the signal to increase from zero to its set value, one in this case. We define a ramp function as follows:<o:p></o:p></span></div>
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<tr><td class="tr-caption" style="text-align: center;">Figure 7: Ramp Function.</td></tr>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4383430891865259289.post-81812036432501449952010-06-04T16:24:00.000+06:002017-11-02T16:03:17.686+06:00The Impulse Function<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="font-family: "arial"; font-size: 100%;">In engineering, we often deal with the idea of an action occurring at a point. Whether it be a force at a point in space or a signal at a point in time, it becomes worth while to develop some way of quantitatively defining this. This leads us to the idea of a unit impulse, probably the second most important function, next to the complex exponential in systems and signals course.<o:p></o:p></span><br />
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<span style="font-size: 100%;"><b><span style="font-family: "arial";">Dirac Delta Function</span></b><b><span style="font-family: "arial";">:<o:p></o:p></span></b></span></div>
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<span style="font-family: "arial"; font-size: 100%;">The <b>Dirac Delta function</b>, often referred to as the unit impulse or delta function is the function that defines the idea of a unit impulse. This function is one that is infinitesimally narrow, infinitely tall, yet integrates to <b>unity</b>, one. Perhaps the simplest way to visualize this is as a rectangular pulse from </span><span style="font-family: "cmmi10"; font-size: 100%;">a – Є/2 </span><span style="font-family: "arial"; font-size: 100%;">to </span><span style="font-family: "cmmi10"; font-size: 100%;">a </span><span style="font-family: "cmr10"; font-size: 100%;">+ </span><span style="font-family: "cmmi10"; font-size: 100%;">Є</span><span style="font-family: "cmr10"; font-size: 100%;"> /2 </span><span style="font-family: "arial"; font-size: 100%;">with a height of 1/</span><span style="font-family: "cmmi10"; font-size: 100%;">Є</span><span style="font-family: "arial"; font-size: 100%;">. As we take the limit of this,</span><br />
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<span style="font-family: "cmmi10"; font-size: 100%;"> </span><span style="font-family: "arial"; font-size: 100%;">we see that the width tends to zero and the height tends to infinity as the total area remains constant at one. The impulse function is often written as </span><span style="font-family: "cmmi10"; font-size: 100%;">δ </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr10"; font-size: 100%;">)</span><span style="font-family: "arial"; font-size: 100%;">.<o:p></o:p></span></div>
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<tr><td class="tr-caption" style="text-align: center;">Figure 1: This is one way to visualize the Dirac Delta Function.</td></tr>
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<tr><td class="tr-caption" style="text-align: center;">Figure 2<br />
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<span style="font-family: "arial"; font-size: 100%;">Figure 2: Since it is quite difficult to draw something that is infinitely tall, we represent the Dirac with an arrow centered at the point it is applied. If we wish to scale it, we may write the value it is scaled by next to the point of the arrow. This is a unit impulse (no scaling).<o:p></o:p></span><br />
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<span style="font-size: 100%;"><b><span style="font-family: "arial";">The Sifting Property of the Impulse:<o:p></o:p></span></b></span></div>
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<span style="font-family: "arial"; font-size: 100%;">The first step to understanding what this unit impulse function gives us is to examine what happens when we multiply another function by it.<o:p></o:p></span></div>
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<span style="font-family: "cmmi10"; font-size: 100%;">f</span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr10"; font-size: 100%;">) δ(</span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr10"; font-size: 100%;">) = </span><span style="font-family: "cmmi10"; font-size: 100%;">f</span><span style="font-family: "cmr10"; font-size: 100%;">(0) δ(</span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr10"; font-size: 100%;">)………..(1)<o:p></o:p></span></div>
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<span style="font-family: "arial"; font-size: 100%;">Since the impulse function is zero everywhere except the origin, we essentially just "pick off" the value of the function we are multiplying by evaluated at zero.<o:p></o:p></span></div>
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<span style="font-family: "arial"; font-size: 100%;">At first glance this may not appear to give use much, since we already know that the impulse evaluated at zero is infinity, and anything times infinity is infinity. However, what happens if we integrate this?<o:p></o:p></span><br />
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<span style="font-size: 100%;"><b><span style="font-family: "arial";">Sifting Property<o:p></o:p></span></b></span></div>
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<tr><td class="tr-caption" style="text-align: center;">Sifting Property</td></tr>
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<span style="font-family: "arial"; font-size: 100%;">It quickly becomes apparent that what we end up with is simply the function evaluated at zero. Had we used δ</span><span style="font-family: "cmmi10"; font-size: 100%;"> </span><span style="font-family: "cmr10"; font-size: 100%;">(t – T) </span><span style="font-family: "arial"; font-size: 100%;">instead of δ</span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr10"; font-size: 100%;">)</span><span style="font-family: "arial"; font-size: 100%;">, we could have "sifted out" f</span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">T</span><span style="font-family: "cmr10"; font-size: 100%;">)</span><span style="font-family: "arial"; font-size: 100%;">. This is what we call the<span style="font-weight: bold;"> </span><b>Sifting Property</b> of the Dirac function, which is often used to define the unit impulse.<o:p></o:p></span><br />
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<span style="font-family: "arial"; font-size: 100%;">The Sifting Property is very useful in developing the idea of convolution which is one of the fundamental principles of signal processing. By using convolution and the sifting property we can represent an approximation of any system's output if we know the system's impulse response and input.<o:p></o:p></span><br />
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<span style="font-size: 100%;"><b><span style="font-family: "arial";">Other Impulse Properties:<o:p></o:p></span></b></span></div>
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<span style="font-family: "arial"; font-size: 100%;">Below we will briefly list a few of the other properties of the unit impulse without going into detail of their proofs - we will leave that up to you to verify as most are straightforward. Note that these properties hold for continuous <b>and</b> discrete time.<o:p></o:p></span><br />
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<span style="font-size: 100%;"><b><span style="font-family: "arial";">Unit Impulse Properties<o:p></o:p></span></b></span></div>
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<tr><td class="tr-caption" style="text-align: center;"><b style="font-size: 16px; text-align: justify;"><span style="font-family: "arial";">Units Impulse Properties</span></b></td></tr>
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<span style="font-size: 100%;"><b><span style="font-family: "arial";">Discrete-Time Impulse (Unit Sample)</span></b><b><span style="font-family: "arial";">:<o:p></o:p></span></b></span></div>
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<span style="font-family: "arial"; font-size: 100%;">The extension of the Unit Impulse Function to discrete-time becomes quite trivial. All we really need to realize is that integration in continuous-time equates to summation in discrete-time. Therefore, we are looking for a signal that sums to zero and is zero everywhere except at zero.<o:p></o:p></span><br />
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<span style="font-size: 100%;"><b><span style="font-family: "arial";">Discrete-Time Impulse<o:p></o:p></span></b></span></div>
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<span style="font-family: "arial"; font-size: 100%;">Figure 3: The graphical representation of the discrete-time impulse function</span>
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<span style="font-family: "arial"; font-size: 10pt;">Looking at the discrete-time plot of any discrete signal one can notice that all discrete signals are composed of a set of scaled, time-shifted unit samples. If we let the value of a sequence at each integer </span><span style="font-family: "cmmi10"; font-size: 10pt;">k </span><span style="font-family: "arial"; font-size: 10pt;">be denoted by </span><span style="font-family: "cmmi10"; font-size: 10pt;">s </span><span style="font-family: "cmr10"; font-size: 10pt;">[</span><span style="font-family: "cmmi10"; font-size: 10pt;">k</span><span style="font-family: "cmr10"; font-size: 10pt;">] </span><span style="font-family: "arial"; font-size: 10pt;">and the unit sample delayed that occurs at </span><span style="font-family: "cmmi10"; font-size: 10pt;">k </span><span style="font-family: "arial"; font-size: 10pt;">to be written as δ</span><span style="font-family: "cmr10"; font-size: 10pt;">[n – K]</span><span style="font-family: "arial"; font-size: 10pt;">, we can write any signal as the sum of delayed unit samples that are scaled by the signal value, or weighted coefficients.<o:p></o:p></span><br />
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<span style="font-family: "arial"; font-size: 100%;">This decomposition is strictly a property of discrete-time signals and proves to be a very useful property.<o:p></o:p></span></div>
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<span style="font-size: 100%;"><b><span style="font-family: "arial";">The Impulse Response</span></b><b><span style="font-family: "arial";">:<o:p></o:p></span></b></span></div>
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<span style="font-family: "arial"; font-size: 100%;">The <b>impulse response</b> is exactly what its name implies - the response of an LTI system, such as a filter, when the system's input is the unit impulse (or unit sample). A system can be completed described by its impulse response due to the idea mentioned above that all signals can be represented by a superposition of signals. An impulse response gives an equivalent description of a system as a transfer function, since they are Laplace Transforms of each other.<o:p></o:p></span><br />
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<span style="font-family: "arial"; font-size: 100%;">Notation: Most texts use δ</span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr10"; font-size: 100%;">) </span><span style="font-family: "arial"; font-size: 100%;">and δ</span><span style="font-family: "cmr10"; font-size: 100%;">[</span><span style="font-family: "cmmi10"; font-size: 100%;">n</span><span style="font-family: "cmr10"; font-size: 100%;">] </span><span style="font-family: "arial"; font-size: 100%;">to denote the continuous-time and discrete-time impulse response, respectively.<o:p></o:p></span></div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4383430891865259289.post-38671956272884316942010-06-04T15:24:00.001+06:002017-11-02T16:07:49.145+06:00System Classifications and Properties<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="font-family: "arial"; font-size: 100%;">In this module some of the basic classifications of systems will be briefly introduced and the most important properties of these systems are explained. As can be seen, the properties of a system provide an easy way to separate one system from another. Understanding these basic differences between systems, and their properties, will be a fundamental concept used in all signal and system courses, such as digital signal processing (<span style="font-weight: bold;">DSP</span>). Once a set of systems can be identified as sharing particular properties, one no longer has to deal with proving a certain characteristic of a system each time, but it can simply be accepted do the systems classification. Also remember that this classification presented here is neither exclusive (systems can belong to several different classifications) nor is it unique (there are other methods of classification). Examples of simple systems can be found here</span><span style="font-family: "arial"; font-size: 100%;">.<o:p></o:p></span><br />
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<span style="font-size: 100%;"><b><span style="font-family: "arial";">Classification of Systems:<o:p></o:p></span></b></span></div>
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<span style="font-family: "arial"; font-size: 100%;">Along with the classification of systems below, it is also important to understand other Classification of Signals.<o:p></o:p></span></div>
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<span style="font-size: 100%;"><b><span style="font-family: "arial";">Continuous vs. Discrete:<o:p></o:p></span></b></span></div>
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<span style="font-family: "arial"; font-size: 100%;">This may be the simplest classification to understand as the idea of discrete-time and continuous time is one of the most fundamental properties to all of signals and system. A system where the input and output signals are continuous is a <b>continuous system</b>, and one where the input and output signals are discrete is a <b>discrete system</b>.<o:p></o:p></span></div>
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<span style="font-size: 100%;"><b><span style="font-family: "arial";">Linear vs. Nonlinear:<o:p></o:p></span></b></span></div>
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<span style="font-family: "arial"; font-size: 100%;">A <b>linear</b> system is any system that obeys the properties of scaling (homogeneity) and superposition (additivity), while a <b>nonlinear</b> system is any system that does not obey at least one of these. To show that a system </span><span style="font-family: "cmmi10"; font-size: 100%;">H </span><span style="font-family: "arial"; font-size: 100%;">obeys the scaling property is to show that<o:p></o:p></span></div>
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<span style="font-family: "cmmi10"; font-size: 100%;">H </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">k(f </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr10"; font-size: 100%;">)) = </span><span style="font-family: "cmmi10"; font-size: 100%;">kH </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">f </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr10"; font-size: 100%;">))………………..(1)<o:p></o:p></span></div>
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<tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/_Jt8jI5P6sEU/S376eqhLNZI/AAAAAAAADzs/v_P1FYCPi_k/s1600-h/1.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="margin-left: auto; margin-right: auto;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5440060804626593170" src="https://2.bp.blogspot.com/_Jt8jI5P6sEU/S376eqhLNZI/AAAAAAAADzs/v_P1FYCPi_k/s400/1.JPG" style="float: left; height: 102px; margin: 0pt 10px 10px 0pt; width: 400px;" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Figure 1: A block diagram demonstrating the scaling property of linearity</td></tr>
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<span style="font-family: "arial"; font-size: 100%;">To demonstrate that a system </span><span style="font-family: "cmmi10"; font-size: 100%;">H </span><span style="font-family: "arial"; font-size: 100%;">obeys the superposition property of linearity is to show that<o:p></o:p></span></div>
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<span style="font-family: "cmmi10"; font-size: 100%;">H </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">f</span><span style="font-family: "cmr7"; font-size: 100%;">1 </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr10"; font-size: 100%;">) + </span><span style="font-family: "cmmi10"; font-size: 100%;">f</span><span style="font-family: "cmr7"; font-size: 100%;">2 </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr10"; font-size: 100%;">)) = </span><span style="font-family: "cmmi10"; font-size: 100%;">H </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">f</span><span style="font-family: "cmr7"; font-size: 100%;">1 </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr10"; font-size: 100%;">)) + </span><span style="font-family: "cmmi10"; font-size: 100%;">H </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">f</span><span style="font-family: "cmr7"; font-size: 100%;">2 </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr10"; font-size: 100%;">))…………(2)<o:p></o:p></span></div>
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<tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/_Jt8jI5P6sEU/S376e6Uit_I/AAAAAAAADz0/3pFhnPPDubc/s1600-h/2.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="margin-left: auto; margin-right: auto;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5440060808868575218" src="https://2.bp.blogspot.com/_Jt8jI5P6sEU/S376e6Uit_I/AAAAAAAADz0/3pFhnPPDubc/s400/2.JPG" style="float: left; height: 102px; margin: 0pt 10px 10px 0pt; width: 400px;" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Figure 2: A block diagram demonstrating the superposition property of linearity</td></tr>
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<span style="font-family: "arial"; font-size: 100%;">It is possible to check a system for linearity in a single (though larger) step. To do this, simply combine the first two steps to get<o:p></o:p></span></div>
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<span style="font-family: "cmmi10"; font-size: 100%;">H </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">k</span><span style="font-family: "cmr7"; font-size: 100%;">1</span><span style="font-family: "cmmi10"; font-size: 100%;">f</span><span style="font-family: "cmr7"; font-size: 100%;">1 </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr10"; font-size: 100%;">) + </span><span style="font-family: "cmmi10"; font-size: 100%;">k</span><span style="font-family: "cmr7"; font-size: 100%;">2</span><span style="font-family: "cmmi10"; font-size: 100%;">f</span><span style="font-family: "cmr7"; font-size: 100%;">2 </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr10"; font-size: 100%;">)) = </span><span style="font-family: "cmmi10"; font-size: 100%;">k</span><span style="font-family: "cmr7"; font-size: 100%;">2</span><span style="font-family: "cmmi10"; font-size: 100%;">H </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">f</span><span style="font-family: "cmr7"; font-size: 100%;">1 </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr10"; font-size: 100%;">)) + </span><span style="font-family: "cmmi10"; font-size: 100%;">k</span><span style="font-family: "cmr7"; font-size: 100%;">2</span><span style="font-family: "cmmi10"; font-size: 100%;">H </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">f</span><span style="font-family: "cmr7"; font-size: 100%;">2 </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr10"; font-size: 100%;">)) </span><span style="font-family: "arial"; font-size: 100%;">……..(3)<o:p></o:p></span></div>
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<span style="font-size: 100%;"><b><span style="font-family: "arial";">Time Invariant vs. Time Variant:<o:p></o:p></span></b></span></div>
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<span style="font-family: "arial"; font-size: 100%;">A <b>time invariant</b> system is one that does not depend on when it occurs: the shape of the output does not change with a delay of the input. That is to say that for a system </span><span style="font-family: "cmmi10"; font-size: 100%;">H </span><span style="font-family: "arial"; font-size: 100%;">where </span><span style="font-family: "cmmi10"; font-size: 100%;">H </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">f </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr10"; font-size: 100%;">)) = </span><span style="font-family: "cmmi10"; font-size: 100%;">y </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr10"; font-size: 100%;">)</span><span style="font-family: "arial"; font-size: 100%;">, </span><span style="font-family: "cmmi10"; font-size: 100%;">H </span><span style="font-family: "arial"; font-size: 100%;">is time invariant if for all </span><span style="font-family: "cmmi10"; font-size: 100%;">T</span><span style="font-family: "arial"; font-size: 100%;"><o:p></o:p></span></div>
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<span style="font-family: "cmmi10"; font-size: 100%;">H </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">f </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t – T</span><span style="font-family: "cmr10"; font-size: 100%;">) = </span><span style="font-family: "cmmi10"; font-size: 100%;">y </span><span style="font-family: "cmr10"; font-size: 100%;">(t – T)…………..(4)<o:p></o:p></span></div>
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<tr><td class="tr-caption" style="text-align: center;">Figure 3</td></tr>
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<span style="font-family: "arial"; font-size: 100%;">Figure 3: This block diagram shows what the condition for time invariance. The output is the same whether the delay is put on the input or the output.</span></div>
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<span style="font-family: "arial"; font-size: 100%;">When this property does not hold for a system, then it is said to be <b>time variant</b>, or time-varying.<o:p></o:p></span></div>
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<span style="font-size: 100%;"><b><span style="font-family: "arial";">Causal vs. Non-causal:<o:p></o:p></span></b></span></div>
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<span style="font-family: "arial"; font-size: 100%;">A <b>causal </b>system is one that is <b>non-anticipative</b>; that is, the output may depend on current and past inputs, but not future inputs. All "realtime" systems must be causal, since they can not have future inputs available to them.<o:p></o:p></span></div>
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<span style="font-family: "arial"; font-size: 100%;">One may think the idea of future inputs does not seem to make much physical sense; however, we have only been dealing with time as our dependent variable so far, which is not always the case. Imagine rather that we wanted to do image processing. Then the dependent variable might represent pixels to the left and right (the "future") of the current position on the image, and we would have a <b>non-causal</b> system.<o:p></o:p></span></div>
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<span style="font-size: 100%;"><b><span style="font-family: "arial";"><span style="font-family: "times new roman"; font-weight: 400; text-align: center;">Figure 4: (a) For a typical system to be causal... (b) ...the output at time </span><span style="font-family: "times new roman"; font-weight: 400; text-align: center;">t</span><span style="font-family: "times new roman"; font-weight: 400; text-align: center;">0</span><span style="font-family: "times new roman"; font-weight: 400; text-align: center;">, </span><span style="font-family: "times new roman"; font-weight: 400; text-align: center;">y </span><span style="font-family: "times new roman"; font-weight: 400; text-align: center;">(</span><span style="font-family: "times new roman"; font-weight: 400; text-align: center;">t</span><span style="font-family: "times new roman"; font-weight: 400; text-align: center;">o</span><span style="font-family: "times new roman"; font-weight: 400; text-align: center;">)</span><span style="font-family: "times new roman"; font-weight: 400; text-align: center;">, can only depend on the portion of the input signal before </span><span style="font-family: "times new roman"; font-weight: 400; text-align: center;">t</span><span style="font-family: "times new roman"; font-weight: 400; text-align: center;">o</span><span style="font-family: "times new roman"; font-weight: 400; text-align: center;">.</span></span></b></span></div>
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<span style="font-size: 100%;"><b><span style="font-family: "arial";">Stable vs. Unstable:<o:p></o:p></span></b></span></div>
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<span style="font-family: "arial"; font-size: 100%;">A <b>stable</b> system is one where the output does not diverge as long as the input does not diverge. There are many ways to say that a signal "diverges"; for example it could have infinite energy. One particularly useful definition of divergence relates to whether the signal is bounded or not. Then a system is referred to as <b>bounded input-bounded output (BIBO) </b>stable if <b>every possible</b> bounded input produces a bounded output.<o:p></o:p></span></div>
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<span style="font-family: "arial"; font-size: 100%;">Representing this in a mathematical way, a stable system must have the following property, where </span><span style="font-family: "cmmi10"; font-size: 100%;">x </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr10"; font-size: 100%;">) </span><span style="font-family: "arial"; font-size: 100%;">is the input and </span><span style="font-family: "cmmi10"; font-size: 100%;">y </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr10"; font-size: 100%;">) </span><span style="font-family: "arial"; font-size: 100%;">is the output. The output must satisfy the condition</span><span style="font-family: "cmr10"; font-size: 100%;"><o:p></o:p></span></div>
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<span style="font-family: "cmmi10"; font-size: 100%;">Іy </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr10"; font-size: 100%;">)І</span><span style="font-family: "cmsy10"; font-size: 100%;"> ≤ </span><span style="font-family: "cmmi10"; font-size: 100%;">M</span><span style="font-family: "cmmi7"; font-size: 100%;">y </span><span style="font-family: "cmmi10"; font-size: 100%;">< α …………..(5)</span><span style="font-family: "cmr10"; font-size: 100%;"><o:p></o:p></span></div>
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<span style="font-family: "arial"; font-size: 100%;">When we have an input to the system that can be described as<o:p></o:p></span></div>
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<span style="font-family: "cmmi10"; font-size: 100%;">Іx </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr10"; font-size: 100%;">) </span><span style="font-family: "cmmi10"; font-size: 100%;">І</span><span style="font-family: "cmsy10"; font-size: 100%;"> ≤ </span><span style="font-family: "cmmi10"; font-size: 100%;">M</span><span style="font-family: "cmmi7"; font-size: 100%;">x </span><span style="font-family: "cmmi10"; font-size: 100%;">< α ………….. (6)<o:p></o:p></span></div>
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<span style="font-family: "cmmi10"; font-size: 100%;">M</span><span style="font-family: "cmmi7"; font-size: 100%;">x </span><span style="font-family: "arial"; font-size: 100%;">and </span><span style="font-family: "cmmi10"; font-size: 100%;">M</span><span style="font-family: "cmmi7"; font-size: 100%;">y </span><span style="font-family: "arial"; font-size: 100%;">both represent a set of finite positive numbers and these relationships hold for all of </span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "arial"; font-size: 100%;">.<o:p></o:p></span></div>
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<span style="font-family: "arial"; font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-family: "arial"; font-size: 100%;">If these conditions are not met, i.e. a system's output grows without limit (diverges) from a bounded input, then the system is <b>unstable</b>. Note that the <b>BIBO</b> stability of a linear time-invariant system (LTI) is neatly described in terms of whether or not its impulse response is absolutely integrable.<o:p></o:p></span></div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4383430891865259289.post-89256316250630983982010-06-04T00:54:00.000+06:002017-11-02T16:15:28.662+06:00Properties of Systems<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="font-size: 100%;"><b><span style="font-family: "arial";">Linear Systems:</span></b><b><span style="font-family: "arial";"><o:p></o:p></span></b></span> </div>
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<span style="font-family: "arial"; font-size: 100%;">If a system is linear, this means that when an input to a given system is scaled by a value, the output of the system is scaled by the same amount.<o:p></o:p></span></div>
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<span style="font-family: "arial"; font-size: 100%;"><o:p> </o:p></span></div>
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<tr><td class="tr-caption" style="text-align: center;">Figure 1(a) and Figure 1(b)</td></tr>
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<span style="font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-family: "arial"; font-size: 100%;">In Figure 1(a) above, an input </span><span style="font-family: "cmmi10"; font-size: 100%;">x </span><span style="font-family: "arial"; font-size: 100%;">to the linear system </span><span style="font-family: "cmmi10"; font-size: 100%;">L </span><span style="font-family: "arial"; font-size: 100%;">gives the output </span><span style="font-family: "cmmi10"; font-size: 100%;">y</span><span style="font-family: "arial"; font-size: 100%;">. If </span><span style="font-family: "cmmi10"; font-size: 100%;">x </span><span style="font-family: "arial"; font-size: 100%;">is scaled by a value α and passed through this same system, as in Figure 1(b), the output will also be scaled by α.</span><span style="font-family: "cmmi10"; font-size: 100%;"><o:p></o:p></span></div>
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<span style="font-family: "arial"; font-size: 100%;">A linear system also obeys the principle of superposition. This means that if two inputs are added together and passed through a linear system, the output will be the sum of the individual inputs' outputs.<o:p></o:p></span></div>
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<span style="font-family: "arial"; font-size: 100%;"><o:p> </o:p></span></div>
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<tr><td class="tr-caption" style="text-align: center;">Figure 1</td></tr>
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<span style="font-size: 16px;">If Figure 1 is true, then the principle of superpo</span><span style="font-size: 16px;">sition says that Figure 2 (Superposition<o:p></o:p></span></div>
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<span style="font-size: 16px;">Principle) is true as well. This holds for linear systems.</span></div>
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<tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/_Jt8jI5P6sEU/TAf7r43lCZI/AAAAAAAAD3E/Aa1zMaxDMKA/s1600/3.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="margin-left: auto; margin-right: auto;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5478624203137943954" src="https://2.bp.blogspot.com/_Jt8jI5P6sEU/TAf7r43lCZI/AAAAAAAAD3E/Aa1zMaxDMKA/s400/3.JPG" style="float: left; height: 53px; margin: 0pt 10px 10px 0pt; width: 400px;" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Figure 2</td></tr>
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<span style="font-family: "arial"; font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-family: "arial"; font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-family: "arial"; font-size: 100%;">That is, if Figure 1 is true, then Figure 2 (Superposition Principle) is also true for a linear system.<o:p></o:p></span></div>
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<span style="font-family: "arial"; font-size: 100%;">The scaling property mentioned above still holds in conjunction with the superposition principle. Therefore, if the inputs x and y are scaled by factors α</span><span style="font-family: "cmmi10"; font-size: 100%;"> </span><span style="font-family: "arial"; font-size: 100%;">and </span><span style="font-family: "cmmi10"; font-size: 100%;">β</span><span style="font-family: "arial"; font-size: 100%;">, respectively, then the sum of these scaled inputs will give the sum of the individual scaled outputs:</span></div>
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<span style="font-size: 16px;"><b><span style="font-family: "arial";">Time-Invariant Systems:</span></b><b><span style="font-family: "arial";"><o:p></o:p></span></b></span></div>
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<span style="font-size: 16px;">A time-invariant system has the property that a certain input will always give the same output, without regard to when the input was applied to the system.</span></div>
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<span style="font-family: "arial"; font-size: 100%;"><o:p> </o:p></span></div>
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<tr><td class="tr-caption" style="text-align: center;">Figure (a)</td></tr>
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<tr><td style="text-align: center;"><a href="http://4.bp.blogspot.com/_Jt8jI5P6sEU/TAf7A1PvpNI/AAAAAAAAD2M/KJSCqV8HDaw/s1600/5.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="margin-left: auto; margin-right: auto;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5478623463431185618" src="https://4.bp.blogspot.com/_Jt8jI5P6sEU/TAf7A1PvpNI/AAAAAAAAD2M/KJSCqV8HDaw/s400/5.JPG" style="float: left; height: 60px; margin: 0pt 10px 10px 0pt; width: 400px;" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Figure (b)</td></tr>
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<span style="font-size: 100%;"><b><span style="font-family: "arial";"><span style="font-family: "times new roman"; font-weight: 400;">Figure (a) shows an input at time </span><span style="font-family: "times new roman"; font-weight: 400;">t </span><span style="font-family: "times new roman"; font-weight: 400;">while Figure (b) shows the same input </span><span style="font-family: "times new roman"; font-weight: 400;">t</span><span style="font-family: "times new roman"; font-weight: 400;">0 </span><span style="font-family: "times new roman"; font-weight: 400;">seconds later. In a time-invariant system both outputs</span><span style="font-family: "times new roman"; font-weight: 400;"> would be identical except that the one in Figure (b) would be delayed by </span><span style="font-family: "times new roman"; font-weight: 400;">t</span><span style="font-family: "times new roman"; font-weight: 400;">0</span><span style="font-family: "times new roman"; font-weight: 400;">.</span></span></b></span></div>
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<tr><td style="text-align: center;"><a href="http://4.bp.blogspot.com/_Jt8jI5P6sEU/TAf7BG0mWPI/AAAAAAAAD2U/qAzsfcA0seU/s1600/6.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="margin-left: auto; margin-right: auto;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5478623468149168370" src="https://4.bp.blogspot.com/_Jt8jI5P6sEU/TAf7BG0mWPI/AAAAAAAAD2U/qAzsfcA0seU/s400/6.JPG" style="float: left; height: 56px; margin: 0pt 10px 10px 0pt; width: 400px;" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: 16px; text-align: justify;">Figure, </span><span style="font-size: 16px; text-align: justify;">x </span><span style="font-size: 16px; text-align: justify;">(</span><span style="font-size: 16px; text-align: justify;">t</span><span style="font-size: 16px; text-align: justify;">) </span><span style="font-size: 16px; text-align: justify;">and </span><span style="font-size: 16px; text-align: justify;">x </span><span style="font-size: 16px; text-align: justify;">(</span><span style="font-size: 16px; text-align: justify;">t - t</span><span style="font-size: 16px; text-align: justify;">0</span><span style="font-size: 16px; text-align: justify;">)</span></td></tr>
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<span style="font-family: "arial"; font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-family: "arial"; font-size: 100%;"><span style="font-family: "arial"; font-size: 100%;">In this figure, </span><span style="font-family: "cmmi10"; font-size: 100%;">x </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr10"; font-size: 100%;">) </span><span style="font-family: "arial"; font-size: 100%;">and </span><span style="font-family: "cmmi10"; font-size: 100%;">x </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t - t</span><span style="font-family: "cmr7"; font-size: 100%;">0</span><span style="font-family: "cmr10"; font-size: 100%;">) </span><span style="font-family: "arial"; font-size: 100%;">are passed through the system TI. Because the system TI is time invariant, the inputs </span><span style="font-family: "cmmi10"; font-size: 100%;">x </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr10"; font-size: 100%;">) </span><span style="font-family: "arial"; font-size: 100%;">and </span><span style="font-family: "cmmi10"; font-size: 100%;">x </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t -</span><span style="font-family: "cmsy10"; font-size: 100%;"> </span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr7"; font-size: 100%;">0</span><span style="font-family: "cmr10"; font-size: 100%;">) </span><span style="font-family: "arial"; font-size: 100%;">produce the same output. The only difference is that the output due to </span><span style="font-family: "cmmi10"; font-size: 100%;">x </span><span style="font-family: "cmr10"; font-size: 100%;">(</span><span style="font-family: "cmmi10"; font-size: 100%;">t -</span><span style="font-family: "cmsy10"; font-size: 100%;"> </span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr7"; font-size: 100%;">0</span><span style="font-family: "cmr10"; font-size: 100%;">) </span><span style="font-family: "arial"; font-size: 100%;">is shifted by a time </span><span style="font-family: "cmmi10"; font-size: 100%;">t</span><span style="font-family: "cmr7"; font-size: 100%;">0</span><span style="font-family: "arial"; font-size: 100%;">.</span></span><br />
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<span style="font-family: "arial"; font-size: 100%;">Whether a system is time-invariant or time-varying can be seen in the deferential equation (or difference equation) describing it. Time-invariant systems are modeled with constant coefficient equations. A constant coefficient deferential (or difference) equation means that the parameters of the system are not changing over time and an input now will give the same result as the same input later.</span><br />
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<span style="font-family: "arial"; font-size: 100%; font-weight: bold;">Linear Time-Invariant (LTI) Systems:</span><span style="font-family: "arial"; font-size: 100%;"><o:p></o:p></span></div>
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<span style="font-family: "arial"; font-size: 100%;">Certain systems are both linear and time-invariant, and are thus referred to as LTI systems.<o:p></o:p></span></div>
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<tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/_Jt8jI5P6sEU/TAf7BaZveiI/AAAAAAAAD2c/StybFsQ2_Xw/s1600/7.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="margin-left: auto; margin-right: auto;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5478623473405229602" src="https://2.bp.blogspot.com/_Jt8jI5P6sEU/TAf7BaZveiI/AAAAAAAAD2c/StybFsQ2_Xw/s400/7.JPG" style="float: left; height: 56px; margin: 0pt 10px 10px 0pt; width: 400px;" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Figure (a) and (b)</td></tr>
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<span style="font-family: "arial"; font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-family: "arial"; font-size: 100%;">Figure: This is a combination of the two c</span><span style="font-family: "arial"; font-size: 100%;">ase</span><span style="font-family: "arial"; font-size: 100%;">s abo</span><span style="font-family: "arial"; font-size: 100%;">ve. Since the input to Figure (b) is a scaled, time-shifted version of the input in Figure (a), so is the output.</span></div>
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<span style="font-family: "arial"; font-size: 10pt;">As LTI systems are a subset of linear systems, they obey the principle of superposition. In the figure below, we see the effect of applying time-invariance to the superposition definition in the linear systems section above.<o:p></o:p></span></div>
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<tr><td class="tr-caption" style="text-align: center;">Superposition in Linear Time-Invariant Systems</td></tr>
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<span style="font-size: 100%;"></span><span style="font-family: "arial"; font-size: 10pt;"><o:p> </o:p></span> <br />
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<span style="font-family: "arial"; font-size: 10pt;"><o:p> </o:p></span><span style="font-size: 100%;"></span></div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4383430891865259289.post-39876202951944877582010-02-20T02:43:00.000+06:002017-11-02T16:20:56.270+06:00Time Domain Analysis of Continuous Time Systems<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="font-family: Arial; font-size: 100%;"><span style="font-weight: bold;">Continuous-Time Linear Systems:</span><o:p></o:p></span><br />
<span style="font-family: Arial; font-size: 100%;">Physically realizable, linear time-invariant systems can be described by a set of linear differential equations (LDEs):<o:p></o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;"><o:p> </o:p></span></div>
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<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: 16px; text-align: justify;">Figure with </span><span style="font-size: 16px; text-align: justify;">input, </span><span style="font-size: 16px; text-align: justify;">f </span><span style="font-size: 16px; text-align: justify;">(</span><span style="font-size: 16px; text-align: justify;">t</span><span style="font-size: 16px; text-align: justify;">) </span><span style="font-size: 16px; text-align: justify;">and an output, </span><span style="font-size: 16px; text-align: justify;">y </span><span style="font-size: 16px; text-align: justify;">(</span><span style="font-size: 16px; text-align: justify;">t</span><span style="font-size: 16px; text-align: justify;">)</span><span style="font-size: 16px; text-align: justify;">.</span></td></tr>
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<span style="font-family: Arial; font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-size: 16px;">Figure: Graphical description of a basic linear time-invariant system with an input, </span><span style="font-size: 16px;">f </span><span style="font-size: 16px;">(</span><span style="font-size: 16px;">t</span><span style="font-size: 16px;">) </span><span style="font-size: 16px;">and an output, </span><span style="font-size: 16px;">y </span><span style="font-size: 16px;">(</span><span style="font-size: 16px;">t</span><span style="font-size: 16px;">)</span><span style="font-size: 16px;">.</span></div>
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<span style="font-family: Arial; font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-size: 100%;">Equivalently,</span></div>
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<span style="font-family: Arial; font-size: 100%;"><o:p> </o:p></span></div>
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<tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/_Jt8jI5P6sEU/TAjG5XKZTuI/AAAAAAAAD3s/K17T1i-Eq3o/s1600/3.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="margin-left: auto; margin-right: auto;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5478847635468668642" src="https://1.bp.blogspot.com/_Jt8jI5P6sEU/TAjG5XKZTuI/AAAAAAAAD3s/K17T1i-Eq3o/s400/3.JPG" style="float: left; height: 76px; margin: 0pt 10px 10px 0pt; width: 332px;" /></a></td></tr>
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<span style="font-family: Arial; font-size: 100%;">With </span><span style="font-family: CMMI10; font-size: 100%;">a</span><span style="font-family: CMMI7; font-size: 100%;">n </span><span style="font-family: CMR10; font-size: 100%;">= 1</span><span style="font-family: Arial; font-size: 100%;">.<o:p></o:p>It is easy to show that these equations define a system that is linear and time invariant. A natural question to ask, then, is how to find the system's output response </span><span style="font-family: CMMI10; font-size: 100%;">y </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">) </span><span style="font-family: Arial; font-size: 100%;">to an input </span><span style="font-family: CMMI10; font-size: 100%;">f </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">)</span><span style="font-family: Arial; font-size: 100%;">. Recall that such a solution can be written as<o:p></o:p></span></div>
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<span style="font-family: CMMI10; font-size: 100%;">y </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">) = </span><span style="font-family: CMMI10; font-size: 100%;">y</span><span style="font-family: CMMI7; font-size: 100%;">i </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">) + </span><span style="font-family: CMMI10; font-size: 100%;">y</span><span style="font-family: CMMI7; font-size: 100%;">s </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">)<o:p></o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;">We refer to </span><span style="font-family: CMMI10; font-size: 100%;">y</span><span style="font-family: CMMI7; font-size: 100%;">i </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">) </span><span style="font-family: Arial; font-size: 100%;">as the zero-input response – the homogeneous solution due only to the initial conditions of the system. We refer to </span><span style="font-family: CMMI10; font-size: 100%;">y</span><span style="font-family: CMMI7; font-size: 100%;">s </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">) </span><span style="font-family: Arial; font-size: 100%;">as the zero-state response – the particular solution in response to the input </span><span style="font-family: CMMI10; font-size: 100%;">f </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">)</span><span style="font-family: Arial; font-size: 100%;">. We now discuss how to solve for each of these components of the system's response.<o:p></o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;"><span style="font-weight: bold;">Finding the Zero-Input Response:</span><o:p></o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;">The zero-input response, </span><span style="font-family: CMMI10; font-size: 100%;">y</span><span style="font-family: CMMI7; font-size: 100%;">i </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">)</span><span style="font-family: Arial; font-size: 100%;">, is the system response due to initial conditions only.<o:p></o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;">Close the switch in the circuit pictured in Figure at time t=0 and then leave everything else alone. The voltage response is shown in Figure<o:p></o:p></span></div>
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<tr><td class="tr-caption" style="text-align: center;">Finding the Zero-Input Response</td></tr>
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<tr><td class="tr-caption" style="text-align: center;">Finding the Zero-State Response<br /></td></tr>
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<span style="font-family: Arial; font-size: 100%;"><span style="font-weight: bold;">Finding the Zero-State Response:</span><o:p></o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;">Solving a linear differential equation<o:p></o:p></span></div>
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<tr><td class="tr-caption" style="text-align: center;">Linear differential equation</td></tr>
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<span style="font-family: Arial; font-size: 100%;">Given a specific input </span><span style="font-family: CMMI10; font-size: 100%;">f </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">) </span><span style="font-family: Arial; font-size: 100%;">is a difficult task in general. More importantly, the method depends entirely on the nature of </span><span style="font-family: CMMI10; font-size: 100%;">f </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">)</span><span style="font-family: Arial; font-size: 100%;">; if we change the input signal, we must completely re-solve the system of equations to find the system response.<o:p></o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;">Convolution helps to bypass these difficulties. We explain how convolution helps to determine the system's output, given only the input </span><span style="font-family: CMMI10; font-size: 100%;">f </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">) </span><span style="font-family: Arial; font-size: 100%;">and the system's impulse response </span><span style="font-family: CMMI10; font-size: 100%;">h</span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">)</span><span style="font-family: Arial; font-size: 100%;">.<o:p></o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;">Before deriving the convolution procedure, we show that a system's impulse response is easily derived from its linear, differential equation (LDE). We will show the derivation for the LDE below, where </span><span style="font-family: CMMI10; font-size: 100%;">m <><span style="font-family: Arial; font-size: 100%;">:<o:p></o:p></span></span></div>
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<span style="font-family: Arial; font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;">We can rewrite as<o:p></o:p></span></div>
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<span style="font-family: CMMI10; font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-family: CMMI10; font-size: 100%;">Q</span><span style="font-family: CMMI7; font-size: 100%;">D </span><span style="font-family: CMR10; font-size: 100%;">[</span><span style="font-family: CMMI10; font-size: 100%;">y </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">)] = </span><span style="font-family: CMMI10; font-size: 100%;">P</span><span style="font-family: CMMI7; font-size: 100%;">D </span><span style="font-family: CMR10; font-size: 100%;">[</span><span style="font-family: CMMI10; font-size: 100%;">f </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">)]</span><span style="font-family: Arial; font-size: 100%;"><o:p></o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;">Where </span><span style="font-family: CMMI10; font-size: 100%;">Q</span><span style="font-family: CMMI7; font-size: 100%;">D </span><span style="font-family: CMR10; font-size: 100%;">[.] </span><span style="font-family: Arial; font-size: 100%;">is an operator that maps </span><span style="font-family: CMMI10; font-size: 100%;">y </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">) </span><span style="font-family: Arial; font-size: 100%;">to the left hand side<o:p></o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;">And </span><span style="font-family: CMMI10; font-size: 100%;">P</span><span style="font-family: CMMI7; font-size: 100%;">D </span><span style="font-family: CMR10; font-size: 100%;">[.] </span><span style="font-family: Arial; font-size: 100%;">maps </span><span style="font-family: CMMI10; font-size: 100%;">f </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">) </span><span style="font-family: Arial; font-size: 100%;">to the right hand side. Impulse response of the system described by the equation is given by:<o:p></o:p></span></div>
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<span style="font-family: CMMI10; font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-family: CMMI10; font-size: 100%;">h </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">) = </span><span style="font-family: CMMI10; font-size: 100%;">b</span><span style="font-family: CMMI7; font-size: 100%;">n</span><span style="font-family: CMMI10; font-size: 100%;"> δ</span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">) + </span><span style="font-family: CMMI10; font-size: 100%;">P</span><span style="font-family: CMMI7; font-size: 100%;">D </span><span style="font-family: CMR10; font-size: 100%;">[</span><span style="font-family: CMMI10; font-size: 100%;">y</span><span style="font-family: CMMI7; font-size: 100%;">n </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">)] μ(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">)<o:p></o:p></span></div>
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<span style="font-family: CMR10; font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;">Where for </span><span style="font-family: CMMI10; font-size: 100%;">m <><span style="font-family: Arial; font-size: 100%;">we have </span><span style="font-family: CMMI10; font-size: 100%;">b</span><span style="font-family: CMMI7; font-size: 100%;">n </span><span style="font-family: CMR10; font-size: 100%;">= 0</span><span style="font-family: Arial; font-size: 100%;">. Also, </span><span style="font-family: CMMI10; font-size: 100%;">y</span><span style="font-family: CMMI7; font-size: 100%;">n </span><span style="font-family: Arial; font-size: 100%;">equals the zero input response with initial conditions.<o:p></o:p></span></span></div>
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<span style="font-family: Arial; font-size: 100%;">{y<sup>n-1</sup> (0)=1,y<sup>n-2</sup> (0)=1,…,y(0)=0}<o:p></o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;"><o:p> </o:p></span></div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4383430891865259289.post-19600494700364047482010-02-20T02:05:00.000+06:002017-11-02T16:22:28.767+06:00Convolution<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="font-family: Arial; font-size: 100%;">As mentioned above, the convolution integral provides an easy mathematical way to express the output of an LTI system based on an arbitrary signal, </span><span style="font-family: CMMI10; font-size: 100%;">x </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">)</span><span style="font-family: Arial; font-size: 100%;">, and the system's impulse response, </span><span style="font-family: CMMI10; font-size: 100%;">h</span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">)</span><span style="font-family: Arial; font-size: 100%;">. The convolution integral is expressed as:<o:p></o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;">Convolution is such an important tool that it is represented by the symbol *, and can be written as<o:p></o:p></span></div>
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<span style="font-family: CMMI10; font-size: 100%;">y </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">) = </span><span style="font-family: CMMI10; font-size: 100%;">x </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">) *</span><span style="font-family: CMSY10; font-size: 100%;"> </span><span style="font-family: CMMI10; font-size: 100%;">h </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">)<o:p></o:p></span></div>
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<span style="font-family: CMR10; font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;">By making a simple change of variables into the convolution integral, </span><span dir="RTL" lang="HE" style="font-family: Arial; font-size: 100%;">ז</span><span style="font-family: Arial; font-size: 100%;">=t-</span><span dir="RTL" lang="HE" style="font-family: Arial; font-size: 100%;">ז</span><span style="font-family: Arial; font-size: 100%;">, we can easily show that convolution is commutative:</span></div>
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<span style="font-family: CMMI10; font-size: 100%;">x </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">) *</span><span style="font-family: CMSY10; font-size: 100%;"> </span><span style="font-family: CMMI10; font-size: 100%;">h </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">) = </span><span style="font-family: CMMI10; font-size: 100%;">h </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">) *</span><span style="font-family: CMSY10; font-size: 100%;"> </span><span style="font-family: CMMI10; font-size: 100%;">x </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">)</span><span style="font-family: Arial; font-size: 100%;"><o:p></o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;"><o:p> </o:p></span></div>
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<span style="font-family: Arial; font-size: 100%;"><span style="font-weight: bold;">Brief Overview of Derivation Steps:</span><o:p></o:p></span></div>
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<span style="font-size: 100%;"><o:p> </o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: Arial; font-size: 100%;">1.<span style="font-family: ";"> </span></span><!--[endif]--><span style="font-family: Arial; font-size: 100%;">An impulse input leads to an impulse response output.<o:p></o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: Arial; font-size: 100%;">2.<span style="font-family: ";"> </span></span><!--[endif]--><span style="font-family: Arial; font-size: 100%;">A shifted impulse input leads to a shifted impulse response output. This is due to the time invariance of the system.<o:p></o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: Arial; font-size: 100%;">3.<span style="font-family: ";"> </span></span><!--[endif]--><span style="font-family: Arial; font-size: 100%;">We now scale the impulse input to get a scaled impulse output. This is using the scalar multiplication property of linearity.<o:p></o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: Arial; font-size: 100%;">4.<span style="font-family: ";"> </span></span><!--[endif]--><span style="font-family: Arial; font-size: 100%;">We can now "sum up" an infinite number of these scaled impulses to get a sum of an infinite number of scaled impulse responses. This is using the additivety attribute of linearity.<o:p></o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: Arial; font-size: 100%;">5.<span style="font-family: ";"> </span></span><!--[endif]--><span style="font-family: Arial; font-size: 100%;">Now we recognize that this infinite sum is nothing more than an integral, so we convert both sides into integrals.<o:p></o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: Arial; font-size: 100%;">6.<span style="font-family: ";"> </span></span><!--[endif]--><span style="font-family: Arial; font-size: 100%;">Recognizing that the input is the function </span><span style="font-family: CMMI10; font-size: 100%;">f </span><span style="font-family: CMR10; font-size: 100%;">(</span><span style="font-family: CMMI10; font-size: 100%;">t</span><span style="font-family: CMR10; font-size: 100%;">)</span><span style="font-family: Arial; font-size: 100%;">, we also recognize that the output is exactly the convolution integral.<o:p></o:p></span></div>
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<span style="font-size: 100%;"><o:p> </o:p></span></div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4383430891865259289.post-17472039246701051692010-02-20T01:47:00.000+06:002017-11-02T16:25:52.534+06:00The Fourier Series<div dir="ltr" style="text-align: left;" trbidi="on">
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A Fourier series is an expansion of a periodic function <i>f </i>(<i>t</i>) in terms of an infinite sum of cosines and sines. </div>
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In other words, any periodic function can be resolved as a summation of constant value and cosine and sine functions:</div>
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The computation and study of Fourier series is known as <i>harmonic analysis</i> and is extremely useful as a way to break up an arbitrary periodic function into a set of simple terms that can be plugged in, solved individually, and then recombined to obtain the solution to the original problem or an approximation to it to whatever accuracy is desired or practical. <o:p></o:p></div>
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Symmetry Considerations:</div>
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<!--[if !supportLists]--><span style="font-family: "symbol";">·<span style="font-family: "; font-size: 7pt;"> </span></span><!--[endif]-->Symmetry functions:<o:p></o:p></div>
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(i) <b>even</b> symmetry<o:p></o:p></div>
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(ii) <b>odd</b> symmetry<o:p></o:p></div>
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<o:p> </o:p></div>
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Even symmetry: Any function <i>f</i> (<i>t</i>) is <b>even</b> if its plot is symmetrical about the vertical axis, i.e. <link href="file:///C:%5CDOCUME%7E1%5CRashad%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml" rel="File-List"></link><!--[if gte mso 9]><xml> <w:worddocument> <w:view>Normal</w:View> <w:zoom>0</w:Zoom> <w:compatibility> <w:breakwrappedtables/> <w:snaptogridincell/> <w:wraptextwithpunct/> <w:useasianbreakrules/> </w:Compatibility> <w:browserlevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style> <!-- /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman";} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;} </style> </div>
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--><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman";} </style> <![endif]--><span style="font-family: "; font-size: 12pt;">f(-t)=f(t)</span><!--[if gte mso 9]><xml> <o:oleobject type="Embed" progid="Equation.3" shapeid="_x0000_i1025" drawaspect="Content" objectid="_1337135467"> </o:OLEObject> </xml><![endif]--><div class="MsoNormal" style="text-align: justify;">
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The examples of <b>even</b> functions are:<o:p></o:p></div>
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The integral of an <b>even</b> function from −<i>A</i> to +<i>A</i> is twice the integral from 0 to +<i>A<o:p></o:p></i></div>
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Odd symmetry: Any function <i>f</i> (<i>t</i>) is <b>odd</b> if its plot is antisymmetrical about the vertical axis, i.e. <link href="file:///C:%5CDOCUME%7E1%5CRashad%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_filelist.xml" rel="File-List"></link><!--[if gte mso 9]><xml> <w:worddocument> <w:view>Normal</w:View> <w:zoom>0</w:Zoom> <w:compatibility> <w:breakwrappedtables/> <w:snaptogridincell/> <w:wraptextwithpunct/> <w:useasianbreakrules/> </w:Compatibility> <w:browserlevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><style> <!-- /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman";} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;} </style> </div>
--><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman";} </style> <![endif]--><span style="font-family: "; font-size: 12pt;">f(-t)=-f(t)</span><!--[if gte mso 9]><xml> <o:oleobject type="Embed" progid="Equation.3" shapeid="_x0000_i1026" drawaspect="Content" objectid="_1337135468"> </o:OLEObject> </xml><![endif]--><div class="MsoNormal" style="text-align: justify;">
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The examples of <b>odd</b> functions are:<o:p></o:p></div>
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The integral of an <b>odd</b> function from −<i>A</i> to +<i>A</i> is zero<i><o:p></o:p></i></div>
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Even and odd functions:</div>
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The product properties of even and odd functions are:<o:p></o:p></div>
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<!--[if !supportLists]--><span style="font-family: "symbol";">·<span style="font-family: "; font-size: 7pt;"> </span></span><!--[endif]-->(even) × (even) = (even)<o:p></o:p></div>
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<!--[if !supportLists]--><span style="font-family: "symbol";">·<span style="font-family: "; font-size: 7pt;"> </span></span><!--[endif]-->(odd) × (odd) = (even)<o:p></o:p></div>
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<!--[if !supportLists]--><span style="font-family: "symbol";">·<span style="font-family: "; font-size: 7pt;"> </span></span><!--[endif]-->(even) × (odd) = (odd)<o:p></o:p></div>
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<!--[if !supportLists]--><span style="font-family: "symbol";">·<span style="font-family: "; font-size: 7pt;"> </span></span><!--[endif]-->(odd) × (even) = (odd)<o:p></o:p></div>
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<tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/_Jt8jI5P6sEU/TAjnkdDCceI/AAAAAAAAD4s/DB7rgz3an0k/s1600/9.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="margin-left: auto; margin-right: auto;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5478883560154886626" src="https://2.bp.blogspot.com/_Jt8jI5P6sEU/TAjnkdDCceI/AAAAAAAAD4s/DB7rgz3an0k/s400/9.JPG" style="float: left; height: 292px; margin: 0pt 10px 10px 0pt; width: 400px;" /></a></td></tr>
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Function defines over a finite interval<o:p></o:p></div>
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<!--[if !supportLists]--><span style="font-family: "symbol";">·<span style="font-family: "; font-size: 7pt;"> </span></span><!--[endif]-->Fourier series only support periodic functions<o:p></o:p></div>
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<!--[if !supportLists]--><span style="font-family: "symbol";">·<span style="font-family: "; font-size: 7pt;"> </span></span><!--[endif]-->In real application, many functions are non-periodic<o:p></o:p></div>
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<!--[if !supportLists]--><span style="font-family: "symbol";">·<span style="font-family: "; font-size: 7pt;"> </span></span><!--[endif]-->The non-periodic functions are often can be defined over finite intervals, e.g.<o:p></o:p></div>
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<tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/_Jt8jI5P6sEU/TAjnk8PI2qI/AAAAAAAAD40/YOQ8bvoz4Jk/s1600/10.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="margin-left: auto; margin-right: auto;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5478883568527137442" src="https://2.bp.blogspot.com/_Jt8jI5P6sEU/TAjnk8PI2qI/AAAAAAAAD40/YOQ8bvoz4Jk/s400/10.JPG" style="float: left; height: 115px; margin: 0pt 10px 10px 0pt; width: 400px;" /></a></td></tr>
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<!--[if !supportLists]--><span style="font-family: "symbol";">·<span style="font-family: "; font-size: 7pt;"> </span></span><!--[endif]-->Therefore, any non-periodic function must be <b>extended to a periodic function</b> first, before computing its Fourier series representation<o:p></o:p></div>
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<!--[if !supportLists]--><span style="font-family: "symbol";">·<span style="font-family: "; font-size: 7pt;"> </span></span><!--[endif]-->Normally, we prefer symmetry (even or odd) periodic extension instead of normal periodic extension, since symmetry function will provide zero coefficient of either <i>an</i> or <i>bn</i> <o:p></o:p></div>
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<!--[if !supportLists]--><span style="font-family: "symbol";">·<span style="font-family: "; font-size: 7pt;"> </span></span><!--[endif]-->This can provide a simpler Fourier series expansion<o:p></o:p></div>
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Half-range Fourier series expansion:</div>
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<!--[if !supportLists]--><span style="font-family: "symbol";">·<span style="font-family: "; font-size: 7pt;"> </span></span><!--[endif]-->The Fourier series of the even or odd periodic extension of a non-periodic function is called as the <i>half-range Fourier series<o:p></o:p></i></div>
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<!--[if !supportLists]--><span style="font-family: "symbol";">·<span style="font-family: "; font-size: 7pt;"> </span></span><!--[endif]-->This is due to the non-periodic function is considered as the half-range before it is extended as an even or an odd function <o:p></o:p></div>
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<!--[if !supportLists]--><span style="font-family: "symbol";">·<span style="font-family: "; font-size: 7pt;"> </span></span><!--[endif]-->If the function is extended as an even function, then the coefficient <i>bn</i>= 0, hence<i><o:p></o:p></i></div>
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<tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/_Jt8jI5P6sEU/TAjnlBm_AKI/AAAAAAAAD48/Iooy5pZ8Gvc/s1600/11.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="margin-left: auto; margin-right: auto;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5478883569969332386" src="https://2.bp.blogspot.com/_Jt8jI5P6sEU/TAjnlBm_AKI/AAAAAAAAD48/Iooy5pZ8Gvc/s400/11.JPG" style="float: left; height: 75px; margin: 0pt 10px 10px 0pt; width: 274px;" /></a></td></tr>
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Which only contains the cosine harmonics.<o:p></o:p></div>
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<!--[if !supportLists]--><span style="font-family: "symbol";">·<span style="font-family: "; font-size: 7pt;"> </span></span><!--[endif]-->Therefore, this approach is called as the <i>half-range Fourier cosine series <o:p></o:p></i></div>
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</div>
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<!--[if !supportLists]--><span style="font-family: "symbol";">·<span style="font-family: "; font-size: 7pt;"> </span></span><!--[endif]-->If the function is extended as an odd function, then the coefficient <i>an</i>= 0, hence<i><o:p></o:p></i></div>
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<tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/_Jt8jI5P6sEU/TAjnlYyJ1jI/AAAAAAAAD5E/mOpZKatn_vc/s1600/12.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="margin-left: auto; margin-right: auto;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5478883576190195250" src="https://1.bp.blogspot.com/_Jt8jI5P6sEU/TAjnlYyJ1jI/AAAAAAAAD5E/mOpZKatn_vc/s400/12.JPG" style="float: left; height: 75px; margin: 0pt 10px 10px 0pt; width: 274px;" /></a></td></tr>
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Which only contains the sine harmonics.<o:p></o:p></div>
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<!--[if !supportLists]--><span style="font-family: "symbol";">·<span style="font-family: "; font-size: 7pt;"> </span></span><!--[endif]-->Therefore, this approach is called as the <i>half-range Fourier sine series <o:p></o:p></i></div>
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<o:p> </o:p></div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4383430891865259289.post-50253500140751077162010-02-20T00:56:00.000+06:002017-11-02T16:27:09.301+06:00Parseval’s Theorem<div dir="ltr" style="text-align: left;" trbidi="on">
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Parserval’s theorem states that the average power in a periodic signal is equal to the sum of the average power in its DC component and the average powers in its harmonics. </div>
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For sinusoidal (cosine or sine) signal,<o:p></o:p></div>
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For simplicity, we often assume <i>R</i> = 1Ω, which yields</div>
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<tr><td style="text-align: center;"><a href="http://3.bp.blogspot.com/_Jt8jI5P6sEU/TAlA3GPfv8I/AAAAAAAAD6M/gRjVpxKv9zE/s1600/3.JPG" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="margin-left: auto; margin-right: auto;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5478981736985509826" src="https://3.bp.blogspot.com/_Jt8jI5P6sEU/TAlA3GPfv8I/AAAAAAAAD6M/gRjVpxKv9zE/s400/3.JPG" style="float: left; height: 64px; margin: 0pt 10px 10px 0pt; width: 110px;" /></a></td></tr>
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For sinusoidal (cosine or sine) signal,</div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-4383430891865259289.post-28575832900306358902010-02-18T22:59:00.000+06:002017-11-02T16:28:24.066+06:00Exponential Fourier series<div dir="ltr" style="text-align: left;" trbidi="on">
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Recall that, from the Euler’s identity,<o:p></o:p> <br />
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Then the Fourier series representation becomes<o:p></o:p></div>
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Then, the coefficient C<i><sub>n</sub></i> can be derived from<o:p></o:p></div>
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<span style="font-family: "symbol";">·<span style="font-family: "; font-size: 7pt;"> </span></span><!--[endif]-->In fact, in many cases, the complex Fourier series is easier to obtain rather than the trigonometrical Fourier series<o:p></o:p></div>
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<!--[if !supportLists]--><span style="font-family: "symbol";">·<span style="font-family: "; font-size: 7pt;"> </span></span><!--[endif]-->In summary, the relationship between the complex and trigonometrical Fourier series are:<o:p></o:p></div>
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