## Saturday, June 5, 2010

### Classification of Signals

Along with the classification of signals below, it is also important to understand the Classification of Systems.
Continuous-Time vs. Discrete-Time:
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.
 Figure 1
Analog vs. Digital:
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.
 Figure 2
Periodic vs. Aperiodic:
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:
f(t) = f(T + t).........(1)
The fundamental period of our function, f(t) is the smallest value of T that the still allows Equation (1) to be true.
 (a) A periodic signal with period To

 (b) An aperiodic signal Figure 3

Causal vs. Anticausal vs. Noncausal:
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).
 (a) A causal signal

 (b) An anticausal signal

 (c) A noncausal signal Figure 4

Even vs. Odd:
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).
 (a) An even signal

 (b) An odd signal Figure 5

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.
f(t)=1/2(f(t)+f(-t))+1/2(f(t)−f(-t)).............(2)
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).
 (a) The signal we will decompose using odd-even decomposition

 (b) Even part: e(t)=1/2(f(t)+f(-t))

 (c) Odd part: o(t)=1/2(f(t)−f(-t)) Figure 6
Deterministic vs. Random:
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).
 (a) Deterministic Signal

 (b) Random Signal Figure 7

Right-Handed vs. Left-Hand Signal:
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
--> t <>1 < α , and a left-handed signal is defined as any signal where f(t)=0 for -->t > t1 > - α. See (Figure 8) for an example. Both figures "begin" at t1 and then extends to positive or negative infinity with mainly nonzero values.
 (a) Right-handed signal
 (b) Left-handed signal Figure 8
Finite vs. Infinite Length:
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
-->t1 <>2 Where t1 > - α and t2 < 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
 Figure 9: Finite-Length Signal.
Note that it only has nonzero values on a set, finite interval.

### Signal Operations

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.

Time Shifting:
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).
 Figure 1: f (t – T) moves (delays) f to the right by T.
Time Scaling:
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.
 Figure 2: f (at) compresses f by a.

Time Reversal:
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.
 Figure 3: Reverse the time axis.