Friday, February 19, 2010

Time Domain Analysis of Continuous Time Systems

Continuous-Time Linear Systems:
Physically realizable, linear time-invariant systems can be described by a set of linear differential equations (LDEs):

Figure: Graphical description of a basic linear time-invariant system with an input, f (t) and an output, y (t).

Equivalently,



With an = 1.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 y (t) to an input f (t). Recall that such a solution can be written as

y (t) = yi (t) + ys (t)

We refer to yi (t) as the zero-input response – the homogeneous solution due only to the initial conditions of the system. We refer to ys (t) as the zero-state response – the particular solution in response to the input f (t). We now discuss how to solve for each of these components of the system's response.

Finding the Zero-Input Response:

The zero-input response, yi (t), is the system response due to initial conditions only.

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

Finding the Zero-State Response:

Solving a linear differential equation

Given a specific input f (t) is a difficult task in general. More importantly, the method depends entirely on the nature of f (t); if we change the input signal, we must completely re-solve the system of equations to find the system response.

Convolution helps to bypass these difficulties. We explain how convolution helps to determine the system's output, given only the input f (t) and the system's impulse response h(t).

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 m <>:

We can rewrite as

QD [y (t)] = PD [f (t)]

Where QD [.] is an operator that maps y (t) to the left hand side

And PD [.] maps f (t) to the right hand side. Impulse response of the system described by the equation is given by:

h (t) = bn δ(t) + PD [yn (t)] μ(t)

Where for m <>we have bn = 0. Also, yn equals the zero input response with initial conditions.

{yn-1 (0)=1,yn-2 (0)=1,…,y(0)=0}


Convolution

As mentioned above, the convolution integral provides an easy mathematical way to express the output of an LTI system based on an arbitrary signal, x (t), and the system's impulse response, h(t). The convolution integral is expressed as:




Convolution is such an important tool that it is represented by the symbol *, and can be written as

y (t) = x (t) * h (t)

By making a simple change of variables into the convolution integral, ז=t-ז, we can easily show that convolution is commutative:

x (t) * h (t) = h (t) * x (t)

Brief Overview of Derivation Steps:

1. An impulse input leads to an impulse response output.

2. A shifted impulse input leads to a shifted impulse response output. This is due to the time invariance of the system.

3. We now scale the impulse input to get a scaled impulse output. This is using the scalar multiplication property of linearity.

4. 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.

5. Now we recognize that this infinite sum is nothing more than an integral, so we convert both sides into integrals.

6. Recognizing that the input is the function f (t), we also recognize that the output is exactly the convolution integral.

The Fourier Series

A Fourier series is an expansion of a periodic function f (t) in terms of an infinite sum of cosines and sines.

In other words, any periodic function can be resolved as a summation of constant value and cosine and sine functions:

The computation and study of Fourier series is known as harmonic analysis 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.

Symmetry Considerations:

· Symmetry functions:

(i) even symmetry

(ii) odd symmetry

Even symmetry: Any function f (t) is even if its plot is symmetrical about the vertical axis, i.e. f(-t)=f(t)

The examples of even functions are:

The integral of an even function from −A to +A is twice the integral from 0 to +A

Odd symmetry: Any function f (t) is odd if its plot is antisymmetrical about the vertical axis, i.e. f(-t)=-f(t)

The examples of odd functions are:

The integral of an odd function from −A to +A is zero

Even and odd functions:

The product properties of even and odd functions are:

· (even) × (even) = (even)

· (odd) × (odd) = (even)

· (even) × (odd) = (odd)

· (odd) × (even) = (odd)

Function defines over a finite interval

· Fourier series only support periodic functions

· In real application, many functions are non-periodic

· The non-periodic functions are often can be defined over finite intervals, e.g.

· Therefore, any non-periodic function must be extended to a periodic function first, before computing its Fourier series representation

· Normally, we prefer symmetry (even or odd) periodic extension instead of normal periodic extension, since symmetry function will provide zero coefficient of either an or bn

· This can provide a simpler Fourier series expansion

Half-range Fourier series expansion:

· The Fourier series of the even or odd periodic extension of a non-periodic function is called as the half-range Fourier series

· 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

· If the function is extended as an even function, then the coefficient bn= 0, hence







Which only contains the cosine harmonics.

· Therefore, this approach is called as the half-range Fourier cosine series

· If the function is extended as an odd function, then the coefficient an= 0, hence






Which only contains the sine harmonics.

· Therefore, this approach is called as the half-range Fourier sine series

Parseval’s Theorem

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.

For sinusoidal (cosine or sine) signal,





For simplicity, we often assume R = 1Ω, which yields




For sinusoidal (cosine or sine) signal,

Thursday, February 18, 2010

Exponential Fourier series

Recall that, from the Euler’s identity,

Then the Fourier series representation becomes

Then, the coefficient Cn can be derived from












· In fact, in many cases, the complex Fourier series is easier to obtain rather than the trigonometrical Fourier series

· In summary, the relationship between the complex and trigonometrical Fourier series are: