LECTURE 22: FOURIER ANALYSIS OF DT SYSTEMS Objectives: Frequency Response Response of a Sinusoid DT MA Filter Filter Design DT WMA Filter Difference Equations Resources: Wiki: Fourier Analysis Wiki: Moving Average USU: The Discrete-Time FT DSPGuide: Moving Average Filters Logix4u: MA Demo MS Equation 3.0 was used with settings of: 18, 12, 8, 18, 12. URL:

Fourier Analysis of Discrete-Time Systems Recall our convolution sum for DT systems: Assume the ordinary DTFT of h[n] exists (absolute summability): The DTFT of h[n] is: We can also write our input/output relations: Note also (applying the time shift property): DT LTI ]i=k,

Response to a Sinusoid Paralleling our derivation for CT systems: Note that the H(ej) is periodic with period 2. Also, since h[n] is real-valued, |H(ejj)| is an even function: Taking the inverse DTFT: As we saw with CT LTI systems, when the input is a sinusoid, the output is a sinusoid at the same frequency with a modified amplitude and phase. If the system were nonlinear, what differences might we see in the output?

Example: Response to a Sinusoid

Linear Constant-Coefficient Difference Equations We can model the input/output behavior of a DT LTI systems using an Nth-order input/output difference equation (also called a digital filter): DT LTI Solution of such equations can be easily computed by solving for y[n]: Let us consider a simple example: Let us assume: (the latter are referred to as initial conditions). The output can be computed using a table: n x[n] x[n-1] x[n-2] y[n] y[n-1] y[n-2] 1 2 3 4 2.5

Difference Equations in MATLAB The solutions to these equations can be easily programmed in MATLAB. Note that the key step is actually a dot product between the equation’s coefficients and the previous samples of the output and input (often referred to as the filter memory). The response to a unit step function can also be computed using the function recur. The unit step function is created by assigning values of “1” to x, followed by the invocation of the recur function that performs the difference equation computations.

Complete Response of a First-Order Equation Consider the first-order linear difference equation: Let us assume that: The first part of the response is due to the initial condition being nonzero. The second part of the response is due to the forcing function, x[n]. Together, they comprise the complete response of the system. We will see that closed-form solutions of these equations can be easily computed using the z-transform, which is very similar to the Laplace transform. The z-transform converts the difference equation to an algebraic equation. Closed-form solutions can also be found using summation tables.

Example: Moving Average (MA) Filter In this example, we will demonstrate that the process of averaging is essentially a lowpass filter. Therefore, many different types of filters, or difference equations, can be used to average. In this example, we analyze what is known as a “moving average” filter: MATLAB CODE: W=0:.01:1; H=(1/2).*(1-exp(-j*2*pi*W))/.(1-exp(-j*pi*W)); magH=abs(H); angH=180*angle(H)/pi;

Example: Filter Design The magnitude of the frequency response of a 3rd-order MA filter: is shown to the right. What is wrong? Can we do better? Optimization of the coefficients, c, d, and e, is a topic known as filter design. We will use three constraints: This generates three equations:

Example: Filter Design (Cont.) Solution of these equations results in: The magnitude response is shown to the right. Suppose we cascade two of these filters. What is the impact? Both of these filters can be shown to have linear phase responses. Both compute “weighted” averages. Which is better? Which is more costly? Can we do better?

Summary Introduced the use of the DTFT to compute the output of a DT LTI system. Demonstrated that the output of a DT LTI system to a sinusoidal input is also sinusoidal. Formalized the concept of a difference equation Introduced the concept of filter design and demonstrated the design of moving average filters. Next: Laplace Transforms What properties will hold for this transform? Why do we need another transform? Where you have applied this?