1. To learn about the frequency response of linear time-invariant (LTI) systems;
2. To learn how to use MATLAB to implement convolution numerically;
3. To explore different methods for measuring the frequency response of an LTI system;
4. To learn about the power of a signal and how to measure power.
The project is split into four parts. This document describes Part 1, which focuses on the definition of the frequency response of an LTI system and how to compute the frequency response from the impulse response.
Lab Report Your lab report for this lab will consist of answers and complete documentation of the questions and exercises in Section 2. Section 1, is designed to get you ready for the problems in Section 2. Do not skip Section 1! Guidelines for preparing the lab report are posted on the Blackboard site. If you have any questions, talk to your lab TA. Each student must do his or her own work on this lab. However, you may ask other students or any of the teaching staff for advice.
Any reasonable suspicion of an honor code violation will be reported. You are allowed to discuss lab exercises with other students, but the submitted work should be original and it should be your own work. For more information about the Honor Code, please see the ECE 201 Lecture Syllabus and the ECE 201 Lab Syllabus. If you have questions, ask a member of the teaching staff.
Systems are used to process signals. As indicated in Figure 1, they accept an input signal x[n], process that signal, and produce the output signal y[n]. Throughout this project, we focus on discrete-time system so that both signals x[n] and y[n] consist of sequences of samples. As a result, the frequencies that we will encounter normalized (digital) frequencies fˆd. Recall that the digital frequencies relate to analog frequencies troughthe relationship fˆd = f . We will assume that analog signals have been oversampled so that 0 ≤ fˆd < 1.
Linear, time-invariant (LTI) systems are a large and very important subset of systems for signal processing. You have explored the concepts of linearity and time-invariance in a previous lab. For any LTI system, the output signal y[n] is computed from the input signal x[n] through a a convolution with the impulse response h[n] of the LTI system,
y[n] = x[n] ∗ h[n] =k=Σ−∞h[k]
An important aspect of LTI systems is their response, i.e., output, when the input signal x[n] is a complex exponential signal of frequency fˆd. When x[n] = exp(j2πfˆdn) is convolved with the filter’s impulse response h[n] according to equation (1) then the output signal y[n] is
y[n] =
k=Σ−∞ k=Σ−∞ k=Σ−∞
h[k] x[n − k]
h[k] exp(j2πfˆd(n − k))
h[k] exp(j2πfˆdn)
exp(−j2πfˆdk)
= exp(j2πfˆdn) k=Σ−∞
h[k] exp(−j2πfˆdk) = exp(j2πfˆdn)
(On the last line, we have defined
The quantity H(ej2πfˆ ) is called the frequency response of the system. Notice that H(ej2πfˆ ) is a complex-
valued scalar that depends on the impulse response h[n] and on the frequency fd. For a given LTI system with impulse response h[n] it is of great interest to know how the frequency response varies with fˆd.
We showed above that when the input signal is a complex exponential signal of frequency fˆd, i.e., x[n] =
exp(j2πfˆdn), then the output signal is y[n] = exp(j2πfˆdn)
This equation provides three important insights:
1. When the inputs signal to an LTI system is a complex exponential signal of frequency fˆd, then the output signal is also a complex exponential signal of the same frequency.
2. The frequency response H(ej2πfˆ ) determines the amplitude and phase of the output signal. From the above, we see that for an input signal with amplitude Ax = 1 and phase φx = 0, H(ej2πfˆ ) is the phasor of the output signal. Therefore, the amplitude of the output signal
3. From these two considerations, it follows that for a complex exponential input signal x[n] = exp(j2πfˆdn), we do not need to perform a convolution to find the output signal y[n]. The frequency of y[n] is the same as the input frequency and the amplitude and phase can be determined from the frequency response
d ). This observation is easily extended to sums of complex exponentials because of linearity.
Because of these observations, the frequency response often provides the most useful characterization of a LTI system. In particular, the magnitude of the frequency response is usually the first thing to look at to understand what a LTI system does.In this lab, we focus on computing and visualizing the frequency response of an LTI system from it’s impulse response.
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