0.3 Lab 3 - frequency analysis  (Page 4/4)

System analysis

Double click the icon labeled CT System Analysis using a Network Analyzer to bring up a system as shown in [link] . This system includes a Network Analyzer model for measuring the frequency response of a system.The Network Analyzer works by generating a weighted chirp signal (shown on the Scope ) as an input to the system-under-test.The analyzer measures the frequency response of the input and output of the system and computes the transfer function.By computing the inverse Fourier transform, it then computes the impulse response of the system.Use this setup to compute the frequency and impulse response of the given fourth order Butterworth filter with a cut-offfrequency of 1Hz. Print the figure showing the magnitude response, the phase response and the impulse response of the system. To use the tallmode to obtain a larger printout, type orient('tall'); directly before you print.

An alternative method for computing the impulse response is to input a step into the system and then to compute the derivative of the output.The model for doing this is given in the CT System Analysis using a Unit Step block. Double click on this icon and compute the impulse response of thefilter using this setup ( [link] ). Make sure that the characteristics of the filterare the same as in the previous setup. After running the simulation, print the graphof the impulse response.

Discrete-time frequency analysis

In this section of the laboratory, we will study the use of the discrete-time Fourier transform.

Discrete-time fourier transform

The DTFT (Discrete-Time Fourier Transform) is the Fourier representation used for finite energy discrete-time signals.For a discrete-time signal, $x\left(n\right)$ , we denote the DTFT as the function $X\left(e^{j\omega }\right)$ given by the expression

Since $X\left(e^{\mathrm{j\omega }}\right)$ is a periodic function of $\omega$ with a period of $2\pi$ , we need only to compute $X\left(e^{\mathrm{j\omega }}\right)$ for $-\pi <\omega <\pi$   .

Write a Matlab function X=DTFT(x,n0,dw) that computes the DTFT of the discrete-time signal x Here n0 is the time index corresponding to the 1st element of the x vector, and dw is the spacing between the samples of the Matlab vector X . For example, if x is a vector of length N , then its DTFT is computed by

where $w$ is a vector of values formed by w=(-pi:dw:pi) .

For the following signals use your DTFT function to

• i Compute $X\left(e^{j\omega }\right)$
• ii Plot the magnitude and the phase of $X\left(e^{j\omega }\right)$ in a single plot using the subplot command.

1. $x\left(n\right)=\delta \left(n\right)$
2. $x\left(n\right)=\delta (n-5)$
3. $x\left(n\right)={(0.5)}^{n}u\left(n\right)$

System analysis

For help on printing Simulink system windows click here .

Double click the icon labeled DT System Analysis to bring up an incomplete block diagram as shown in [link] . It is for a model that takes a discrete-time sine signal, processesit according to a difference equation and plots the multiplexed input and output signals in a graph window.Complete this block diagram such that it implements the following difference equation given in "Magnitude and Phase of Discrete-Time Systems" of the background exercises.

You are provided with the framework of the setup and the building blocks that you will need.You can change the values of the Gain blocks by double clicking on them. After you complete the setup, adjust the frequency of Sine Wave to the following frequencies: $\omega =\pi /16$  ,   $\omega =\pi /8$  , and $\omega =\pi /4$  . For each frequency,make magnitude response measurements using the input and output sequences shown in the graph window.Compare your measurements with the values of the magnitude response $\left|H\right(e^{j\omega }\left)\right|$ which you computed in the background exercises at these frequencies.

An alternative way of finding the frequency response is taking the DTFT of the impulse response.Use your DTFT function to find the frequency response of this system from its impulse response. The impulse response was calculated in "Magnitude and Phase of Discrete-Time Systems" of the background exercises. Plot the impulse response, andthe magnitude and phase of the frequency response in the same figure using the subplot command.