# ELEC 310 - Signal Analysis II

## Assignment 3

• For the 3 filters of Assignment 1, find the transfer function H(z). Write H(z) as a ratio of polynomials and identify the poles and zeros. Choose a = 0.9.
• For the 3 filters of Assignment 1, find the impulse response h(n) in two ways:
• work out algebraically using the inverse z-transform, and evaluate the first few terms numerically.
• Use the POI software to place the poles and zeros in the complex plane, and let the program calculate the impulse response. in POI, drag zeros on the z-plane from the top left, and drag poles from the top right. Under the tab "analysis", choose "analysis graph", and then in the new window under the tab "graph" choose "impulse response". Under the tab "file" choose "save values" which will save the impulse response values in a file.
Compare the results of algebra with POI. The sampling rate fs is not defined in Assignment 1, the POI software lets you choose the sampling rate. Choose fs = 44,100 Hz used for music CDs.
• POI also calculates the frequency response H(z) with z along the unit circle z = exp(jw_hat) = exp(j2pif/fs) for |f| less thanl fs/2. The sampling rate fs is not defined in Assignment 1, the POI software lets you choose the sampling rate. For fs = 44,100 Hz, calculate the frequency response for each of the 3 filters of Assignment 1. Do this two ways:
• Use the POI software to place the poles and zeros in the complex plane, and let the program calculate the frequency response.
• work out the algebraic expression for the frequency response and plot using Matlab.
Compare the results from POI and Matlab.
• Example Matlab code for plotting frequency response filtdemo.m
• A linear time invariant filter is described by the difference equation y[n]=0.8y[n-1]-0.8x[n]+x[n-1].
• Find H(z) and identify the poles and zeros.
• Find an expression for the frequency response, assuming fs=44,100 Hz.
• Show that the magnitude of the frequency response is 1 for all values of f.
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