Section 3: Frequency Response & Input-Output Relationship

The system function of an ITI system is defined by its frequency response H(e)

1+²

H(e)=

1-1.85 cos

+0.83

18.

Note that the frequency response of the system is the ratio of the output to input. This can be

inferred from the duality property between time domain convolution and its equivalent frequency

domain multiplication.

H(el") =

Y(e)

X(eu)

5. Using freqz with N=512, make plots of the magnitude and phase responses H(e)

and ZH(e) for 0 ≤ ≤n. Specify what type of filter this system represents (.e.

lowpass, highpass, bandpass, etc.).

6. As you know, an ITI system can be characterized by its input-output relationship in addition

to its definition using the impulse response denoted by h[n] (discrete time domain

representation) or the frequency response denoted by H(e) or H(w). [Same notations

since both are function of frequency o-Frequency domain representation.

Find the input-output relationship of the given LTI system.

Hint: You should get something similar to the difference equation presented in the previous

problem.

2

7. Plot the impulse response h[n] in Matlab.

Hint: To get the impulse response of the system, you have to excite the system with a unit

impulse sequence. In other terms, the input excitation x[n] that should be applied to the

system is the unit impulse function 8[n]. Accordingly, the output to the system will be the

convolution of h[n] with 8[n] and which will result in simply h[n]. Hence, plotting the

output y[n] is equivalent to plotting the system impulse response h[n].

Note that there is an alternative method to find the discrete time impulse response h[n] by

directly applying inverse Fourier Transform to the frequency response H(e) but this

approach is more difficult and requires lot of algebraic computations.