3.1 Rational Z-transforms

As noted in your textbook, the common Z-transform pairs can be written as a ratio of two polynomials in z^-1 or z. That is H(z) ND((zz)) b0 b1z11LbaMNzzMN

a0 a1z L

If we know that a system can be described by H(z), its output to a known input X(z) can be computed by multiplying H(z) X(z). On the other hand, if we know the input X(z) and the output Y(z), the system can be identified. The time domain solution can be easily obtained by using partial-fraction expansions.

Problem 1.

Consider the system:

z² z H(z) z₂ 3z2

How many poles and zeros do you have?

Poles _______

Zeros _______

Use the command roots in Matlab to determine the numerical values of these poles and zeros. Use roots([1 1 0]) for the zeros and roots([1 3 2]) for the poles.

Poles =

Zeros =

Describing the same system in terms of increasing order of z^-1, we have 1z1

H(z)1 1 2z2 3z

Use roots([1 1]) for the zeros and roots([1 3 2]) for the poles. Do you have the same answer?

Poles =

Zeros =

Compute h(n) using the poles and zeros found with Matlab.

h(n) =

Problem 2

Use the filter command in Matlab to generate and plot the impulse response h[n] of the following second order difference equation. Plot h[n] in the range of 10 n 100.

y[n] 1.8cos(/16)y[n-1] + 0.81y[n-2] = x[n] + 0.5x[n-1]

Determine the impulse response using the inverse Z-transform and confirm your results.

3.2 Frequency response of difference equations

The frequency response of a system h(n) can be obtained by evaluating the z-transform of the system on the unit circle. That is, if a system is described by

K

(z z_k )

H(z) A ^k_M¹ ,

(z p_m )

m1

the contribution of each pole and each zero to the magnitude in the frequency domain depends on the length of the vector from the pole or zero to the point e ^jw . Thus, taking the magnitude and evaluating at e ^jw yields:

K K

| e ^jw z_k | v_k

| H(e jw)| A kM1

| e ^jw p_m | u_m

m1 m1

as shown in the next figure.

The phase is given by

K M

H(e ^jw) A(e ^jw z_k )(e ^jw p_k )

k1 m1

Problem 3.

Plot the poles and zeros of the system in Problem 2.

Using the command freqz, plot the magnitude and phase of the frequency response of this system. If the system is to be defined as a filter, what kind of filter it is?

Filter is ____________________________

Do the same for the following filters:

• y[n] + 0.13y[n-1] + 0.52y[n-2] + 0.3y[n-3] = 0.16x[n] 0.48x[n-1] + 0.48x[n-2]

0.16x[n-3]

H(z) =

____________________________________________

Filter is ____________________________

• y[n] 0.268y[n-2] = 0.634x[n] 0.634x[n-2]

H(z) =

____________________________________________

Filter is ____________________________

• y[n] + 0.268y[n-2] = 0.634x[n] + 0.634x[n-2]

H(z) =

____________________________________________

Filter is ____________________________

• 10y[n] 5y[n-1] + 2y[n-2] = x[n] 5x[n-1] + 10x[n-2]

H(z) =

____________________________________________

Filter is ____________________________

Without taking into consideration the cut-off frequencies, draw the poles and zeros of the simplest low-pass and high-pass filters you can think of. That is, use a minimum number of poles and zeros. Explain your results.

You will be given an acoustic signal in the file BoatWhale.mat. Note that the signal is already in Matlab format. The samples correspond to a recording using a hydrophone of a whale sound but it also records the sound of the boat (Fs = 22050Hz). Design a filter that can attenuate the sound of the boat. Please include the pole-zero diagram and your methodology.

Matlab instructions that can help you:

To listen to the audio use sound

To obtain a polynomial from roots use the command poly The command tf can give you the pole-zero plot.

Hint, the sound of the boat has a good number of harmonics.