1)

1) Find the Z-transforms of the following signals:

a) x[n] = [1] ≡ δ[n]

b) x[n] = [0 1] ≡ δ[n−1]

c) x[n] = [2 2 2 …] ≡ 2 u[n]

d) x[n] = [1 2 3] ≡ δ[n] + δ[n−1] + 3δ[n−2]

2) Find the coefficients a, b, c and d below to make H(z) an All-pass system:

Plot the linear-scale amplitude response of H(z). Also plot and calculate the zeros and poles of H(z).

How the poles and zeros are related to each other?

II. From lecture notes we find that in order to make an allpass system the multipliers in the denominator are the multipliers of the nominator in reversed order. This means also that the poles must be the inverses of the zeros to make H(z) an allpass system.

So d = 0, a = 8, b = -5 and c = 3.

Num=[3 -5 8];

Den=[8 -5 3];

Figure;

Freqz(Num,Den);

Figure;

Zplane(Num,Den)

Zeros=roots(Num);

Poles=roots(Den);

3) Consider the signal:

a) Generate and plot x[n] using Matlab.

b) What are the discrete frequencies present in x[n]?

c) If the sampling frequency is 2000, what continuous sines do x[n] represent?

d) If the sampling frequency is 500, what continuous sines do x[n] represent?

III.

n = 0:100;

xn = sin(pi/6*n) - sin(pi/12*n) + 2*sin(pi/8*n);

plot(xn)

%(c)

pi/6 / (2*pi) * 2000

pi/8 / (2*pi) * 2000

pi/12 / (2*pi) * 2000

%(d)

pi/6 / (2*pi) * 500

pi/8 / (2*pi) * 500

pi/12 / (2*pi) * 500

a) The discrete frequencies present are π/6, π/8 and π/12.

b) Sines of frequencies 167Hz, 125Hz and 83Hzi.

c) Sines of frequencies 42Hz, 31Hz and 21Hz.

a) Present the direct form implementation of the system given by the difference equation:

. Next implement the system both as a cascade and as a

parallel connection of two sub-systems.

(Hint: for the parallel implementation, consider partial fractional expansion of the

transfer function).

b) Implement the system as a cascade of two

sub-systems. (Hint: z = 1 is a root of the polynomial 1−3z + 4z2 −2z3).

IV. Taking Z-transform on both sides of the difference equation and organizing a bit we get

from which we can solve the zeros (none) and poles (two). The poles are 2 and 1 and so the transfer function can be formulated as follows

This implementation is shown in figure 1. The transfer function can be processed further into a sum of two functions by partial fraction expansion. . This is shown in figure 2.

(Figure 1 - above)

(Figure 2 - above)

This can also be formulated as follows

5) Assume g[n]=[1 2 3 2 1] is the common term for h1[n], h2[n], h3[n] and h4[n] which are type I, type

type III and type IV linear phase FIR filter. Use matlab and implement h1, h2, h3 and h4 and plot

frequency response for each, plus zero-poles and impulse response.

g=[1 2 3 2 1];

h1=g;

figure;

freqz(h1)

figure;

impz(h1);

figure;

zplane(h1)

f2=[1/2 1/2];

h2=conv(f2,g);

figure;

freqz(h2)

figure;

impz(h2);

figure;

zplane(h2)

f3=[1/2 0 -1/2];

h3=conv(f3,g);

figure;

freqz(h3)

figure;

impz(h3);

figure;

zplane(h3)

f4=[1/2 -1/2];

h4=conv(f4,g);

figure;

freqz(h4)

figure;

impz(h4);

figure;

zplane(h4)