Page 2 ECS707P (2018)

Question 1: Transfer functions of digital systems [25 marks]

(a) A linear digital filter is described by the following difference equation:

y(n) = 2x(n) + 4 exp(j⇡/2)x(n 1) + y(n 1) + 0.75y(n 2) (1)

Find all poles and zeros of this filter in the form a+ jb and plot them in a fully labelled

pole-zero diagram. [7 marks]

(b) A second filter is described by the following difference equation:

y(n) = 2x(n) + 3x(n 1) + y(n 1) + 0.75y(n 2) (2)

Calculate the impulse response h(n) for this digital filter for all integer values of n,

and plot its values for n 2 {0, 1, 2, 3, 4}. [4 marks]

(c) Define BIBO stability, in words and by giving a mathematical expression to define it.

Is the system defined by equation (2) stable in a BIBO sense? Show your work and

reasoning as part of your answer. [4 marks]

(d) Draw a block diagram of the system defined by equation (2). [4 marks]

(e) Find the region of convergence of the transfer function specified by equation (2).

Discuss whether it shows the system to be stable or unstable. [3 marks]

(f) Find the expression for the frequency response of the phase for a filter having the

following transfer function,

H(z) =

z(z 2)

z2 1 , |z| > 1.

(You do not need to plot it nor calculate values at specific points.) [3 marks]

ECS07P (2018) Page 3

Question 2: DFT, convolution and system properties [25 marks]

(a) Calculate the inverse DFT of X(k) = {1,3, 2, 2}.

[8 marks]

(b) (i) Predict the length and (ii) calculate the values of the following linear convolution

product, using the “convolution machine” technique as covered in the lectures:

{2,1, 1} ⇤ {1,3, 2, 2}.

[7 marks]

(c) Define and explain the following system properties, with reference to the impulse

response h(n) of a digital signal processor:

1. causality

2. memory

3. stability

4. time invariance

[4 marks]

(d) An N -point DFT needs to be designed that calculates the spectrum of a signal with a

sampling rate fs = 15 kHz, such that the minimum guaranteed frequency resolution

is f = 1.5 Hz. Determine:

1. the number of points that is needed by a general DFT algorithm to achieve this;

2. the actual resolution achieved by an FFT-type implementation of the DFT;

3. the maximum frequency in the spectrum of a baseband signal that can be

represented by the sampled signal.

(Hint: An FFT algorithm is a DFT with a power-of-two number of frequency samples.)

[6 marks]

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Page 4 ECS707P (2018)

Question 3: Digital filtering [25 marks]

Consider two digital brickwall LTI filters, identified with letters A and B.

• Filter A is a linear phase lowpass filter with a normalised cutoff frequency of 0.5⇡

radians/sample. Its gain is 1.

• Filter B is a linear phase highpass filter with a normalised cutoff frequency of 0.4⇡

radians/sample. Its gain is 1.

(a) Sketch the shape of the magnitude response of each filter over the normalised

frequency range [⇡, ⇡]. [4 marks]

(b) Draw the magnitude response of the filter that results from a cascade of filter A

followed by B, assuming their phase delays are equal. [3 marks]

(c) Draw the magnitude response of the filter that results from a cascade of filter B

followed by A followed by B, assuming their phase delays are equal. [3 marks]

(d) Consider uniformly sampling a real continuous white noise signal x(t) at a sampling

rate of 1 kHz. You then pass the sampled signal through filter A followed by filter B,

and finally reconstruct the output by ideal (sinc) interpolation, creating the continuous

real signal xAB(t). In what frequency bands of (1000, 1000) Hz will xAB(t) have no

energy? [10 marks]

(e) Denote the frequency response of filters A and B by HA(ej!) and HB(ej!). Find the

phase response of filter B such that the magnitude response of the filter that results

from A and B in parallel is zero everywhere. (Hint: Recall that sending a phasor of

frequency ! through filter A will produce the output |HA(ej!)|ej![n\HA(ej!)].)

[5 marks]

ECS707P (2018) Page 5

Question 4: Digital filtering, part two [25 marks]

For some of the following questions, consider the digital structure shown below.

(a) Show your work as you find the transfer function of this filter. Specify its region of

convergence! [5 marks]

(b) For what values of k1 will this filter be stable? [2 marks]

(c) Choose a value k1 6= 0 for which the filter will be stable, and find and plot the poles

and zeros of the filter. [4 marks]

(d) What kind of filter is this, e.g., lowpass and FIR, lowpass and IIR, etc. Explain your

answer. [4 marks]

(e) Devise a different digital filter structure that has the same transfer function.

[4 marks]

(f) Describe any two methods for FIR digital filter design. [6 marks]

End of questions

Turn over

Page 6 ECS707P (2018)

Appendix: Useful formulae

linear convolution of two sequences x[n], h[n] : (x ? h)[n] =

1X

m=1

x[m]h[nm]

Z-transform of sequence x[n] : X(z) =

1X

n=1

x[n]zn

DTFT of sequence x[n] : X(ej!) =

1X

n=1

x[n]ej!n

DFT of length-N sequence x[n] : X[k] =

N1X

n=0

x[n]ej2⇡nk/N

inverse DFT of length-N sequence X[k] : x[n] =

1

N

N1X

k=0

X[k]ej2⇡nk/N

Bilinear transformation : s =

2

T

1 z1

1 + z1

sum of the first N -terms of a geometric series of ↵ : SN =

N1X

n=0

↵n =

1 ↵N

1 ↵

Appendix: Useful signals

Kroneker delta sequence [n] =

8<:1, n = 00, else

Step function µ[n] =

8<:1, n 00, else

Table 3: Properties of the z-Transform

Property Sequence Transform ROC

x[n] X(z) R

x1[n] X1(z) R1

x2[n] X2(z) R2

Linearity ax1[n] + bx2[n] aX1(z) + bX2(z) At least the intersection

of R1 and R2

Time shifting x[n− n0] z−n0X(z) R except for the

possible addition or

deletion of the origin

Scaling in the ejω0nx[n] X(e−jω0z) R

z-Domain zn0x[n] X

(

z

z0

)

z0R

anx[n] X(a−1z) Scaled version of R

(i.e., |a|R = the

set of points {|a|z}

for z in R)

Time reversal x[−n] X(z−1) Inverted R (i.e., R−1

= the set of points

z−1 where z is in R)

Time expansion x(k)[n] =

{

x[r], n = rk

0, n "= rk

X(zk) R1/k

for some integer r (i.e., the set of points z1/k

where z is in R)

Conjugation x∗[n] X∗(z∗) R

Convolution x1[n] ∗ x2[n] X1(z)X2(z) At least the intersection

of R1 and R2

First difference x[n]− x[n− 1] (1− z−1)X(z) At least the

intersection of R and |z| > 0

Accumulation

∑n

k=−∞ x[k]

1

1−z−1X(z) At least the

intersection of R and |z| > 1

Differentiation nx[n] −z dX(z)dz R

in the z-Domain

Initial Value Theorem

If x[n] = 0 for n < 0, then

x[0] = limz→∞X(z)

Table 4: Some Common z-Transform Pairs

Signal Transform ROC

1. δ[n] 1 All z

2. u[n] 11−z−1 |z| > 1

3. −u[−n− 1] 11−z−1 |z| < 1

4. δ[n−m] z−m All z except

0 (if m > 0) or

∞ (if m < 0)

5. αnu[n] 11−αz−1 |z| > |α|

6. −αnu[−n− 1] 11−αz−1 |z| < |α|

7. nαnu[n] αz

−1

(1−αz−1)2 |z| > |α|

8. −nαnu[−n− 1] αz

−1

(1−αz−1)2 |z| < |α|

9. [cosω0n]u[n]

1−[cosω0]z−1

1−[2 cosω0]z−1+z−2

|z| > 1

10. [sinω0n]u[n]

[sinω0]z−1

1−[2 cosω0]z−1+z−2

|z| > 1

11. [rn cosω0n]u[n]

1−[r cosω0]z−1

1−[2r cos ω0]z−1+r2z−2

|z| > r

12. [rn sinω0n]u[n]

[r sinω0]z−1

1−[2r cos ω0]z−1+r2z−2

|z| > r