EENG21000 SIGNALS AND SYSTEMS 2019
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January/February 2019
EENG21000
SIGNALS AND SYSTEMS
Q1 (a) Determine which of the following properties hold and which do not hold for systems (i) and (ii). Justify you answers.
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Properties |
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Systems |
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1. |
Memoryless |
(1 mark) |
i) |
y\n\ = n x\n\ |
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2. |
Causal |
(1 mark) |
ii) |
*(o = E-i) |
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3 |
BIBO Stable |
(1 mark) |
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4. |
Linear |
(2 marks) |
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5. |
Time invariant |
(2 marks) |
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(14 marks)
(b) Consider the cascade interconnection of three causal LTI systems shown in Figure 1. The impulse response h2[n] is: h2[n] = 〃[〃] - u\n - 2] where u[n] is the discrete-time unit step function.
Figure 1
The overall impulse response is as shown in Figure 2:
Figure 2
i) Find the impulse response h^[n]
ii) Find the response of the overall system to the input: x\n\ = d\n\ - 3[〃-1]
(6 marks)
Q2 (a) A discrete-time linear system is described by the following difference equation
y[n] — 0.9y[n-1] + 0.81y[n-2] = x\n\ + 1. lx[n-l]
What is the transfer function of this system? Comment on the stability of this system (3 marks)
(b) Sketch a block diagram for the system described by the above difference equation (3 marks)
(c) Determine the frequency response of this system and sketch the normalised amplitude response. (5 marks)
(d) Sketch the pole-zero plot of the following transfer functions. By considering the geometric interpretation of the magnitude of the Fourier transform from the pole-zero plot, determine whether the corresponding system has an approximately lowpass, bandpass, or highpass frequency response.
z_1
g)=f
1 +服T
X(Z)= ] 16 i + 64 2
iii) X(z)=
(9 marks)
Q3 The block diagram of a continuous time system is shown in Figure 3 where X(s) and Y(s) are the Laplace transforms of the input and output signals respectively.
Figure 3
(a) Find the system transfer function (4 marks)
(b) Determine the value of 妇 and ^2 when the poles of this system are at s = 1 and s = -5. Comment on the stability of the system (5 marks)
(c) Draw the system pole-zero plot and comment on the stability for ki=6 and k?=25 (5 marks)
(d) Find the impulse response of this system for ki=6 and k2=25 (6 marks)
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Signal 5(t) |
Transform 1 |
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u(t) |
1 s |
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-u(-t) |
1 s |
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tn-1 u(t) (n-1)!" tn-1 (n-1)! ' |
1 / 1 S7 |
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e 项 u(t) |
1 s + a |
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-e 项 u(-t) |
1 s + a |
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tn-1 ——e ~at u(t) (n-1)! tn-1 —-t—e 项 u(-t) (n-1)! ' 5(t -T) |
1 (s + a)n 1 (s + a)n e-sT |
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[COSTO ot]u(t) |
s s2 + TO 0 |
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[sin® gt]u(t) |
TO0 2 2 s2 + TO 0 |
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[e_atcos(D0t]u(t) |
s + a (s + a)2 + to 0 |
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[e_atsin® ot]u(t) |
TO 0 (s + a) + to 0 |
Theorem
F(s)
心切 ，(t) + f2(t)] 碓一"(圳 L[f(t-T)]
ft)]
= Jf(t)e-stdt
0- kF(s)
Fi(s) + F2(S)
F(s + a)
sF(s)-f(0-)
Definition
Linearity Theorem
Linearity Theorem
Frequency Shift Theorem Time Shift Theorem
Scaling Theorem
Differentiation Theorem
t
L Jf(T)dx
_0-
f(3)
f(0+)
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s2F(s)-sf(0-)-f(0-) |
Differentiation Theorem |
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n snF(s)-Z sn-kfk-1(0-) k=1 |
Differentiation Theorem |
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Fs) s |
Integration Theorem |
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lim sF(s) st0 |
Final Value Theorem |
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lim sF(s) ST8 |
Initial Value Theorem |
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Properties of the Z-Transform |
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fk |
F(z) |
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1. d |
1, |
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2. 1, |
A Z-1 尸 |
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3. k, |
Z W Z T 尸 |
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4. kn, |
[-土） ’（1 - Z」）」 |
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5. ak, |
(1 -OZ T )T |
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6. knak, |
[-Zdz)" I尸 |
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7.丄， k |
-1n(1 —z t ) |
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8. cos 弘, |
(1 — Z_1 cosa)(1 - 2z _'cosa + Z _2 ) 1 |
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9. sin 弘 |
Z t sina(1 — 2z _1 cosa + z ~2) 1 |
2023-01-09