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信号系统代写 | SIGNALS & SYSTEMS 5CCS2SAS COURSEWORK 2

这个作业是完成脉冲响应、傅里叶变换等信号相关的问题

SIGNALS & SYSTEMS
5CCS2SAS
COURSEWORK 2

Question 1:
The impulse response of a discrete-time linear time invariant (LTI)
system is given by
ℎ[𝑛] = 2 [
sin(
𝜋
4
𝑛)
𝜋𝑛 ] cos(
𝜋
2
𝑛)
a) Plot the discrete-time Fourier transform of ℎ[𝑛] over
– 𝜋 < 𝜔 < 𝜋.
[15 marks]
b) Find the output 𝑦[𝑛] of this system to the input
𝑥[𝑛] =
1
2
𝑒
𝑗
𝜋
6
𝑛 + 𝑒
𝑗
𝜋
3
𝑛 + 𝑒
𝑗
𝜋
2
𝑛 + 𝑒
𝑗
2𝜋
3
𝑛 +
1
2
𝑒
𝑗
5𝜋
6
𝑛 −
1
2
𝑒
−𝑗
𝜋
6
𝑛 − 𝑒
−𝑗
𝜋
3
𝑛
− 𝑒
−𝑗
𝜋
2
𝑛 − 𝑒
−𝑗
2𝜋
3
𝑛 −
1
2
𝑒
−𝑗
5𝜋
6
𝑛
[10 marks]
Question 2:
Consider the continuous-time signal
𝑥(𝑡) =
sin(10𝜋𝑡)
𝜋𝑡
+
sin(20𝜋𝑡)
𝜋𝑡
a) Draw the Fourier transform of 𝑥(𝑡). [15 marks]
b) Find and draw the Fourier transform of 𝑦(𝑡) = 𝑥(𝑡) cos(30𝜋𝑡).
[10 marks]
2
Question 3:
The impulse response ℎ(𝑡) of a linear-time invariant (LTI) system and a
signal 𝑥(𝑡) are given as:
ℎ(𝑡) = {
1, − 5 ≤ 𝑡 ≤ 5
0, otherwise
𝑥(𝑡) = {
1, 0 ≤ 𝑡 ≤ 10
0, otherwise
a) Using convolution in time domain find and sketch the output of
this system to the input signal 𝑧(𝑡) = 𝑥(𝑡 + 5).
Note: Details and calculations leading to the final result must be
written.
[15 marks]
b) Find and sketch the output of this LTI system to the input 𝑥(𝑡).
[5 marks]
c) Is this system causal? Why?
[5 marks]
Question 4:
In a causal discrete-time linear-time invariant system the relation
between the input 𝑥[𝑛] and the output 𝑦[𝑛] is given by the following
difference equation:
𝑦[𝑛] −
1
4
𝑦[𝑛 − 1] = 𝑥[𝑛]
Find the Fourier series representation of the output 𝑦[𝑛] for the input:
𝑥[𝑛] = cos (
𝜋
4
𝑛) + 2cos (
𝜋
2
𝑛)
[25 marks]


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