final assignment signal spring 2020 - guc
TRANSCRIPT
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German University in Cairo
Faculty of IET
Prof. Dr. Ahmed El-Mahdy
COMM 401 Signal & System Theory
Final Assignment
Deadline: (31/05/2020)
Full Name:
Student's ID:
Tutorial Number:
TA Name:
.
CHEATING CASES ARE GRADED ZERO!
Question
Number
1 2 3 4 5 6 7 8 9 Total
Obtained
score
Maximum
Score
2
:Directions
You should send the assignment with the cover page.
Solve in the pages following each problem.
Solution must be individual, grouping is totally refused.
For the practical question, you can use MATLAB. If you do not have it ) or Octave (very similar to MALAB-same commands). https://www.gnu.org/software/octave/. Write down the code used and indicate the final results for each part. All codes and figures should be printed from Matlab itself.
The deadline for submitting the assignment will be: May 31/05/2020. The system will not accept any delayed submission
Important note: Substitute with the numbers a, b, and c from the first step in your solution where a, b, and c are given as:
If you have two zeros in your ID, convert the zero to 1
Example: ID=46-2004, then a=2, b=1, c=4
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Problem (1):
(a) Check the linearity, causality, time invariance, and stability of the following system
and show your answer by equations: π¦(π‘) = 2ππ₯(π‘)cos (π‘ + π)
(b) Check if the following systems are invertible or not and show your answer:
(i)π¦(π‘) = ln (ππ₯(π‘)) (ii) π¦(π‘) = sin (π₯(π‘)) (iii)π¦(π‘) =πππ₯(π‘)
ππ‘
If the system is invertible determine the inverse system
--------------------------------------------------- Solution--------------------------------------------
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Problem (2):
(a) Find the convolution of )(tx and h(t) where:
π₯(π‘) = 2π π’(π‘)π’ (π
8β π‘) and β(π‘) = 2π πππ (
2π π‘
π) π’(π‘) π’ (
π
8β π‘)
(b) Consider the interconnection of the Linear Time-Invariant (LTI) systems shown in
the Figure and have impulse responses: β1[π] = ππΏ[π β 1] and
β2[π] = β3[n]=[n+1] c u[n]
.][1 nh to the system ][nhinvsystem (i) Determine the impulse response of the inverse
(ii) Determine (without drawing) the overall impulse response of the system ][nh .
(iii) Determine the response of the system shown in the figure to the input:
π₯[π] =a πΏ[π + 2] + ππΏ[π β 1]
--------------------------------------------------- Solution--------------------------------------------
+
+
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7
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Problem (3):
Given the periodic signal x[n] with period N=6c as shown in the following figure. The amplitude of the impulses are: 1 and 2b as shown. Find the Fourier series coefficients of this signal.
--------------------------------------------------- Solution-----------------------------------------------
-6c 0 6c 12c
1
2b
1
2b
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Problem (4):
Find Fourier series coefficients of the following periodic continuous time
signal with period T=2c:
--------------------------------------------------- Solution-----------------------------------------------
(a)
aπβπ‘ a
a-
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Problem (5):
(a) Derive Fourier transform of the following signal: π₯(π‘) = 2ππβbπ‘
2a u(t). Then plot the magnitude and phase responses.
(b) Find the inverse Fourier transform of the signal:
π(ππ) = ππ ππ sin(π) π sin (2π)
π2
--------------------------------------------------- Solution-----------------------------------------------
(a)
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Problem (6):
(a) Use the duality property to find Fourier transform of: π¦(π‘) =2ππ‘
(1+π‘2)2
(b) The signal π₯(π‘) = ππ ππ(π‘π1)
π‘ is applied to the input of an ideal bandpass
filter with frequency response:
π»(ππ) = {π ππ1 β€ |π| β€ ππ2
0 ππ‘βπππ€ππ π
Where ππ1 = π1/2 and ππ2 = π1
(i) Plot the frequency spectrum of the output signal π(ππ).
(ii) Find y(t) using the properties of Fourier transform.
--------------------------------------------------- Solution-----------------------------------------------
(a)
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Problem (7):
(a) A signal π(π‘) = 2ππππ 3(2ππ103π‘ + 0.3π) is to be sampled. Find the
minimum sampling angular frequency required.
(b) The signal π₯(π‘) =ππ ππ(π π‘)
π π‘ is modulated by multiplication of c(t) to produce
y(t) as shown in the following figure:
Find and draw Fourier transform of y(t) if π(t) = ππππ (2πππ‘) (Use the
Fourier transform)
------------------------------------------ Solution------------------------------------------------
(a)
x(t)
)(tc
y(t)
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Problem (8): (MATLAB Problem)
Write a Matlab Function that plots the of the following signal in the discrete time domain for
any value of the shift π and for any range of π:
π₯(π β π) = {π(π β π)ππβπ , π β₯ π
0, π < π
Problem (9): (MATLAB Problem)
Subplot the following signals in the same figure choose an appropriate range for π:
- π₯(π) = b cos (0.1bππ +π
6)
- π¦(π) = 2π sin (0.1b ππ +π
3)
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