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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

3

Problem (1):

(a) Check the linearity, causality, time invariance, and stability of the following system

and show your answer by equations: 𝑦(𝑡) = 2𝑏𝑥(𝑡)cos (𝑡 + 𝑐)

(b) Check if the following systems are invertible or not and show your answer:

(i)𝑦(𝑡) = ln (𝑏𝑥(𝑡)) (ii) 𝑦(𝑡) = sin (𝑥(𝑡)) (iii)𝑦(𝑡) =𝑐𝑑𝑥(𝑡)

𝑑𝑡

If the system is invertible determine the inverse system

--------------------------------------------------- Solution--------------------------------------------

4

5

Problem (2):

(a) Find the convolution of )(tx and h(t) where:

𝑥(𝑡) = 2𝑎 𝑢(𝑡)𝑢 (𝑇

8− 𝑡) and ℎ(𝑡) = 2𝑎 𝑐𝑜𝑠 (

2𝜋 𝑡

𝑇) 𝑢(𝑡) 𝑢 (

𝑇

8− 𝑡)

(b) Consider the interconnection of the Linear Time-Invariant (LTI) systems shown in

the Figure and have impulse responses: ℎ1[𝑛] = 𝑏𝛿[𝑛 − 1] and

ℎ2[𝑛] = ℎ3[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:

𝑥[𝒏] =a 𝛿[𝑛 + 2] + 𝑏𝛿[𝑛 − 1]

--------------------------------------------------- Solution--------------------------------------------

+

+

6

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)

a𝑒−𝑡 a

a-

10

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Problem (5):

(a) Derive Fourier transform of the following signal: 𝑥(𝑡) = 2𝑐𝑒−b𝑡

2a u(t). Then plot the magnitude and phase responses.

(b) Find the inverse Fourier transform of the signal:

𝑋(𝑗𝜔) = 𝑗𝜔 𝑎𝑐 sin(𝜔) 𝑏 sin (2𝜔)

𝜔2

--------------------------------------------------- Solution-----------------------------------------------

(a)

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Problem (6):

(a) Use the duality property to find Fourier transform of: 𝑦(𝑡) =2𝑏𝑡

(1+𝑡2)2

(b) The signal 𝑥(𝑡) = 𝑐𝑠𝑖𝑛(𝑡𝑇1)

𝑡 is applied to the input of an ideal bandpass

filter with frequency response:

𝐻(𝑗𝜔) = {𝑏 𝜔𝑐1 ≤ |𝜔| ≤ 𝜔𝑐2

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

Where 𝜔𝑐1 = 𝑇1/2 and 𝜔𝑐2 = 𝑇1

(i) Plot the frequency spectrum of the output signal 𝑌(𝑗𝜔).

(ii) Find y(t) using the properties of Fourier transform.

--------------------------------------------------- Solution-----------------------------------------------

(a)

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Problem (7):

(a) A signal 𝑚(𝑡) = 2𝑏𝑐𝑜𝑠3(2𝜋𝑐103𝑡 + 0.3𝑏) is to be sampled. Find the

minimum sampling angular frequency required.

(b) The signal 𝑥(𝑡) =𝑏𝑠𝑖𝑛(𝜋 𝑡)

𝜋 𝑡 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 𝑐(t) = 𝑏𝑐𝑜𝑠(2𝜋𝑐𝑡) (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 𝑘 and for any range of 𝑛:

𝑥(𝑛 − 𝑘) = {𝑐(𝑛 − 𝑘)𝑒𝑛−𝑘 , 𝑛 ≥ 𝑘

0, 𝑛 < 𝑘

Problem (9): (MATLAB Problem)

Subplot the following signals in the same figure choose an appropriate range for 𝑛:

- 𝑥(𝑛) = b cos (0.1b𝜋𝑛 +𝜋

6)

- 𝑦(𝑛) = 2𝑛 sin (0.1b 𝜋𝑛 +𝜋

3)

-----------------------------------