[Solved] CSE348 Homework1

you are only required to remember the formulas for the summation of geometric series and the complex representation of sin(x), ie, (1)

• Let us have an LTI system. I enter _[0,1](t) to the system as input and I receive e^2tU(t) as output. If I enter 3_[4,5](t) 7_[8,9](t) as input, what will be the output?
• Convolve t_[0,2](t) with itself.
• Prove that

(2)

• Let f(t) = t_[0,2](t) + 2_[2,3](t) + (5 t)_[3,5](t).

a) Plot f(t)

b) Plot 2f(2t + 1) + 1

• Convolve (t + 5)_[5,0](t) + (5 t)_[0,5](t) with

a)

(3)

b)

X

(t 8k) (4)

k=

Plot your results.

• Consider a LTI system whose impulse response is e^5tU(t). If I enter _[2,5](t) to this system as input, what will be its output?
• Find the Fourier transform of t_[0,2](t) + U(t). Show all your work.

2

• Find the fourier transform of 2 + (t 3). Show all your work.
• Consider a signal f(t) whose Fourier Transform is f() = _[20,20]().

Let us modulate this signal with cos(230t).

a) Draw the modulated signal in frequency domain. b) What would you do to demodulate this signal?

• Let us sample f(t) given in Question 9 with a) 5 Hz.

b) 30 Hz.

In both cases, draw the sampled signal in frequency domain. In which case(s) we can recover the original signal from the sampled signal? Note: = 2f, where f is frequency, measured in Hertz.

• Convolve 2 0 5 3 1 -4 6 with itself.
• Filter the signal

1 2 4 1 2 0 4 3

1 1 1 1

1 0 4 2

with the filter

1 0 1 0 2 0 1 0 1

Use periodic boundary conditions.