Description
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 χ[0,1](t) to the system as input and I receive e−2tU(t) as output. If I enter 3χ[4,5](t) − 7χ[8,9](t) as input, what will be the output?
- Convolve tχ[0,2](t) with itself.
- Prove that
(2)
- Let f(t) = tχ[0,2](t) + 2χ[2,3](t) + (5 −t)χ[3,5](t).
…a) Plot f(t)
…b) Plot 2f(2t + 1) + 1
- Convolve (t + 5)χ[−5,0](t) + (5 −t)χ[0,5](t) with
…a)
(3)
…b)
∞
X
δ(t− 8k)                                                             (4)
k=−∞
Plot your results.
- Consider a LTI system whose impulse response is e−5tU(t). If I enter χ[2,5](t) to this system as input, what will be its output?
- Find the Fourier transform of tχ[0,2](t) + U(t). Show all your work.
2
- Find the fourier transform of 2 + δ(t− 3). Show all your work.
- Consider a signal f(t) whose Fourier Transform is f(ω) = χ[−20,20](ω).
Let us modulate this signal with cos(2Ï€30t).
…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: ω = 2πf, 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.
