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?
2) Convolve tχ[0,2](t) with itself.
3) Prove that
(2)
4) 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
5) 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.
6) 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?
7) Find the Fourier transform of tχ[0,2](t) + U(t). Show all your work.
2
8) Find the fourier transform of 2 + δ(t− 3). Show all your work.
9) Consider a signal f(t) whose Fourier Transform is f(ω) = χ[−20,20](ω).
Let us modulate this signal with cos(2π30t).
...a) Draw the modulated signal in frequency domain. ...b) What would you do to demodulate this signal?
10) 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.
11) Convolve 2 0 5 3 1 -4 6 with itself.
12) 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.
