0.3 Modelling corruption  (Page 6/11)

Calculate the Fourier transform of $\delta (t-t_0)$ from the definition. Now calculate it using the time shift property [link] . Are they the same?Hint: They had better be.

Use the definition of the IFT [link] to show that

Mimic the code in specdelta.m to find the spectrum of the discrete delta function when:

1. The delta does not occur at the start of x . Try x[10]=1 , x[100]=1 , and x[110]=1 . How do the spectra differ? Can you use the time shift property [link] to explain what you see?
2. The delta changes magnitude x . Try x[1]=10 , x[10]=3 , and x[110]= 0.1 . How do the spectra differ? Can you use the linearity property [link] to explain what you see?

Mimic the code in specdelta.m to find the magnitude spectrum of the discrete delta function when:

1. The delta does not occur at the start of x . Try x[10]=1 , x[100]=1 , and x[110]=1 . How do the spectra differ? Can you use the time shift property [link] to explain what you see?
2. The delta changes magnitude x . Try x[1]=10 , x[10]=3 , and x[110]= 0.1 . How do the spectra differ? Can you use the linearity property [link] to explain what you see?

Modify the code in specdelta.m to find the phase spectrum of a signal that consists of a delta function when the nonzero termis located at the start ( x(1)=1 ), the middle ( x(100)=1 ) and at the end ( x(200)=1 ).

Mimic the code in specdelta.m to find the spectrum of a train of equally spaced pulses. For instance, x(1:20:end)=1 spaces the pulses 20 samples apart, and x(1:25:end)=1 places the pulses 25 samples apart.

1. Can you predict how far apart the resulting pulses in the spectrum will be?
2. Show that
   $$\sum _{k=-\infty }^{\infty }\delta (t-k{T}_s)\iff \frac{1}{T_s}\sum _{n=-\infty }^{\infty }\delta (f-n{f}_s),$$
   where $f_s=1/T_s$ . Hint: Let $w\left(t\right)=1$ in [link] .
3. Now can you predict how far apart the pulses in the spectrum are? Your answer should be in terms of how farapart the pulses are in the time signal.