- a) Consider a time domain signal, x=[1 1 1 1];
Construct a plot of the magnitude of the Discrete
Time Fourier Series (DTFS) of this signal, x.
This can be done using the DFT (FFT) function in Octave:
x=[1 1 1 1]; plot(abs(fft(x)));

Explain this plot. - b) Consider a discrete time signal: x[n]=1, 0<=n<=3; x[n]=0, otherwise. Determine the Discrete Time Fourier Transform (DTFT) for this signal. Sketch a graph of this function.
- c) Consider a discrete time periodic signal, constructed by periodizing
the above signal: y[n]=sum(x[n-lN]), for integer l, and integer period N.
Suppose N=1000. Construct a plot of the magnitude of the Discrete
Time Fourier Series (DTFS). This can be done using the DFT (FFT)
function in Octave:
y=[1 1 1 1 zeros(1,N-4)]; plot(abs(fft(y)));

How does this plot relate to the plots in parts (a) and (b). - d) Repeat part (c) above, with N=8:
y=[1 1 1 1 zeros(1,4)]; plot(abs(fft(y)));

How does this plot relate to the plots in parts (a), (b), and (c).

- a) find the FT [F(w)] of f(t) = exp(-(t^2)/Z) where Z = 4 hint multiply f(t) by exp(-w^2)*exp(w^2) = 1 plot this function.[w:rad/s][t:s] %%%%ans = F(w) = 2*exp(-w^2) b) f[n] = f(Ts*n) where n is an integer and Ts is a constant. write an equation for the DTFT of f[n] in terms of F(w); c) g(t) = f(t-50) for 0<=t<=100 write a progam to sample g(t) in the range of 0<=t<=100 with the sampling rate Fs = 3 Hz to get gs[n] calculate the DFT of this vector. Gs[k] = DFT(gs[n]) To do this use the octave command Gs = fft(gs) plot Gs (the magnitude of Gs). Explain (in words) how F(w) and Gs[k] and the DTFT of f[n} are related. be sure to address the complex nature Gs and include concepts of aliasing in time and frequency in your discussion. try varying the function f(t) by varying Z (try values ranging from 0.1 to 100). what do you observe? d)Zero Padding in time: write a script to generate: gszt[n] = gs[n] 0<=n<=300; 0 for 300 < n <= 900 compare the magnitude of Gszt[k] = fft(gsz) to that of Gs[k]. what is the effect of zero paddingin time? f) Zero Padding in Frequency: write a script to generate: Gszf[n] = Gs[n} for 0<=n<=150; 0 for 150 < n <=750; Gs[n-600] for 750