Question

Problem 4. Make a plot of the DTFT magnitude, $|S(\omega)|$, versus digital frequency $\omega$ at 201 equally-spaced frequencies spanning the interval $0 \le \omega \le 2\pi$. Superimpose on the same graph the corresponding 10-point DFT values plotted with dot-markers (see example graph in Fig. 1(a)). Make the same plot, but now plot the spectra versus the DFT index, that is, the normalized frequency variable, $k = Nw/(2\pi)$, which takes the integer values $k = 0, 1, ..., N - 1$, at the $N$ DFT frequencies $\omega_k$ (see example graph in Fig. 1(b)). Based on this plot and the fact that $s(n)$ is real-valued, can you explain the presence of only $\omega_1, \omega_3, \omega_5$ in the expression in (5)? Problem 5. Verify numerically, using the same set of frequencies $\omega$ of Problem 4, that $S(\omega)$ computed from (2) agrees with that computed via freqz. Moreover, verify numerically that $S_k$ computed by (3) agrees with that computed using the Matlab function fft.

          Problem 4. Make a plot of the DTFT magnitude, $|S(\omega)|$, versus digital frequency $\omega$ at 201
equally-spaced frequencies spanning the interval $0 \le \omega \le 2\pi$. Superimpose on the same graph the
corresponding 10-point DFT values plotted with dot-markers (see example graph in Fig. 1(a)).
Make the same plot, but now plot the spectra versus the DFT index, that is, the normalized
frequency variable, $k = Nw/(2\pi)$, which takes the integer values $k = 0, 1, ..., N - 1$, at the $N$
DFT frequencies $\omega_k$ (see example graph in Fig. 1(b)). Based on this plot and the fact that $s(n)$ is
real-valued, can you explain the presence of only $\omega_1, \omega_3, \omega_5$ in the expression in (5)?
Problem 5. Verify numerically, using the same set of frequencies $\omega$ of Problem 4, that $S(\omega)$
computed from (2) agrees with that computed via freqz.
Moreover, verify numerically that $S_k$ computed by (3) agrees with that computed using the
Matlab function fft.

Problem 4. Make a plot of the DTFT magnitude, |S(ω)|, versus digital frequency ω at 201
equally-spaced frequencies spanning the interval 0 ≤ω≤ 2π. Superimpose on the same graph the
corresponding 10-point DFT values plotted with dot-markers (see example graph in Fig. 1(a)).
Make the same plot, but now plot the spectra versus the DFT index, that is, the normalized
frequency variable, k = Nw/(2π), which takes the integer values k = 0, 1, ..., N - 1, at the N
DFT frequencies (see example graph in Fig. 1(b)). Based on this plot and the fact that s(n) is
real-valued, can you explain the presence of only ω1, ω3, ω5 in the expression in (5)?
Problem 5. Verify numerically, using the same set of frequencies ω of Problem 4, that S(ω)
computed from (2) agrees with that computed via freqz.
Moreover, verify numerically that Sk computed by (3) agrees with that computed using the
Matlab function fft.

Added by Dwayne C.

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