$4\mathrm{.}55$ repeat Example $4\mathrm{.}7,$ using zero padding and the MATLAB commands fft and fftshift to sample and plot Y$($en$)$ at $512$ points on $-\backslash$pi $<mi\; id="EditorElement\_954665">\backslash </mi>$Omega pi for each case.

$4\mathrm{.}55$ Repeat Example $4\mathrm{.}7,$ using zero padding and the

MATLAB commands $fft$ and fftshift to

sample and plot $Y(e^i)$ at $512$ points on

for each case. Example $4\mathrm{.}7$ Application: Windowing Data It is common in data$-$processing ap$-$

plications to have access only to a portion of a data record. In this example, we use the

multiplication property to analyze the effect of truncating a signal on the DTFT$.$ Consid$-$

er the signal

$x[n]=cos(\frac{7}{16}n)+cos(\frac{9}{16}n).$

Evaluate the effect of computing the DTFT$,$ using only the $2M+1$ values $x[n],|n|M.$

Solution: The DTFT of $x[n]$ is obtained from the FS coefficients of $x[n]$ and Eq$.(4\mathrm{.}8)$ as

$x(e^i)=(+\frac{9}{16})+(+\frac{7}{16})+(-\frac{7}{16})+(-\frac{9}{16}),$

which consists of impulses at $+-7\frac{}{16}$ and $+-9\frac{}{16}.$ Now define a signal $y[n]=x[n]w[n],$

where

$w[n]=\{\begin{array}{cc}1,& |n|M\\ 0,& |n|>M\end{array}$

Multiplication of $x[n]$ by $w[n]$ is termed windowing, since it simulates viewing $x[n]$ through

a window. The window $w[n]$ selects the $2M+1$ values of $x[n]$ centered on $n=0.$ We

compare the DTFTs of $y[n]=x[n]w[n]$ and $x[n]$ to establish the effect of windowing.

The discrete$-$time multiplication property Eq$.(4\mathrm{.}13)$ implies that

$Y(e^j)=\frac{1}{2}\{W(e^{i(+9\frac{}{16})})+W(e^{i(+7\frac{}{16})})+W(e^{j(-7\frac{}{16})})+W(e^{i(-916)})\},$

where the DTFT of the window $w[n]$ is given by

$W(e^j)=\frac{sin(\frac{2M+1}{2})}{sin(\frac{}{2})}.$

We see that windowing introduces replicas of $W(e^j)$ centered at the frequencies $7\frac{}{16}$ and

$9\frac{}{16},$ instead of the impulses that are present in $x(e^j).$ We may view this state of affairs

as a smearing or broadening of the original impulses: The energy in $Y(e^{in})$ is now smeared

over a band centered on the frequencies of the cosines. The extent of the smearing depends

on the width of the mainlobe of $W(e^j),$ which is given by $4\frac{}{2M+1}.($See Figure $3\mathrm{.}30.)$

Figure $4\mathrm{.}15($a$)-($c$)$ depicts $Y(e^{jn})$ for several decreasing values of $M.$ If $M$ is large

enough so that the width of the mainlobe of $W(e^i)$ is small relative to the separation be$-$

tween the frequencies $7\frac{}{16}$ and $9\frac{}{16},$ then $Y(e^j)$ is a fairly good approximation to

$x(e^i).$ This case is depicted in Fig. $4\mathrm{.}15($a$),$ using $M=80.$ However, as $M$ decreases and

the mainlobe width becomes about the same as the separation between frequencies $7\frac{}{16}$

and $9\frac{}{16},$ the peaks associated with each shifted version of $W(e^{jI})$ begin to overlap and

merge into a single peak. This merging is illustrated in Fig. $4\mathrm{.}15($b$)$ and $($c$)$ by using the val$-$

ues $M=12$ and $M=8,$ respectively.