In this problem we introduce the concept of windowing, which is of great
importance both in the design of LTI systerns and in the spectral analysis of
signals. Windowing is the operation of taking a signal \(x[n]\) and multiplying
it by a finite-duration window. \(\operatorname{signal} w[n] .\) That is
$$p[n]=x[n] w[n]$$ Note that \(p[n]\) is also of finite duration. The importance
of windowing in spectral analysis stems from the fact that in numerous
applications one wishes to compute the Fousier transform of a signal that has
been measured Since in practice we can measure a signal \(x[n]\) only over a
finte time interval (the time window), the actual signal available for
spectral analysis is $$p[n]=\left\\{\begin{array}{ll}
x[n], & -M \leq n \leq M \\
0, & \text { otherwise }
\end{array}\right.$$ where \(-M \leq n \leq M\) is the time window. Thus,
\\[
p[n]=x[n] w[n]
\\] where \(w[n]\) is the rectangular window; that is,
\\[
w[n]=\left\\{\begin{array}{ll}
1, & -M \leq n \leq M \\
0, & \text { otherwise }
\end{array}\right.
\\] Windowing also plays a role in LTI system design. Specifically, for a
variety of reasons (such as the potential utility of the FFT algorithm; see
Problem P5.54), it is often advantageous to design a system that has an
impulse response of finite duration to achieve some desired signal-processing
objective. That is, we often begin with a desired frequency response
\(H\left(e^{j \omega}\right)\) whose inverse transform \(h[n]\) is an impulse
response of infinite (or at least excessively long) duration. What is required
then is the construction of an impulse response \(g[n]\) of finite duration
whose transform \(G\left(e^{j \omega}\right)\) adequately approximates
\(H\left(e^{j 00}\right),\) One general approach to choosing \(g[n]\) is to find a
window function \(w[n]\) such that the transform of \(h[n] w[n]\) meets the
desired specifications for \(G\left(e^{j \omega}\right)\)
Clearly, the windowing of a signal has an effect on the resulting spectrum. In
this problem, we illustrate that effect.
(a) To gain some understanding of the effect of windowing, consider windowing
the signal $$x[n]=\sum_{k=-\infty}^{\infty} \delta[n-k]$$ using the
rectangular window signal given in eq. (PS.55-1).
(i) What is \(X\left(e^{j \omega}\right) ?\)
(ii) Sketch the tansform of \(p[n]=x\\{n] w[n]\) when \(M=1\)
(iii) Do the same for \(M=10\)
(b) Next, consider a signal \(x[n]\) whose Fourier transform is specified by
$$X\left(e^{j \omega}\right)=\left\\{\begin{array}{ll}
1, & |\omega|<\pi / 4 \\
0, & \pi / 4<|\omega| \leq \pi
\end{array}\right.$$ Let \(p[n]=x[n] w[n],\) where \(w[n]\) is the rectangular
window of eq. \((P 5.55-1) .\) Roughly sketch \(P\left(e^{j \omega}\right)\) for
\(M=4,8,\) and 16
(c) One of the problems with the use of a rectangular window is that it
intrioduces ripples in the transforn \(P\left(e^{\prime \omega}\right)\). (This
is in fact directly related to the Gibbs phenomenon.) For that reason, a
variety of other window signals have been developed. These signals are
tapered; that is, they go from 0 to 1 more gradually than the abrupt
transition of the rectangular window, The result is a reduction in the
amplitude of the ripples in \(P\left(e^{j \omega}\right)\) at the expense of
adding a bit of distortion in terms of further smoothing of \(X\left(e^{\prime
\omega}\right)\)
To illustrate the points just made, consider the signal \(x[n]\) described in
part
(b), and let \(p[n]=x[n] w[n],\) where \(w[n]\) is the triangular or Bartlet
window:
that is, $$w[n]=\left\\{\begin{array}{ll}
1-\frac{|n|}{n+1}, & -M \leq n \leq M \\
0, & \text { otherwise }
\end{array}\right.$$ Roughly sketch the Fourier transform of \(p[n]=x[n] w[n]\)
for \(M=4,8,\) and
16\. [Hint: Note that the triangular signal can be obtained as a convolution
of a rectangular signal with itself. This fact Jeads to a convenient
expression for \(W\left(e^{J \omega}\right)\) (d) Let \(p[n]=x[n] w[n],\) where
\(w[n]\) is a raised cosine signal known as the \(H a n\) ning wradow: i.e.,
$$w[n]=\left\\{\begin{array}{ll}
\frac{1}{2}[1+\cos (\pi n / M)], & -M \leq n \leq M \\
0, & \text { otherwise }
\end{array}\right.$$ Roughly skecch \(P\left(e^{j \omega}\right)\) for \(M=4,8,\)
and 16
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