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Problems
The sequence x[n] = (-1)' is obtained by sampling the continuous-time sinusoidal signal x(t) =
cos oot at 1-ms intervals, i.e.,
Determine three distinct possible values of wo.
Consider the system in Figure P16.2.
(a) Sketch X,(w) for -97 : co 5 9x for the following values of wo.
(i) o = 7r
(ii) wo = 27
(iii) wo = 3w
(iv) wo = 57
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(b) For which of the preceding values of wo is x,(t) identical?
In the system in Figure P16.3, x(t) is sampled with a periodic impulse train, and a reconstructed
signal x,(t) is obtained from the samples by lowpass filtering.
The sampling period T is 1 ms, and x(t) is a sinusoidal signal of the form x(t) cos(27rfot + 0).
For each of the following choices of fo and 0, determine x,(t).
(a) fo = 250 Hz, 0 = 7r/4
(b) fo = 750 Hz, 0 = -r/2
(c) fo = 500 Hz, 0 = r/2

Figure P16.4 gives a system in which the sampling signal is an impulse train with alternating sign.
The Fourier transform of the input signal is as indicated in the figure.

Consider the system in Figure P16.5-1.
Figures P16.5-2 and P16.5-3 contain several Fourier transforms of x(t) and x,(t). For each input
spectrum X(w) in Figure P16.5-2, identify the correct output spectrum X,(w) from Figure
P16.5-3.

Suppose we sample a sinusoidal signal and then process the resultant impulse train, as shown
in Figure P16.6-1.

The result of our processing is a value Q.As w changes, Q may change. Determine which of the
plots in Figures P16.6-2 and P16.6-3 are possible candidates for the variation of Q as a
function of w.

Optional Problems
The sampling theorem as we have derived it states that a signal x(t) must be sampled at a rate
greater than its bandwidth (or, equivalently, a rate greater than twice its highest frequency).
This implies that if x(t) has a spectrum as indicated in Figure P16.7-1, then x(t) must be
sampled at a rate greater than 2W2. Since the signal has most of its energy concentrated in a
narrow band, it seems reasonable to expect that a sampling rate lower than twice the highest
frequency could be used. A signal whose energy is concentrated in a frequency band is often
referred to as a bandpass signal.There are a variety of techniques for sampling such signals,
and these techniques are generally referred to as bandpasssampling
To examine the possibility of sampling a bandpass signal at a rate less than the total
bandwidth, consider the system shown in Figure P16.7-2. Assuming that wi > (W 2 - wi), find
the maximum value of T and the values of the constants A, W", and Wb such that x,(t) = x(t).

In Problem P16.7 we considered one procedure for bandpass sampling and reconstruction.
Another procedure when x(t) is real consists of using complex modulation followed by
sampling. The sampling system is shown in Figure P16.8-1.

With x(t) real and with X(w) nonzero only for wi < wI < W2 , the modulating frequency wo is
chosen as wo = 2(w1 + W2), and the lowpass filter H1 (w) has cutoff frequency 1(c2 -2
(a) For X(w) as shown in Figure P 16.8-2, sketch X,(w).
(b) Determine the maximum sampling period T such that x(t) is recoverable from xP(t).
(c) Determine a system to recover x(t) from x,(t).

Given the system in Figure P16.9-1 and the Fourier transforms in Figure P16.9-2, determine A
and find the maximum value of T in terms of W such that y(t) = x(t) if s(t) is the impulse train

Consider the system in Figure P16.10-1.
Given the Fourier transform of x,(t) in Figure P16.10-2, sketch the Fourier transform of two
different signals x(t) that could have generated x,(t)

(a) If X(w) = 0 for Io > W, find the maximum value of T, We, and A such that x,(t) = x(t).
(b) Let X 1 (w) = 0 for I > 2W and X 2(w) = 0 for Iw > W. Repeat part (a) for the following.
(i) x(t) = x1(t) * x 2(t)
(ii) x(t) = x1(t) + x 2(t)
(iii) x(t) = x 1(t)x 2(t)
(iv) x(t) = x 1 (lOt)

Solutions

From the convolution theorem
Hence, it is straightforward to find X,(x)

(ii) -7n -6n -5r For wo = 2r:

The signal x(t) = cos(wet + 0), where wo = 2wfo, can be written as
x(t) = lei'eiot + le -'0e -jWOt
and the spectrum of x(t) is given by
X(w) = rej'"(w - wo) + re ~j'0 (w + wo)
The spectrum of p(t) is given by

Therefore, the spectrum of x,(t) is
and the spectrum of X,(W) is given by

Since the input x,(t) cannot be distinguished for certain values of w,the output also should not
be distinguishable for certain values of w. Hence, Q(w) must be periodic in w. Therefore, Figure
P16.6-3 is a possible candidate, but Figure P16.6-2 is not.

The spectra of x, 2(t), where T = have generated x,(t): ir/W, given in Figures S16.10-1 and S16.10-
2, could

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