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In the system shown in Figure P15.1-1, x(t) is used to modulate an impulse train
carrier. The signal x,(t) then corresponds to an impulse train of samples of x(t).
Under appropriate conditions, x(t) can be recovered from x,(t) with an ideal lowpass
filter.
ForX(w) and H(w) as indicated in Figure P15.1-2, sketch X,(w) and X,(W). Indicate
specifically whether in this case x,(t) is equal to (or proportional to) x(t).
P15.1
Problem Matlab Assignment Help

P15.2
Consider the discrete-time modulation system in Figure P15.2-1. Let X(Q) be given
as in Figure P15.2-2. Sketch

P15.3
The system in Figure P15.3 is equivalent to a linear, time-invariant system with
frequency response Determine and sketch
P15.4
A discrete-time pulse amplitude modulation system is shown in Figure P15.4,
where p[n] and X(Q) are as indicated.

(a) Sketch P(Q) and Y(Q).
(b) Describe a system to recover x[n] from y[n].
(c) Discuss how this system could be used to time-division-multiplex two signals

P15.5
In the system in Figure P15.5, s(t) is a rectangular pulse train as indicated.
Determine H(w) so that y(t) = x(t), assuming that no aliasing has occurred.

Optional Problems
Consider the discrete-time system shown in Figure P15.6. The input sequence x[n]
is multiplied by and the product is taken as the input to an LTI system. The final
output y[n] is then obtained as the product of the output of the LTI system multiplied
by
P15.7

Determine the maximum sampling interval T such that w(t) is recoverable from
w,(t) through the use of an ideal lowpass filter.

P15.8
A discrete-time filter bank is to be implemented by using a basic lowpass filter and
appropriate complex exponential amplitude modulation as indicated in Figure P15.8-1.
(a) With H(Q) an ideal lowpass filter, as shown in Figure P15.8-2, the ith channel of the
filter bank is to be equivalent to a bandpass filter with frequency response shown in
Figure P15.8-2. Determine the values of ai and f#i to accomplish this.

P15-6
P15.9
Consider the modulation system in Figure P15.9.

Suppose we know that X(w) is bandlimited to + wc and that s(t) is an arbitraryperiodic
function with period T.
(a) Draw a possible Fourier transform of s(t). Consider the case when S(W)I=o is zero
and the case when it is not zero.
(b) What is the range of T such that Y(o) will have regions equal to zero?
(c) For a typical value of T found in part (b), determine how to recover x(t) from y(t).
P15.10
Consider the modulation system in Figure P15.10-1.

Solutions
S15.1 Recall that the Fourier transform of a train of impulses p(t) is P(w), as shown in
Figure S15.1-1.

Since Xr(w) = X,(xw)H(w), it is as indicated in Figure S15.1-3.

S15.2
By the modulation theorem,

From the lecture we know that the system in Figure S15.3-1 is equivalent to a
filter with response centered at

As an example, consider x[n] with Fourier transform as in Figure S15.3-4.
Then, after multiplication by (-1)", the resulting signal has the Fourier transform
given in Figure S15.3-5.

Finally, multiplying by (- 1)' again yields the spectrum in Figure S15.3-7.

Thus, the spectrum of y[n] is given by the sum of the spectrum in Figure S15.3-8
and as shown in Figure S15.3-8.
S15.4

S15-6
(b) To recover x[n] from y[n], we can filter y[n] with given as in Figure S15.4-3.
S15.5

We note that s(t) is a periodic signal. Therefore, S(w) is composed of impulses
centered at /T for integer k. The impulse at w = 0 has area given by
, where ao is the zeroth Fourier series coefficient of s(t):
Thus, S(w) is as shown in Figure S15.5-1.

Solutions to Optional Problems
S15.6
(a) Consider the labeling of the system in Figure S15.6.

and the system is linear.
S15-8

and the system is not time-invariant.
(b) From part (a),
Therefore, the system is time-invariant.

S15.7 In general, w(t) is recoverable from w,(t) if W,(w) contains repeated
versions of W(w) that do not overlap, i.e., that have no aliasing, as shown in
Figure S15.7.
Thus, since the length of a convolution of two signals is the sum of the individual
lengths,

From the preceding observations,
S15.8
(b) Consider i = 0, 1. Then the corresponding filters are as given in Figure
S15.8.

For no overlap and complete coverage of the frequency band, we need
S15.9
S15-10

Of course, other impulses may also be zero.
(b) Y(w) will be equal to a sum of the shifted and scaled versions of X(w).
Specifically,
where an is the nth Fourier series coefficient of one period of s(t). For some region
Y(w) to be zero, successive terms in the sum in eq. (S15.9-1) cannot overlap. Thus,
the maximum T is such that
(c) In general, we need to find some n such that an # 0. Then we use an ideal real
bandpass filter to isolate the nth term of the sum in eq. (S15.9-1). The resulting signal
r(t) has Fourier transform R(w) given by

(remember the effect of modulating by a cosine signal). Suppose we multiply r(t) by
S15.10

(c) Two possible choices are given in Figures S15.10-3 and S15.10-4.

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