Signal Processing Assignment Help

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Signal Processing Assignment Help

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Problem

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P15.1 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).

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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

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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.

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(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

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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.

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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

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

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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.

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P15-6

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

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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.

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Solutions

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

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

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S15.2

By the modulation theorem,

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S15.3

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

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As an example, consider x[n] with Fourier transform S15.3-4.

as in Figure

Then, after multiplication by (-1)", the resulting signal has the Fourier transform given in Figure S15.3-5.

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

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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

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

S15.5

given as in

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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.

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

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and the system is linear.

S15-8

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

Therefore, the system is time-invariant.

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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,

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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 need

S15.9

S15-10

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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. (S15.91). The resulting signal r(t) has Fourier transform R(w) given by

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

S15.10

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

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