Signals and Systems Assignment Help

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Discrete-Time Sampling
Recommended Problems
P19.1
Consider Figures P19.1-1 and P19.1-2, and determine X(Q), P(Q), x,[n), and
X,(Q).
P19.2
x[n] has a transform X(Q). Determine in terms of X(Q) the transforms
of the signals in parts (a) and (b).
(a) x[n/2]1, n even, (a)x[n]=0,) n odd
(b) (b) xd[n] = x[2n], i.e., xd[n] is x[n] decimated.
Problem
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P19-2
(c) If X(Q) is as given in Figure P19.2, sketch X,(Q) and Xd(Q) for parts (a)
and (b).
P19.3
Consider the system in Figure P19.3-1
(a) If p[n] is given by Figure P19.3-2 sketch P(Q) for N =1, 2, and
L, an arbitrary integer.

(b) For each of the discrete-time spectra in Figures P19.3-3 and P19.3-4, determine
the maximum sampling period N such that x[n] is reconstructible from its samples
x,[n] using an ideal lowpass filter
In each case, specify the associated cutoff frequencies for the lowpass
filter.

P19.4
Suppose the signal x(t) is processed as shown in Figure P19.4-1.
(a) The system in Figure P19.4-1 can be replaced by the one in Figure P19.4-2. Find
T1 .
(b) Let X(w) be given as in Figure P19.4-3. Find X(Q) and Y(Q).

P19.5
As discussed in Section 8.7 and illustrated in Figure 8.40 of the text as well as in
Figure P19.5-1 below, the procedure for interpolation or upsampling by an integer
factor N can be thought of as the cascade of two operations. The first system,
system A, corresponds to inserting (N - 1) zero sequence values between each
sequence value of x[n], so that
For exact bandlimited interpolation, H(Q) is an ideal lowpass filter.
(a) Determine whether system A is linear.
(b) Determine whether system A is time-invariant.
(c) For Xd(2) as sketched in Figure P19.5-2, with N = 3, sketch X(Q).
(d) For N = 3, Xd(Q) as in Figure P19.5-2, and H(Q) appropriately chosen for exact
bandlimited interpolation, sketch X(Q).

P19.6
Consider the sampling systems in Figure P19.6-
1.
Let x(t) andX(w) be given as in Figure P19.6-2.

Let y(t) and Y(w) be given as in Figure P19.6-3.
(a) Draw x,(t) and Y,(w).
(b) Find X,(w) and y,(t).
(c) Is y,(t) periodic? Does Y,(w) reflect this property? Optional Problems
P19.7 Consider a discrete-time sequence x[n] from which we form two new
sequences, x,[n] and xd[n], where x,[n] corresponds to sampling x[n] with
sampling period 2 and xd[n] corresponds to decimatingx[n] by a factor of 2, so
that
(a) If x[n] is as illustrated in Figure P19.7-1, sketch the sequences x,[n] and xd[n].

(b) If X(Q) is as shown in Figure P19.7-2, sketch X,(Q) and Xd(Q).
Consider the system in Figure P19.8-1, where X(Q) is as shown in Figure P19.8-
2.

There is a range of values for N such that, with an appropriate choice for H(Q), y[n] will
equal x[n]. For each allowable positive integer value of N,
(a) Draw X,(Q).
(b) Find an appropriate H(Q) such that y[n] = x[n].
P19.9 Consider the system with input x[n] and output y[n] related by
(a) For the sequence x[n] in Figure P19.9, sketch y[n].
(b) Express the system as a combination of filtering and decimation. 6
P19.10
Consider the system in Figure P19.10, where
Find a constraint on h[n] such that y[kN] = x[k], for all k.

x[n] is given by
x[n] = (-1)" = e'"
Hence, the Fourier transform of x[n] is
Now p[n] can be written as
Hence, its Fourier transform is given by
It is clear that x,[n] = p[n]. Hence
X,(Q) = P(Q)
S19.2
(a) x,[n] is x[n] "stretched" by interspersing with zeros, as indicated
in Figure S19.2-1.
Solutions

'X(Q/2) is indicated in Figure S19.2-3. Therefore, is as shown in Figure S19.2-4.
(a) For N = 1, p[n] = 1. Hence
as shown in Figure S19.3-1.

For N = 2,
Hence
shown in Figure S19.3-2.
ForN =L
Hence
shown in Figure S19.3-3.

(b) X,(Q), the spectrum of x,[n], is proportional to the periodic convolution of P(Q) and
X(Q). Consequently, with P(Q) as indicated in Figure S19.3-3 and X(Q) as indicated in
Figure S19.3-4, X,(Q) is shown in Figure S19.3-5. In order that x[n] be reconstructible
from x,[n] using an ideal lowpass filter, aliasing must be avoided, which requires that

(i) Qm = 37r/10. Therefore, to avoid aliasing,
Since N must be an integer, we require that N s 3. For N = 3, the cutoff frequency of the
lowpass filter must be greater than 3r/10 and less than
(ii) Q, = 37r/ 5 . To avoid aliasing,
Since N must be a positive integer, this requires that N = 1, i.e., x[n] cannot be
sampled.
S19.4
(a) The sampling period Ti is 3 ms for the system in Figure P19.4-2 to be
equivalent to the one in Figure P19.4-1.
(b) (b) X(Q) is sketched in Figure S19.4-1.

From the result of part (a), Y(Q) is as shown in Figure S19.4-2.
S19.5

So system A is linear. (b) Take xd,[n] as shown in Figure S19.5-1, with
N = 4.
Then x,,[n] is as shown in Figure S19.5-2.
Take Xd2[n] = Xd,[n + 1]. Then x,,[n] is as shown in Figure S19.5-3.
Hence, system A is not time-invariant.

(d) X(Q) is as shown in Figure S19.5-5 for exact bandlimited interpolation.
(a) x,(t) is sketched in Figure S19.6-1, and Y,(w) is sketched in Figure
S19.6-2.

(b) X,(w) is sketched in Figure S19.6-3, and y,(t) is sketched in Figure S19.6-4.

(c) Yes, y,(t) is periodic and this is reflected in Y,(w), which contains
impulses.
Solutions to Optional Problems
S19.7
This is sketched in Figure S19.7-1.
Similarly, xd[n] = x[2n], as shown in Figure S19.7-2.
(b) X,(Q) is obtained as follows:
and

which are shown in Figures S19.7-3 and S19.7-4. See Problem
P19.2(b).
S19.8 (a) We know that the Fourier transform of p[n] is given by
Aliasing will be just avoided when the sampled spectra will look as shown in
Figure S19.8-1.

Hence, we require that
Consequently, aliasing is avoided if 1 S N 3. X,(Q) for N = 1, 2, and 3 are shown in
Figure S19.8-2.

(b) An appropriate H(Q) is shown in Figure S19.8-3.
Hence, y[n] can be sketched as in Figure S19.9.

(b) If
z[n] = [x[nj + x[n + 11 + x[n + 2]], for all n
and
y[n] = z[3n],
we have expressed the processing as a combination of filtering and decimati
If h[O] = 1 and h[n] = 0 for n = kN, k # 0, it is easy to see that the samples
xo[n] that came from x[n] will be unaffected. Hence,
y[kN] = x[k], for all k
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