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Unformatted text preview: 508 Chapter 7 The Discrete Fourier Transform: its Properties and Applications 7.29 Frequency—domain sampling The signal x(n) = al’”, —1 < a < 1 ha transform
1 — a2 X =—————
(w) 1—2acosa)+a2 J (a) Plot X (m) for 0 g a) 5 271"; a = 0.8. Reconstruct and plot X (m) from
X(27rk/N), 0 5 k 5 N — 1 for (b) N = 20
(c) N = 100 ((1) Compare the spectra obtained in parts (b) and (c) with the origin:
X (co) and explain the differences. (e) Illustrate the time—domain aliasing when N = 20. .equency analysis of amplitudemodulated discretetime signal The d
1gna1 x(n) = cos 27rf1n + cos 2nf2n where f1 = ﬁ and f2 = %, modulates the amplitude of the carrier
xc(n) = cos 27:an
Where fa 2 %)§. The resulting amplitudemodulated signal is
xam(n) = x(n) cos 271an (a) Sketch the signals x(n), xc(n), and xam(n), 0 5 n g 255. (b) Compute and sketch the 128—point DFT of the signal xam (n) , 0 S n
(c) Compute and sketch the 128—point DFT of the signal xam (n), 0 5 n
((1) Compute and sketch the 256p0int DFT of the signal xam(n), 0 5‘ n (e) Explain the results obtained in parts (b) through (d), by deriving the
the amplitudemodulated signal and comparing it with the experim‘r 7.31 The sawtooth waveform in Fig. P731 can be expressed in the form ‘1
series as ' 2 1 1 1
x(t)=— sinnt——sin2m+—sin371t——sin47rt~
7T 2 3 4 (3) Determine the Fourier series coefﬁcients ck. (b) Use an N point subroutine to generate samples of this signal in thei
using the ﬁrst six terms of the expansion for N = 64 and N = 11
signal x(t) and the samples generated, and comment on the results Chapter 10 Problems 737 it By computing the frequency response H ((0), show that the discrete—tirne system you = 1601) — X(n  1) is a good approximation of a differentiator at low frequencies. x(n) = A cos(a)on + 0) the window method with a Hamming window to design a 21tap differentiator I
I
I
I
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I
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I 7! he the bilinear transformation to convert the analog ﬁlter with system function s+0.1 H = ——
(s) (s + 0.1)2 + 9
:o a digital IIR ﬁlter. Select T = 0.1 and compare the location of the zeros in H (z) th the locations of the zeros obtained by applying the impulse invariance method
the conversion of H (s). mvert the analog bandpass ﬁlter designed in Example 10.4.1 into a digital ﬁlter by
ms of the bilinear transformation. Thereby derive the digital ﬁlter characteristic
tained in Example 10.4.2 by the alternative approach and verify that the bilinear
nsformation applied to the analog ﬁlter results in the same digital bandpass ﬁlter. lideal analog integrator is described by the system function H, (s) = 1/s. A gital integrator with system function H (2) can be obtained by use of the bilinear
nsformation. That is, T 1 + z‘1 .
H“) = 5 1 _ 2—1 E Ha(s)s=(2/T)(1—z‘1)/(1+z_1) ‘ Write the difference equation for the digital integrator relating the input x (n) to
the output y(n). Roughly sketch the magnitude Ha (j S2)[ and phase 9 ($2) of the analog integrator. Chapter 10 Problems 741 9) Compare the complexity of implementing the FIR ﬁlter in part (3) versus the elliptic ﬁlter obtained in part (d). Assume that the FIR ﬁlter is implemented in
the direct form and the elliptic ﬁlter is implemented as a cascade of two
ﬁlters. Use storage requirements and the number of
point in the comparison of complexity. —pole
multiplications per output he impulse response of an analog ﬁlter is shown in Fig. P1020. gure P1020 ) Let h(n) = ha (nT), where T = 1, be the impulse response of a discretetime ﬁlter. Determine the system function H (z) and the frequency response H (co)
for this FIR ﬁlter. 3 Sketch (roughly) [H (co) and compare this frequency response characteristic with
[H4 0 ml  paring some of the characteristics of analog and
giital implementations of the singlepole low—pass analog system Ha(s) = a hat : ~01):
s+oz© 0 e What is the gain at dc? At what radian frequency is the analog frequency re
sponse 3 dB down from its dc value? At what frequency is the analog frequency response zero? At what time has the analog impulse response decayed to 1 / e of
its initial value? Give the digital system function H (z) for the impulse—invariant design for this
ﬁlter. What is the gain at dc? Give an expression for the 3dB radian frequency.
At what (real—valued) frequency is the response zero? How many samples are there in the unit sample timedomain response before it has decayed to 1/6 of
its initial value? “Prewarp” the parameter a and perform the bilinear transformation to obtain
‘the digital system function H (z) from the analog design. What is the gain at
G‘dc? At what (realvalued) frequency is the response zero? Give an expression for the 3dB radian frequency. How many samples are there in the unit sample
{timedomain response before it has decayed to 1 /e of its initial value? 744 Chapter 10 Design of Digital Filters 10.24 Figure P1024 shows a digital ﬁlter designed using the frequency3am} Figure P10.24 (a) Sketch a zplane pole—zero plot for this ﬁlter.
(b) Is the ﬁlter lowpass, highpass, or bandpass?
(c) Determine the magnitude response H(a)) at the frequencies a k=0,1,2,3,4,5,6.
(d) Use the results of part (c) to sketch the magnitude response for (
conﬁrm your answer to part (b).
@ An analog signal of the form xa (t) = a(t) cos 20007” is bandlimitei 900 g F g 1100 Hz. It is used as an input to the system shown in Fig, A/D x(n) w(n) v01) D/A
X403) converter H(w) COHVGIIC'I i i R = 2500 cos (0.8 7m) =1
xTx Figure P10.25 504 Chapter 7 The Discrete Fourier Transform: its Properties and Applications
7.12 Consider a ﬁniteduration sequence 1601) = {(T), 1, 2, 3, 4} (3) Sketch the sequence S(n) with sixpoint DFT
S(k)=W2*X(k), k=0,1,...,6 (b) Determine the sequence y(n) with sixpoint DFT Y (k) = 9th (k)[.
(c) Determine the sequence v(n) with sixpoint DFT V(k) = 3X(k)l.‘ 7.13 Let xp (n) be a periodic sequence with fundamental period N . Consider t1
DFTs: mm 21:1 X1 (k) xpm) 333—7} X300 (3) What is the relationship between X1 (k) and X3 (k) ?
(b) Verify the result in part (a) using the sequence xp(n)={1,2,1,%,1,2,1,2} 7.14 Consider the sequences X101) = {?,1,2.3,4}, X201) = {9,1,0,0,0}. 80!) = {1,0,0 and their ﬁvepoint DFTs. (a) Determine a sequence y(n) so that Y (k) = X1(k)X2(k). (b) Is there a sequence x3 (n) such that S (k) = X; (k) X3 (k)?
7.15 Consider a causal LTI system with system function 1 W) = m The output y(n) of the system is known for 0 5 n 5 63. Assumingt
available, can you develop a 64—point DFT method to recover the seq‘ 0 5 n 5 63? Can you recover all values of x(n) in this interval?
.he impulse response of an LTI system is given by h(n) = (30;) — 3380
determine the impulse response g(n) of the inverse system, an enginee the N point DFT H (k), N 2 4kg, of h(n) and then deﬁnes g(n) as the i]
of G(k) = 1/H(k), k = 0, 1, 2, . . . , N — 1. Determine g(n) and the i
h(n) x g(n), and comment on Whether the system with impulse response
inverse of the system with impulse response [101). ...
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 Winter '09
 Eltanany

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