Digital signal processing by thomas j cavicchi solucionario

Lines = result from dimpulse; stems = results from analytical calculation.Unit sample response: num chosen for freqz.. Lines = result from dimpulse; stems = results from analytical calcu

Digital Signal Processing

Solutions Manual

Thomas J Cavicchi Grove City College

John Wiley & Sons, Inc.

© 2000

http://www elsolucionario.blogspot.com

LIBROS UNIVERISTARIOS

Y SOLUCIONARIOS DE MUCHOS DE ESTOS LIBROS

LOS SOLUCIONARIOS CONTIENEN TODOS LOS EJERCICIOS DEL LIBRO RESUELTOS Y EXPLICADOS

DE FORMA CLARA VISITANOS PARA

Digital Signal Processing

Solutions Manual Thomas J Cavicchi Table of Contents

Begins on page

Chapter 2 1

Chapter 3 50

Chapter 4 120

Chapter 5 220

Chapter 6 283

Chapter 7 408

Chapter 8 494

Chapter 9 582

Chapter 10 658

Note from author to instructors:

I will place corrections and/or improvements to these solutions on my Web page for this book.

To view these, you will need the following login and password:

password: kh7yb2q5 login: faculty

If you have corrections, comments, or suggestions, please email me at the address also shown on the main webpage for this book.

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200400

600Solid = exact expn for Y(exp(jw)); = DFT{zero-padded y(n) = nx(n)}

x(n) = [u(n)-u(n-30)]

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< yc(0) = 0 and yc`(0) = -4, which check.

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(b) Note that the first difference of the given x(n) is zero for n = 0, 1 for n

 [1, M-1], and zero otherwise, except for -(M-1) at n = M Thus the DFT of thefirst difference, by direct evaluation of the DFT sum (using geometric sumidentity) and subtracting off the n = 0 term and adding on the n = M term:

except the denominator is squared However, the result of part (a) multiplies

calculation or use of the property gives the same result for the DFT of the firstdifference of x(n) Matlab plots using "fft" and using the analytical resultindeed show that the result obtained by numerically computing the DFT sum are thesame as the analytical result in (*) that does not require computing a sum

O stems = fft result; * stems = analytical result.

Mag of DFT of first difference of discrete-time finite-dur ramp.

Al i asi ng of spectrum of boxcar (T = 1 s) for dt = 0.1 s.

norm al i zed radi an frequency (rad/sam pl e)

Anal og frequency W correspondi ng to sam pl es (rad/s)

Sam pl ed | FT{tri angl e}| : T = 1 sec, L = 4, N = 64, dW = 2pi (0.125 Hz)

by the sampling property of the Dirac delta function (as applied to

convolution with a shifted Dirac delta) Thus,

exist in practical reality However, we can mimic its effects by recalling

Thus, we could form a finite-valued, finite-width "impulse" train, whose

sampling circuit with electronic switching could approximate this time signal, which in turn could perfectly well be filtered by the analog

= sinc(Bt/)t), but the above results are valid for a general analog filter

From the convolution theorem and formula 20 (i)in Appendix 4A, any T-periodicfunction g(t) having nonzero dc value [so that G(k=0) is nonzero] multiplyingx(t) can be used for “sampling“ x(t), except that the notion of “samples” ismuddled if that function is not a narrow-width-pulse train (see, e.g., R E.Zeimer, W H Tranter, D R Fannin, “Signals and Systems: Continuous and

produces a replica of the spectrum of x(t) at intervals equal to the

fundamental frequency of g(t) the zero-centered replication and thus can thus be extracted by lowpass filtering x(t)g(t)

x(t) We thus do not really need the pulse-width extremely small unless we want toreally call it reconstruction from true samples alone [as opposed to

reconstruction of x(t) from segments of x(t)] The height of the pulses iseven less important, as it can of course be absorbed into the passband gain ofthe LPF The on/off-multiplication of the narrow pulse train can be easilyimplemented using electronic switching Note, however, that the zero-orderhold does CANNOT be used for ideal reconstruction, for it distorts the zero-centered (baseband) spectrum (see Fig 6.16b,c) The reason is that the ZOHamounts to a time-domain convolution {see (6.11)] rather than a time-domainmultiplication, so in the frequency domain we have a distorting multiplication

of the baseband spectrum

331

involves all orders (powers of t in the Taylor series expansion) Hence thecentral difference formula cannot be a true second differentiator for

characteristic that holds for most functions when )t is small

(d) The frequency response of this system is [see (9.34) - (9.35) for theproof],

evaluated at the input (digital) frequency

second derivative of the sine function is a scaled version of the sine

function

332

6.39) (a) Let z = x+jy, and of course assume )t > 0 Then with

s = (z-1)/(z)t) = F+jS,

we know F < 0 iff Re{(z-1)/z} < 0, or Re{[x-1+jy]/[x+jy]} < 0,

adding and subtracting ¼ to complete the square (and noting the denominator isalways $ 0),

This inequality describes a disk centered on {½,0} with radius ½ This disk

transforms into a stable H(z) Note, however, that the frequency response (jSaxis) is not at all retained (i.e., mapped to the unit circle), except

approximately so for very low frequencies (z near 1) Thus, compared with theBLT, this mapping is not recommended

or

y(n) - y(n-1) = a)t{x(n) - y(n)}

Taking the z-transform,

The process generalizes by using the cascade decomposition; each s is replaced

which is the exterior of a circle of radius 1/)t centered on {1/)t, 0} in the

right-half plane (unstable) and H(z) be stable This happens as long as thoseRHP poles are outside the circle of radius 1/)t centered on {1/)t, 0}

Contrast this with the result in part (a) that the left half of the s-planemaps to the interior of the circle of radius ½ centered on {½, 0} in the z-plane

6.40) (a) With s = F + jS = (z-1)/)t, F < 0 translates to

Re{z-1} < 0, or Re{z} < 1

The region Re{z} < 1 has no relation to the unit circle centered on the

into a stable H(z) Therefore, this transformation is usually to be avoided

in the discretization of an analog filter or other analog system [but see part(b)]

With y(n+1) - y(n) = )tx(n) or y(n+1) = y(n) + )tx(n), we have the usual

rectangular rule integration

(b) Solving for z, we have z = 1 + s)t, or

which is the interior of a circle of radius 1/)t centered on {-1/)t,0} Thus,

if all the poles of the analog system are within this circle, the forward

H(z)

6.41) The trapezoidal rule says

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Ti m e t (s)

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Si m ul ati on usi ng predi ctor-corrector; dt = 0.3 s

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Digital frequency f

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x(n) = triang Stems = |X(k)|; other (identical) curves are: |X(exp(j2pif))| via

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Stems = |XL(k)| via L-point FFT of L-repetitions-repeated h(n).

Solid = |HL(exp(j2pif))| using property, - = using direct sum calc of DTFT{hL(n)}

Digital frequency f

DFTF def, by analyt calc., by zero-padding, & by interp{DFT}; N = 16, M = 1024

x(n) = triang Stems = |X(k)|; other (identical) curves are: |X(exp (j2 pif))|... data-page="336">

6.39) (a) Let z = x+jy, and of course assume )t > Then with

s = (z-1)/(z)t) = F+jS,

we know F < iff Re{(z-1)/z} < 0, or Re{[x-1+jy]/[x+jy]} < 0,

adding... data-page="334">

by the sampling property of the Dirac delta function (as applied to

convolution with a shifted Dirac delta) Thus,

exist in practical reality However, we can mimic its effects by