CHAPTER 12 Applications of Discrete-Time Signals and Systems
12.5 What Have We Accomplished? Where Do We Go from Here?
In this chapter we have seen how the theoretical results presented in the third part of the book relate to digital signal processing, digital control, and digital communications. The Fast Fourier Transform made possible the establishment and significant growth of digital signal processing as a technical area. The next step for you could be to get into more depth in the theory and applications of digital signal processing, preferably including some theory of random variables and processes, toward statis- tical signal processing, speech, and image processing. We have shown you also the connection of the discrete-time signals and systems with digital control and communications. Deeper understanding of these areas would be an interesting next step. You have come a long way, but there is more to learn.
A P P E N D I X
U s e f u l F o r m u l a s
Trigonometric Relations Reciprocal
csc(θ)= 1 sin(θ) sec(θ)= 1
cos(θ) cot(θ)= 1
tan(θ) Pythagorean Identity
sin2(θ)+cos2(θ)=1 Sum and Difference of Angles
sin(θ±φ)=sin(θ)cos(φ)±cos(θ)sin(φ) sin(2θ)=2 sin(θ)cos(θ)
cos(θ±φ)=cos(θ)cos(φ)∓sin(θ)sin(φ) cos(2θ)=cos2(θ)−sin2(θ)
Multiple Angle
sin(nθ)=2 sin((n−1)θ)cos(θ)−sin((n−2)θ) cos(nθ)=2 cos((n−1)θ)cos(θ)−cos((n−2)θ)
Signals and Systems Using MATLAB®. DOI: 10.1016/B978-0-12-374716-7.00017-x
c2011, Elsevier Inc. All rights reserved. 743
Products
sin(θ)sin(φ)= 1
2[cos(θ−φ)−cos(θ+φ)]
cos(θ)cos(φ)=1
2[cos(θ−φ)+cos(θ+φ)]
sin(θ)cos(φ)= 1
2[sin(θ+φ)+sin(θ−φ)]
cos(θ)sin(φ)= 1
2[sin(θ+φ)−sin(θ−φ)]
Euler’s Identity
ejθ =cos(θ)+jsin(θ) j=√
−1 cos(θ)=ejθ+e−jθ
2 sin(θ)=ejθ−e−jθ 2j tan(θ)= −j
"
ejθ −e−jθ ejθ +e−jθ
#
Hyperbolic Trigonometry Relations
Hyperbolic cosine: cosh(α)= 1
2(eα+e−α) Hyperbolic sine: sinh(α)= 1
2(eα−e−α) cosh2(α)−sinh2(α)=1
Calculus
Derivatives (u,vfunctions ofx;α,βconstants) duv
dx =udv dx+vdu
dx dun
dx =nun−1du dx
Integrals
Z
φ(y)dx= Z φ(y)
y0 dy, wherey0= dy dx Z
udv=uv− Z
vdu
Z
xndx= xn+1
n+1 n6= −1, integer
APPENDIX: Useful Formulas 745
Z
x−1dx=log(x) Z
eaxdx=eax
a a6=0 Z
xeaxdx= eax
a2(ax−1) Z
sin(ax)dx= −1 acos(ax) Z
cos(ax)dx= 1 asin(ax) Z sin(x)
x dx=
∞
X
n=0
(−1)n x2n+1
(2n+1)(2n+1)! integral of sinc function Z ∞
0
sin(x) x dx=
Z ∞
0
sin(x) x
2
dx= π 2
[1] A. Antoniou.Digital Filters. New York: McGraw-Hill, 1979.
[2] E. T. Bell.Men of Mathematics. New York: Simon and Schuster, 1965.
[3] J. Belrose. Fessenden and the early history of radio science.http://www.ieee.ca/millennium/radio/radio radioscientist.
html, accessed 2010.
[4] J. Bingham. Multicarrier modulation for data transmission: An idea whose time has come.IEEE Communications Magazine, May 1990: 5–14.
[5] N. K. Bose.Digital Filters. Salem, MA: Elsevier, 1985.
[6] J. L. Bourjaily.http://www-personal.umich.edu/∼jbourj/money.htm, accessed 2010.
[7] C. Boyer.A History of Mathematics. New York: Wiley, 1991.
[8] R. Bracewell.The Fourier Transform and Its Application. Boston: McGraw-Hill, 2000.
[9] M. Brain. How CDs work.http://electronics.howstuffworks.com/cd7.htm, accessed 2010.
[10] W. Briggs and V. Henson.The DFT. Philadelphia: Society for Industrial and Applied Mathematics, 1995.
[11] O. Brigham.The Fast Fourier Transform and Its Applications. Englewood Cliffs, NJ: Prentice-Hall, 1988.
[12] D. Cheng.Analysis of Linear Systems. Reading, MA: Addison-Wesley, 1959.
[13] L. Chua, C. Desoer, and E. Kuh.Linear and Nonlinear Circuits. New York: McGraw-Hill, 1987.
[14] J. Cooley. How the FFT gained acceptance. InA History of Scientific Computing,edited by S. Nash. New York:
Association for Computing Machinery, Inc. Press, 1990, pp. 133–140.
[15] J. W. Cooley and J. W. Tukey. An algorithm for the machine calculation of complex Fourier series.Mathematics of Computation, 19, April 1965: 297.
[16] L. W. Couch.Digital and Analog Communication Systems. Upper Saddle River, NJ: Pearson/Prentice-Hall, 2007.
[17] E. Craig.Laplace and Fourier Transforms for Electrical Engineers. New York: Holt, Rinehart, and Winston, 1966.
[18] E. Cunningham.Digital Filtering. Boston: Houghton Mifflin, 1992.
[19] G. Danielson and C. Lanczos. Some improvements in practical Fourier analysis and their applications to X-ray scattering from liquids.J. Franklin Institute, 1942: 365–380.
[20] C. Desoer and E. Kuh.Basic Circuit Theory. New York: McGraw-Hill, 1969.
[21] R. Dorf and R. Bishop.Modern Control Systems. Upper Saddle River, NJ: Prentice-Hall, 2005.
[22] G. Franklin, J. Powell, and M. Workman.Digital Control of Dynamic Systems. Reading, MA: Addison-Wesley, 1998.
746 Signals and Systems Using MATLAB®. DOI: 10.1016/B978-0-12-374716-7.00025-9
c2011, Elsevier Inc. All rights reserved.
Bibliography 747
[23] G. Johnson. Claude Shannon, Mathematician, Dies at 84. New York Times, February 27, 2001.
[24] R. Gabel and R. Roberts.Signals and Linear Systems. New York: Wiley, 1987.
[25] S. Goldberg.Introduction to Difference Equations. New York: Dover, 1958.
[26] R. Graham, D. Knuth, and O. Patashnik.Concrete Mathematics: A Foundation for Computer Science.Reading, MA:
Addison-Wesley, 1994.
[27] S. Haykin and M. Moher.Communication Systems. Hoboken, NJ: Wiley, 2009.
[28] S. Haykin and M. Moher.Modern Wireless Communications. Upper Saddle River, NJ: Pearson/Prentice-Hall, 2005.
[29] S. Haykin and M. Moher.Introduction to Analog and Digital Communications. Hoboken, NJ: Wiley, 2007.
[30] S. Haykin and B. Van Veen.Signals and Systems. New York: Wiley, 2003.
[31] M. Heideman, D. Johnson, and S. Burrus. Gauss and the history of the FFT.IEEE Acoustics, Speech and Signal Processing (ASSP) Magazine, vol. 1, pp. 14–21, Oct. 1984.
[32] K. Howell.Principles of Fourier Analysis. Boca Raton, FL: Chapman & Hall, CRC Press 2001.
[33] IEEE. Nyquist biography.http://www.ieee.org/web/aboutus/history center/biography/nyquist.html, accessed 2010.
[34] Intel. Microprocessor quick reference guide.http://www.intel.com/pressroom/kits/quickreffam.htm, accessed 2010.
[35] Intel. Moore’s law.http://www.intel.com/technology/mooreslaw/index.htm, accessed 2010.
[36] L. Jackson.Signals, Systems, and Transforms. Reading, MA: Addison-Wesley, 1991.
[37] E. I. Jury.Theory and Application of the Z-Transform Method. New York: Wiley, 1964.
[38] E. Kamen and B. Heck.Fundamentals of Signals and Systems. Upper Saddle River, NJ: Pearson/Prentice-Hall, 2007.
[39] R. Keyes. Moore’s law today.IEEE Circuits and Systems Magazine, 8, 2008: 53–54.
[40] B. P. Lathi.Modern Digital and Analog Communication Systems. New York: Oxford University Press, 1998.
[41] B. P. Lathi.Linear Systems and Signals. New York: Oxford University Press, 2002.
[42] E. Lee and P. Varaiya.Structure and Interpretation of Signals and Systems. Boston: Addison-Wesley, 2003.
[43] W. Lehr, F. Merino, and S. Gillet. Software radio: Implications for wireless services, industry structure, and public policy.http://itc.mit.edu, accessed 2002.
[44] D. Luke. The origins of the sampling theorem.IEEE Communications Magazine, April 1999:106–108.
[45] M. Bellis. The invention of radio.http://inventors.about.com/od/rstartinventions/a/radio.htm, accessed 2010.
[46] D. McGillem and G. Cooper.Continuous and Discrete Signals and System Analysis. New York: Holt, Rinehart, and Wiston, 1984.
[47] Z. A. Melzak.Companion to Concrete Mathematics. New York: Wiley, 1973.
[48] S. Mitra.Digital Signal Processing. New York: McGraw-Hill, 2006.
[49] C. Moler. Cleve’s Corner—The Origins of MATLAB. http://www.mathworks.com/company/newsletters/news notes/
clevescorner/dec04.html, accessed 2010.
[50] National. Op-amp history.http://www.analog.com/library/analogdialogue/.../39.../web chh final.pdf, accessed 2008.
[51] F. Nebeker.Signal Processing—The Emergence of a Discipline, 1948 to 1998. New Brunswick, NJ: IEEE History Center, 1998. (This book was especially published for the 50th anniversary of the creation of the IEEE Signal Processing Society in 1998).
[52] MIT news. MIT Professor Claude Shannon dies; was founder of digital communications. http://web.mit.edu/
newsoffice/2001/shannon.html, accessed 2009.
[53] K. Ogata.Modern Control Engineering. Upper Saddle River, NJ: Prentice-Hall, 1997.
[54] A. Oppenheim and R. Schafer.Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1975.
[55] A. Oppenheim and R. Schafer.Discrete-Time Signal Processing. Upper Saddle River, NJ: Prentice-Hall, 2010.
[56] A. Oppenheim and A. Willsky.Signals and Systems. Upper Saddle River, NJ: Prentice-Hall, 1997.
[57] A. Papoulis.Signal Analysis. New York: McGraw-Hill, 1977.
[58] PBS.org. Who invented radio?http://www.pbs.org/tesla/ll/ll whoradio.html, accessed 2010.
[59] C. Phillips, J. Parr, and E. Riskin.Signals, Systems and Transforms. Upper Saddle River, NJ: Pearson/Prentice-Hall, 2003.
[60] R. Prasad and S. Hara. Overview of multi-carrier CDMA. IEEE Communications Magazine, vol. 35, pp. 126–133, Dec. 1997.
[61] J. Proakis and M. Salehi.Communication Systems Engineering. Upper Saddle River, NJ: Prentice-Hall, 2002.
[62] Qualcomm. Who we are: History.http://www.qualcomm.com/who we are/history.html, accessed 2010.
[63] L. Rabiner and B. Gold.Theory and Application of Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1975.
[64] GNU Radio. The GNU software radio.http://www.gnu.org/software/gnuradio/, accessed 2008.
[65] Jaycar Electronics Reference Data Sheet. Understanding decibels. http://www.jaycar.com.au/images uploaded/decibels.pdf, accessed 2009.
[66] D. Slepian. On bandwidth.Proceeding of the IEEE, 64, March 1976. 292–300.
[67] S. Smith. The Scientist and Engineer’s Guide to Digital Signal Processing (http://www.dspguide.com). California Technical Publishing, San Diego, CA, 1997.
[68] S. Soliman and M. Srinath.Continuous and Discrete Signals and Systems. Upper Saddle River, NJ: Prentice-Hall, 1998.
[69] A. Stanoyevitch.Introduction to Numerical Ordinary and Partial Differential Equations Using MATLAB. New York:
Wiley, 2005.
[70] Statemaster.com. Spread spectrum.http://www.statemaster.com/encyclopedia/Spread-spectrum, accessed 2009.
[71] H. Stern and S. Mahmoud.Communication Systems—Analysis and Design. Upper Saddle River, NJ: Pearson/Prentice- Hall, 2004.
[72] M. Van Valkenburg.Analog Filter Design. New York: Oxford University Press, 1982.
[73] Wikipedia. Euler’s identity.http://en.wikipedia.org/wiki/Euler’s identity, accessed 2009.
[74] Wikipedia. Hedy Lamarr.http://en.wikipedia.org/wiki/Hedy Lamarr, accessed 2009.
[75] Wikipedia. Nikola Tesla.http://en.wikipedia.org/wiki/Nikola Tesla, accessed 2009.
[76] Wikipedia. Oliver Heaviside.http://en.wikipedia.org/wiki/Oliver Heaviside, accessed 2009.
[77] Wikipedia. Field-programable gate array.http://en.wikipedia.org/wiki/Field-programmable gate array, accessed 2008.
[78] M. R. Williams.A History of Computing Technologies. New York: Wiley/IEEE Computer Society Press, 1997.
Index
Ex, 80
F(s)=L[f(t)], 169 Fs, 437
F()=F[f(t)], 305, 344–346 F()=F[f[n]], 587 F(z)+Z[f[n]], 512, 523 H(s)=L[y(t)]/L[x(t)], 197 N, 77
Px, 85 Ts, 456 Xk, 256 1, 441
0=2π/T0, 256
, 656
s, 423 δ(t), 89 δTs(t), 423 ω, 423 τ, 73 x, 458 (nTs), 442 ej`0t, 247 h(t), 149 n, 452 r(t), 90 u(t), 89 x[n], 452, 454 xe[n], 464, 465 xo[n], 465 xe(t), 76 xo(t), 76 yzi(t), 130, 215 yzs(t), 130, 215
A
absolutely summable impulse response, 501, 535–536, 680 absolutely summable signals, 575,
576, 628 advanced signal, 324
amplitude modulation (AM), 87
demodulation, 380 envelope receiver, 381 single sideband, 382–383 suppressed carrier, 379–380 tunable band-pass filter, 379 analog
signal, 9, 67–71 signal, definition, 67
analog communication systems, 730 analog control systems, 363
actuator, 366
cruise control system, 367–369 feedback, 363
open-loop and closed-loop, 364–365
positive and negative feedback, 363
proportional controller, 366 proportional plus integral (PI)
controller, 367 stability and stabilization,
369–371 transducer, 366 analog filtering, 390
basics, 390–393
Butterworth low-pass design, 391, 393–396
Chebyshev low-pass design, 396–402
Chebyshev polynomials, 396 eigenfunction property, 390 factorization, 391, 393–394, 399 frequency transformations,
402–404 loss function, 392 low-pass specifications, 392 magnitude and frequency
normalization, 393 magnitude-squared
function, 391 specifications, 391–393
analog Fourier series
absolutely uniform convergence, 265–270
coefficients, 247 coefficients from Laplace,
255–265
complex exponential, 245–248 convergence, 265–270 DC component, 251 even and odd signals, 279 fundamental frequency, 246,
253, 256
fundamental period, 246 harmonics, 251 linearity, 282–283 line spectrum, 250, 255 mean-square approximation,
266
Parseval’s theorem, 248–250 product of signals, 284 time and frequency shifting,
270–273 time reversal, 280 trigonometric, 251–255 analog Fourier transform
amplitude modulation, 314 convolution, 327–329 differentiation and integration,
346–350
direct and inverse, 299, 301 duality, 310–313
frequency shifting, 313–314 Laplace ROC, 302, 304 linearity, 304–305 periodic signals, 317–320 shifting in time, 345
spectrum and line spectrum, 318 symmetry, 322–327
analog frequency, 619 analog LTI systems
BIBO stability, 153–156 749
analog LTI systems (continued) causality, 143–145 complete response, 216 continuous-time, 119 convolution integral, 136–143 eigenfunction property, 167,
240, 273
frequency response, 240, 327 impulse response, 138 impulse response, transfer
function, and frequency response, 329
represented by differential equations, 214–221 steady-state response, 214 transfer function, 213 transient response, 214 unit-step response, 218, 219 zero-input response, 133, 214 zero-state response, 133, 214 analog systems
causality, 143–145 DC source, 329 passivity, 154 stability, 153 windowing, 331
analog-to-digital converter (ADC), 68, 420
anti-aliasing filter, 430 application-specific integrated
circuit (ASIC), 5
approximate solution of differential equations, 559
B
band-limited signal, 423 basic analog signals
ramp, 90–92 triangular pulse, 90 unit-impulse, 88 unit-step, 89
basic discrete-time signals, 465–478 complex exponentials, 596 damped sinusoid, 466 discrete sinusoids, 469–471 basic signal operations
adder, 72
advancing and delaying, 73 constant multiplier, 71 modulation, 72 reflection, 72 time scaling, 71 windowing, 71 BIBO stability of discrete
systems, 501
bilinear transformation, 654–656 warping, 656
block diagrams, 148, 150 bounded-input bounded-output
(BIBO) stability, 153–156, 499–501
C
causal
sinusoid, 82, 110 causality
discrete LTI systems, 498 discrete signal, 497–498 discrete systems, 497–500 causal systems and signals, 507–508 channel noise, 379
circular shifting, 607–609 cognitive radio, 6–8
compact-disc (CD) player, 5–6 complex variable function, 23–24 complex variables, 20, 23–24 computer-control systems, 8–9 connection ofs-plane and
z-plane, 513 continuous-time
signal, 67–85
convolution integral, 136–133 commutative property, 148 distributive property, 149 Fourier, 327
graphical computation, 145–147 Laplace, 221
convolution sum, 487–494, 526–537
commutative property, 148 deconvolution, 229 noncausal signals, 533
D
delayed signal, 73
difference equations, 18–19, 550–561
digital communications, 709 orthogonal frequency-division
multiplexing (OFDM), 710 PCM, 710
spread spectrum, 710
time-division multiplexing, 730 digital signal processing, 710–722
FFT, 711–715 FFT algorithm, 711
digital signal processor (DSP), 5 digital-to-analog converter, 5,
68, 420
discrete complex exponentials, 466–469
discrete filtering analog signals, 640 bilinear transformation, 640 Butterworth LPF, 658–664 Chebyshev LPF, 666–672 direct, cascade, and parallel IIR
realizations, 698 eigenfunction, 639 FIR design, 681
FIR realizations, 699–700 FIR window design, 681 frequency scales, 652–653 frequency-selective filters, 641 frequency specifications, 659 group delay, 643
IIR and FIR, 643–647 IIR design, 672 linear phase, 641–643 loss function, 648–650 rational frequency
transformations, 672–676 realization, 689–700 time specifications, 652–653 windows for FIR design,
681–683 discrete filters
FIR, 643–647 IIR, 643–647
discrete Fourier series, 599–601 circular representation, 598–599 circular shifting, 607–609 complex exponential, 599–601 periodic convolution, 609–614 Z-transform, 601–602 discrete Fourier transform (DFT),
614–627
fast Fourier transform (FFT), 614 linear and circular
convolution, 624 discrete frequency, 454, 471 discrete LTI systems
causality, 498
response to periodic signals, 273–278
discrete sinusoid, 444 discrete systems
autoregressive (AR), 482 autoregressive moving average
(ARMA), 484 BIBO stability, 500–501 causality and stability, 497–501 convolution sum, 487–494 difference equation
representation, 486–487
Index 751
moving average (MA), 481–482 nonlinear system, 498 time-invariance, 498 discrete-time Fourier transform
(DTFT), 572–596 convergence, 591
convolution sum, 595–596 downsampling and
upsampling, 582 eigenfunctions, 573–575 Parseval’s theorem, 585–587 sampled signal, 578–580 symmetry, 589–595
time and frequency shifts, 628 time-frequency duality, 628 time-frequency supports,
580–585 Z-transform, 573–575 discrete-time signals
absolutely summable, 575, 576, 628
basic, 465–478 definition, 452 Fibonacci sequence, 453 finite energy, 458–461 finite power, 458–461 inherently discrete-time, 452 sample index, 452
sinusoid, 469–472 square summable, 458 discrete transfer function, 655
E
energy, 80
discrete-time signals, 458–461 Euler’s identity, 23–24, 87 even signal, 279, 461–465
F
Fibonacci sequence difference equation, 453 field-programmable gate array
(FPGA), 5
filtering, 276–278, 327–344 analog, 390–408 median filter, 495 filters
anti-aliasing, 430 passband, 332 RC high-pass filter, 336 RC low-pass filter, 277 finite calculus, 9
finite difference, 12–13 summations, 13–16
FIR filters and convolution sum, 528, 529, 531, 533 Fourier basis, 247
four-level quantizer, 441, 442 frequency, harmonically related, 83 frequency aliasing, 424
frequency modulation (FM), 87 frequency response, poles and zeros,
342, 343
G
Gibb’s phenomenon, 266, 267 filtering, 334
graphical convolution sum, 530
H
hybrid system, 119
I
ideal filters band-pass, 332 high-pass, 332 linear phase, 332 low-pass, 332 zero-phase, 333
ideal impulse sampling, 421–428 inverse Laplace
with exponentials, 209 partial fraction expansion, 198 two-sided, 212–214
inverse Z-transform, 542–563 inspection, 542
long-division method, 542–543 partial fraction expansion,
544–546
positive powers of z, 545, 546
L
Laplace transform
convolution integral, 196–197 derivative, 189
integration, 193–194 inverse, 169, 197–214 linearity, 185–188 one-sided, 176–197 proper rational, 198 region of convergence (ROC),
166, 172–176 transfer function, 214, 223 two-sided, 166–176 length of convolution sum, 721 L’Hopital’s rule, 101, 306, 433 LTI systems, superposition, 135–136
M
magnitude line spectrum, 249 Matlab
analog Butterworth and Chebyshev filter design, 414 analog Butterworth filtering, 414 control toolbox, 375
decimation and interpolation, 585
DFT and FFT, 577 discrete filter design, 644 DTFT computation, 577 FFT computation, 717 filter design, 405–408 Fourier series computation,
603–604 functions, 36
general discrete filter design, 646 numerical computations, 30 phase computation, 591 phase unwrapping, 592 plotting, 39–41
saving and loading, 41–43 symbolic computations, 43–53 vectorial operations, 33–35 vectors and matrices, 30–33
N
negative frequencies, 323 nonlinear filtering, median
filter, 495
nonzero initial conditions, 552 normality, 247
Nyquist sampling rate, 431 Nyquist sampling theorem, 431
O
odd signal, 75–77 one-sided Z-transform, 511 orthogonality, 248
P
Parseval’s relation and sampling, 427
periodic convolution, 609–614, 624 periodic discrete sinusoids, 454, 456 phase line spectrum, 249, 250, 253,
257, 259, 261, 263, 265 phase modulation (PM), 87, 378,
386
phasors, sinusoidal steady state, 24–26, 28
poles and zeros, 172–176
poles and zeros of Z-transforms, 511, 549, 551, 564