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The Discrete Fourier Series 145 The following idfs function implements the synthesis equation. function [xn] = idfs(Xk,N) % Computes Inverse Discrete Fourier Series % ---------------------------------------- % [xn] = idfs(Xk,N) % xn = One period of periodic signal over 0
146 Chapter 5 THE DISCRETE FOURIER TRANSFORM a. By applying the analysis equation (5.3), ˜X(k) = N−1 ∑ n=0 L−1 L−1 ∑ ∑ ) n ˜x(n)e −j 2π N nk = e −j 2π N nk = (e −j 2π N k n=0 ⎧ ⎨ L, k =0, ±N,±2N,... = ⎩ 1 − e −j2πLk/N , otherwise 1 − e−j2πk/N The last step follows from the sum of the geometric terms formula (2.7) in Chapter 2. The last expression can be simplified to n=0 1 − e −j2πLk/N e−jπLk/N e jπLk/N − e −jπLk/N = 1 − e−j2πk/N e −jπk/N e jπk/N − e −jπk/N = e −jπ(L−1)k/N sin (πkL/N) sin (πk/N) or the magnitude of ˜X(k) isgiven by ⎧ ⎪⎨ L, k =0, ±N,±2N,... ∣ ∣ ˜X(k) = ⎪ ⎩ sin (πkL/N) ∣ sin (πk/N) ∣ , otherwise b. The MATLAB script for L =5and N = 20: >> L = 5; N = 20; k = [-N/2:N/2]; % Sq wave parameters >> xn = [ones(1,L), zeros(1,N-L)]; % Sq wave x(n) >> Xk = dfs(xn,N); % DFS >> magXk = abs([Xk(N/2+1:N) Xk(1:N/2+1)]); % DFS magnitude >> subplot(2,2,1); stem(k,magXk); axis([-N/2,N/2,-0.5,5.5]) >> xlabel(’k’); ylabel(’Xtilde(k)’) >> title(’DFS of SQ. wave: L=5, N=20’) The plots for this and all other cases are shown in Figure 5.2. Note that since ˜X(k) isperiodic, the plots are shown from −N/2 toN/2. c. Several interesting observations can be made from plots in Figure 5.2. The envelopes of the DFS coefficients of square waves look like “sinc” functions. The amplitude at k =0is equal to L, while the zeros of the functions are at multiples of N/L, which is the reciprocal of the duty cycle. We will study these functions later in this chapter. □ 5.1.2 RELATION TO THE z-TRANSFORM Let x(n) beafinite-duration sequence of duration N such that { Nonzero, 0 ≤ n ≤ N − 1 x(n) = 0, Elsewhere (5.8) Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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Digital Signal Processing Using MAT
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Contents PREFACE xi 1 INTRODUCTION
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CONTENTS vii 6 IMPLEMENTATION OF DI
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CONTENTS ix 11.3 Suppression of Nar
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Preface From the beginning of the 1
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PREFACE xiii implementation. The di
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PREFACE xv ACKNOWLEDGMENTS We are i
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CHAPTER 1 Introduction During the p
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Overview of Digital Signal Processi
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A Brief Introduction to MATLAB 5 It
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A Brief Introduction to MATLAB 7 Va
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A Brief Introduction to MATLAB 9 us
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A Brief Introduction to MATLAB 11 o
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A Brief Introduction to MATLAB 13 o
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A Brief Introduction to MATLAB 15 T
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Applications of Digital Signal Proc
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Applications of Digital Signal Proc
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Brief Overview of the Book 21 In al
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Discrete-time Signals 23 In MATLAB
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Discrete-time Signals 25 For exampl
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Discrete-time Signals 27 n = min(mi
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Discrete-time Signals 29 Solution a
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Discrete-time Signals 31 Sequence i
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Discrete-time Signals 33 The quanti
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Discrete-time Signals 35 Rectangula
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Discrete Systems 37 In DSP we will
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Discrete Systems 39 every input seq
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Convolution 41 2 Input Sequence 1.5
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Convolution 43 x(k) and h(k) x(k) a
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Convolution 45 Hence y(n) ={6, 31,
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Difference Equations 47 Note that t
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Difference Equations 49 Solution Fr
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Difference Equations 51 8 Output Se
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Problems 53 IIR filter If the impul
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Problems 55 1. Modify the evenodd f
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Problems 57 2. x(n) ={1, 1, 0, 1, 1
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CHAPTER 3 The Discrete-time Fourier
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The Discrete-time Fourier Transform
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The Properties of the DTFT 67 TABLE
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The Properties of the DTFT 69 8. Mu
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The Properties of the DTFT 71 60 Ma
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The Properties of the DTFT 73 2 Rea
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The Frequency Domain Representation
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Sampling and Reconstruction of Anal
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Page 114 and 115: Sampling and Reconstruction of Anal
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Page 184 and 185: Properties of the Discrete Fourier
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Page 206 and 207: The Fast Fourier Transform 187 5.6
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The Fast Fourier Transform 195 Let
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The Fast Fourier Transform 197 N =2
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The Fast Fourier Transform 199 comp
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Problems 201 Clearly, ˜x 3(n) isdi
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Problems 203 determine the impulse
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Problems 205 and use this property.
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Problems 207 function x3 = circonvf
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Problems 209 where l is an integer.
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Problems 211 2. From the following
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Basic Elements 213 characteristics
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IIR Filter Structures 215 FIGURE 6.
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IIR Filter Structures 217 FIGURE 6.
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IIR Filter Structures 219 for i = 1
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IIR Filter Structures 221 6.2.6 PAR
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IIR Filter Structures 223 Brow = r(
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IIR Filter Structures 225 □ EXAMP
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IIR Filter Structures 227 a1 = 1.00
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FIR Filter Structures 229 FIGURE 6.
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FIR Filter Structures 235 6.3.8 MAT
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Lattice Filter Structures 239 1.000
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Lattice Filter Structures 241 There
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Lattice Filter Structures 243 FIGUR
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Lattice Filter Structures 245 FIGUR
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Lattice Filter Structures 247 funct
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Lattice Filter Structures 249 Solut
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Representation of Numbers 251 new f
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Representation of Numbers 253 rule
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Representation of Numbers 255 Solut
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Representation of Numbers 257 Ten
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Representation of Numbers 259 negat
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Representation of Numbers 261 Two
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Representation of Numbers 263 For t
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Representation of Numbers 265 IEEE
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The Process of Quantization and Err
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Quantization of Filter Coefficients
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Problems 289 z −1 x(n) K 2 2 0.5
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Problems 291 x(n) −1/2 1/2 −2/3
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Problems 293 0.5 1 −1 z −1 2 1
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Problems 295 1. Direct form 2. Line
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Problems 297 P6.24 MATLAB provides
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Problems 299 4. Explain why the mag
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Problems 301 P6.38 An IIR bandstop
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CHAPTER 7 FIR Filter Design We now
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Preliminaries 305 1 + δ 1 1 1 −
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Properties of Linear-phase FIR Filt
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Window Design Techniques 323 where
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Window Design Techniques 325 Rectan
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Window Design Techniques 327 of att
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Window Design Techniques 329 Hammin
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Window Design Techniques 331 where
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Window Design Techniques 333 To dis
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Window Design Techniques 335 Ideal
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Window Design Techniques 337 Soluti
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Window Design Techniques 339 The id
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Window Design Techniques 341 □ EX
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Window Design Techniques 343 Ideal
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Frequency Sampling Design Technique
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Optimal Equiripple Design Technique
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Problems 375 h(n) 0.8 0.6 0.4 0.2 0
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Problems 377 where coefficients ˜c
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Problems 379 Plot the impulse respo
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Problems 381 2.02 1.98 Hr(ω) 1 0 0
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Problems 383 P7.34 Design a minimum
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Problems 385 P7.38 Design a minimum
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Some Preliminaries 387 8.1 SOME PRE
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Some Preliminaries 389 8.1.2 PROPER
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Some Special Filter Types 391 The c
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Some Special Filter Types 393 Magni
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Some Special Filter Types 395 Imagi
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Some Special Filter Types 397 (a) F
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Some Special Filter Types 399 circl
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Characteristics of Prototype Analog
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Analog-to-Digital Filter Transforma
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Lowpass Filter Design Using MATLAB
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Frequency-band Transformations 449
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Problems 461 1 0.8913 Magnitude Res
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Problems 463 1. Determine the value
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Problems 465 function [b,a] =afd(ty
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Problems 467 function [b,a] = mzt(c
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Problems 469 P8.35 Design a highpas
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Problems 471 Using this function, d
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Problems 473 1 0.9 Equiripple |H (e
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Introduction 475 x a (t) ADC F s =
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Decimation by a Factor D 477 as h(n
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Decimation by a Factor D 479 given
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Decimation by a Factor D 481 |X(ω
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Decimation by a Factor D 483 The ou
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Decimation by a Factor D 485 % (c)
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Interpolation by a Factor I 487 Sol
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Interpolation by a Factor I 489 Ide
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Interpolation by a Factor I 491 % (
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Sampling Rate Conversion by a Ratio
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FIR Filter Designs for Sampling Rat
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FIR Filter Structures for Sampling
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Problems 531 P9.3 Consider a signal
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Problems 533 1. Compute and plot th
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Problems 535 3. Let x(n) =cos(0.3π
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Problems 537 3. Using the fir2 func
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Analysis of A/D Quantization Noise
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Round-off Effects in IIR Digital Fi
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Round-off Effects in FIR Digital Fi
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Problems 591 P10.6 Consider the 1st
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Problems 593 which is implemented i
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Applications in Adaptive Filtering
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LMS Algorithm for Coefficient Adjus
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System Identification or System Mod
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Suppression of Narrowband Interfere
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Adaptive Channel Equalization 603 T
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Adaptive Channel Equalization 605 F
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CHAPTER 12 Applications in Communic
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Pulse-Code Modulation 609 called a
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Differential PCM (DPCM) 611 the sig
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Differential PCM (DPCM) 613 digits
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Adaptive PCM and DPCM (ADPCM) 615 a
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Adaptive PCM and DPCM (ADPCM) 617 F
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Delta Modulation (DM) 619 FIGURE 12
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Delta Modulation (DM) 621 FIGURE 12
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Linear Predictive Coding (LPC) of S
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Linear Predictive Coding (LPC) of S
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Dual-tone Multifrequency (DTMF) Sig
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Dual-tone Multifrequency (DTMF) Sig
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Binary Digital Communications 631 F
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Spread-Spectrum Communications 633
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BIBLIOGRAPHY [1] J. W. Cooley and J
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INDEX ’operator, 25 &operator, 26
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INDEX 639 adaptive delta modulation
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INDEX 641 finite-precision numerica
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INDEX 643 round-off effects, 580-59
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INDEX 645 DTFT formula, 152-153 err
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INDEX 647 optimal equiripple design
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INDEX 649 Real-valued exponential s
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INDEX 651 Sinusoidal signals, 83-84
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