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Problems 205 and use this property. Develop a MATLAB function to implement this approach with the following format. function [X1,X2] = real2dft(x1,x2,N) % DFTs of two real sequences % [X1,X2] = real2dft(x1,x2,N) % X1 = n-point DFT of x1 % X2 = n-point DFT of x2 % x1 = sequence of length
206 Chapter 5 THE DISCRETE FOURIER TRANSFORM P5.22 A 512-point DFT X(k) ofareal-valued sequence x(n) has the following DFT values: X(0)=20+jα; X(5)=20+j30; X(k 1)=−10 + j15; X(152) = 17 + j23; X(k 2)=20− j30; X(k 3)=17− j23; X(480) = −10 − j15; X(256) = 30 + jβ and all other values are known to be zero. 1. Determine the real-valued coefficients α and β. 2. Determine the values of the integers k 1, k 2, and k 3. 3. Determine the energy of the signal x(n). 4. Express the sequence x(n) inaclosed form. P5.23 Let x(n) beafinite length sequence given by x(n) = { ...,0, 0, 0, 1, 2, −3, 4, −5, 0,... ↑ Determine and sketch the sequence x((−8 − n)) 7R 7 (n) where R 7 (n) = { 1, 0 ≤ n ≤ 6 0, elsewhere P5.24 The circonvt function developed in this chapter implements the circular convolution as a matrix-vector multiplication. The matrix corresponding to the circular shifts {x((n − m)) N ; 0 ≤ n ≤ N − 1} has an interesting structure. This matrix is called a circulant matrix, which is a special case of Toeplitz matrix introduced in Chapter 2. 1. Consider the sequences given in Example 5.13. Express x 1(n) asacolumn vector x 1 and x 2((n − m)) N as a circulant matrix X 2 with rows corresponding to n =0, 1, 2, 3. Characterize this matrix X 2. Can it completely be described by its first row (or column)? 2. Determine the circular convolution as X 2x 1 and verify your calculations. P5.25 Develop a MATLAB function to construct a circulant matrix C given an N-point sequence x(n). Use the toeplitz function to implement matrix C. Your subroutine function should have the following format: } function [C] = circulnt(x,N) % Circulant Matrix from an N-point sequence % [C] = circulnt(x,N) % C = circulant matrix of size NxN % x = sequence of length
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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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Problems 97 % check >> error = max(
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Problems 99 Remember that the above
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Problems 101 P3.16 Foralinear, shif
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CHAPTER 4 The z-Transform In Chapte
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The Bilateral z-Transform 105 Im{z}
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Important Properties of the z-Trans
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Inversion of the z-Transform 113 wh
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Inversion of the z-Transform 115 be
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Inversion of the z-Transform 117 He
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System Representation in the z-Doma
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Solutions of the Difference Equatio
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Problems 135 4. x(n) =n 2 (2/3) n
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Problems 137 2. Using (4.32), deter
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Problems 139 It is also known that
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CHAPTER 5 The Discrete Fourier Tran
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The Discrete Fourier Series 143 den
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The Discrete Fourier Series 145 The
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The Discrete Fourier Series 147 DFS
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Sampling and Reconstruction in the
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Page 174 and 175: The Discrete Fourier Transform 155
Page 184 and 185: Properties of the Discrete Fourier
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Page 206 and 207: The Fast Fourier Transform 187 5.6
Page 208 and 209: The Fast Fourier Transform 189 Usin
Page 210 and 211: The Fast Fourier Transform 191 and
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Page 214 and 215: The Fast Fourier Transform 195 Let
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Page 218 and 219: The Fast Fourier Transform 199 comp
Page 220 and 221: Problems 201 Clearly, ˜x 3(n) isdi
Page 222 and 223: Problems 203 determine the impulse
Page 226 and 227: Problems 207 function x3 = circonvf
Page 228 and 229: Problems 209 where l is an integer.
Page 230 and 231: Problems 211 2. From the following
Page 232 and 233: Basic Elements 213 characteristics
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Page 236 and 237: IIR Filter Structures 217 FIGURE 6.
Page 238 and 239: IIR Filter Structures 219 for i = 1
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Page 244 and 245: IIR Filter Structures 225 □ EXAMP
Page 246 and 247: IIR Filter Structures 227 a1 = 1.00
Page 248 and 249: FIR Filter Structures 229 FIGURE 6.
Page 254 and 255: FIR Filter Structures 235 6.3.8 MAT
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Page 260 and 261: Lattice Filter Structures 241 There
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Page 266 and 267: Lattice Filter Structures 247 funct
Page 268 and 269: Lattice Filter Structures 249 Solut
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Page 272 and 273: 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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