Chapter 1 Discrete-time Signals and Linear Systems 1.1 - 1.239
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1
1.2 Examples of Signals . . . . . . . . . . . . . . . . . . . . . . . . . 1.1
1.2.1 Speech signal . . . . . . . . . . . . . . . . . . . . . . . 1.1
1.2.2 ECG signal . . . . . . . . . . . . . . . . . . . . . . . . . 1.2
1.2.3 Electroencephalogram (EEG) signal . . . . . . . . . . . . 1.3
1.3 Classification of Signals . . . . . . . . . . . . . . . . . . . . . . . 1.3
1.3.1 Continuous-time, discrete-time and digital Signals . . . . 1.3
1.3.2 Deterministic and random signals . . . . . . . . . . . . . 1.9
1.4 System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10
1.4.1 Continuous-time system . . . . . . . . . . . . . . . . . . 1.11
1.4.2 Discrete-time system . . . . . . . . . . . . . . . . . . . . 1.11
1.5 Mathematical Model of Continuous-time Systems . . . . . . . . . . 1.12
1.5.1 Modeling of mechanical and electrical elements . . . . . 1.12
1.5.2 Electrical elements . . . . . . . . . . . . . . . . . . . . . 1.13
1.6 Examples of Continuous-Time System Models . . . . . . . . . . . 1.14
1.6.1 RL circuit . . . . . . . . . . . . . . . . . . . . . . . . . 1.14
1.6.2 Mass - Spring - Dashpot system . . . . . . . . . . . . . . 1.15
1.7 Examples of Discrete-Time System Models . . . . . . . . . . . . . 1.15
1.7.1 Population model . . . . . . . . . . . . . . . . . . . . . 1.15
1.7.2 Savings account . . . . . . . . . . . . . . . . . . . . . . 1.16
1.7.3 Amortization . . . . . . . . . . . . . . . . . . . . . . . . 1.16
1.8 Signal Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.18
1.9 Advantages and Limitations of Digital Signal Processing . . . . . . 1.20
1.9.1 Advantages . . . . . . . . . . . . . . . . . . . . . . . . . 1.20
1.9.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . 1.21
1.10 Applications of DSP . . . . . . . . . . . . . . . . . . . . . . . . . 1.21
1.11 Elementary Continuous-time Signals . . . . . . . . . . . . . . . . . 1.22
1.11.1 Unit step function . . . . . . . . . . . . . . . . . . . . . 1.22
1.11.2 Unit ramp function . . . . . . . . . . . . . . . . . . . . . 1.22
1.11.3 Impulse function . . . . . . . . . . . . . . . . . . . . . . 1.23
1.11.4 Sinusoidal signal . . . . . . . . . . . . . . . . . . . . . . 1.23
1.11.5 Real exponential signals . . . . . . . . . . . . . . . . . . 1.24
1.11.6 Complex exponential signal . . . . . . . . . . . . . . . . 1.25
1.12 Continuous-time Periodic Signals . . . . . . . . . . . . . . . . . . 1.26
1.13 Representation of Discrete-time Signals . . . . . . . . . . . . . . . 1.28
1.13.1 Graphical representation . . . . . . . . . . . . . . . . . . 1.28
1.13.2 Functional representation . . . . . . . . . . . . . . . . . 1.28
1.13.3 Tabular representation . . . . . . . . . . . . . . . . . . . 1.29
1.13.4 Sequence representation . . . . . . . . . . . . . . . . . . 1.29
1.14 Elementary Discrete-time signals . . . . . . . . . . . . . . . . . . . 1.29
1.14.1 Unit step sequence . . . . . . . . . . . . . . . . . . . . . 1.29
1.14.2 Unit ramp sequence . . . . . . . . . . . . . . . . . . . . 1.30
1.14.3 Unit-sample sequence(unit impulse sequence) . . . . . . 1.30
1.14.4 Exponential sequence . . . . . . . . . . . . . . . . . . . 1.31
1.14.5 Sinusoidal signal . . . . . . . . . . . . . . . . . . . . . . 1.31
1.14.6 Complex exponential signal . . . . . . . . . . . . . . . . 1.32
1.15 Classification of Discrete-Time Signals . . . . . . . . . . . . . . . 1.33
1.15.1 Energy signals and power signals . . . . . . . . . . . . . 1.33
1.15.2 Periodic and aperiodic signals . . . . . . . . . . . . . . . 1.36
1.15.3 Symmetric(even) and antisymmetric (odd) signals . . . . 1.42
1.15.4 Causal and noncausal signals . . . . . . . . . . . . . . . 1.43
1.16 Operation on Signals . . . . . . . . . . . . . . . . . . . . . . . . . 1.43
1.16.1 Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.43
1.16.2 Time reversal . . . . . . . . . . . . . . . . . . . . . . . . 1.44
1.16.3 Time Scaling . . . . . . . . . . . . . . . . . . . . . . . . 1.45
1.16.4 Scalar multiplication . . . . . . . . . . . . . . . . . . . . 1.47
1.16.5 Signal multiplier . . . . . . . . . . . . . . . . . . . . . . 1.47
1.16.6 Addition operation . . . . . . . . . . . . . . . . . . . . . 1.47
1.17 Sampling and Aliasing . . . . . . . . . . . . . . . . . . . . . . . . 1.48
1.18 Discrete-Time System . . . . . . . . . . . . . . . . . . . . . . . . . 1.52
1.19 Classification of Discrete-Time systems . . . . . . . . . . . . . . . 1.52
1.19.1 Static and dynamic systems . . . . . . . . . . . . . . . . 1.52
1.19.2 Causal and noncausal systems . . . . . . . . . . . . . . . 1.53
1.19.3 Linear and non-linear systems . . . . . . . . . . . . . . . 1.54
1.19.4 Time variant and time-invariant systems . . . . . . . . . 1.56
1.20 Representation of an Arbitrary Sequence . . . . . . . . . . . . . . . 1.58
1.21 Impulse Response and Convolution Sum . . . . . . . . . . . . . . . 1.60
1.22 Properties of Convolution . . . . . . . . . . . . . . . . . . . . . . . 1.61
1.23 Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.61
1.24 FIR and IIR Systems . . . . . . . . . . . . . . . . . . . . . . . . . 1.62
1.25 Stable and Unstable Systems . . . . . . . . . . . . . . . . . . . . . 1.63
1.26 Interconnection of LTI Systems . . . . . . . . . . . . . . . . . . . . 1.71
1.26.1 (i) Parallel connection of systems . . . . . . . . . . . . . 1.71
1.26.2 (ii) Cascade connection of two systems . . . . . . . . . . 1.72
1.27 Correlation of Two Sequences . . . . . . . . . . . . . . . . . . . . 1.75
1.27.1 Cross-correlation . . . . . . . . . . . . . . . . . . . . . . 1.76
1.27.2 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . 1.76
1.27.3 Properties of crosscorrelation and autocorrelation sequences 1.77
1.27.4 Computation of correlation . . . . . . . . . . . . . . . . 1.78
1.27.5 Correlation of power and periodic signals . . . . . . . . . 1.79
1.28 Inverse System and Deconvolution . . . . . . . . . . . . . . . . . . 1.80
1.28.1 Inverse system . . . . . . . . . . . . . . . . . . . . . . . 1.80
1.28.2 Deconvolution . . . . . . . . . . . . . . . . . . . . . . . 1.80
1.29 Time Response Analysis of Discrete-time Systems . . . . . . . . . 1.83
1.29.1 Natural response (zero input response) . . . . . . . . . . 1.85
1.29.2 Forced response (zero state response) . . . . . . . . . . . 1.88
1.29.3 Total response . . . . . . . . . . . . . . . . . . . . . . . 1.91
1.30 Impulse Response . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.93
1.31 Step Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.99
1.32 Frequency Analysis of Discrete-Time Signals . . . . . . . . . . . . 1.100
1.33 Discrete Frequency Spectrum and Frequency Range . . . . . . . . . 1.103
1.34 Discrete-Time Fourier Transform . . . . . . . . . . . . . . . . . . . 1.107
1.34.1 Existence of discrete-time Fourier transform . . . . . . . 1.109
1.34.2 Properties of discrete-time Fourier transform . . . . . . . 1.110
1.35 Frequency Response Analysis of Discrete-time systems . . . . . . . 1.128
1.35.1 Frequency Response of First-order system . . . . . . . . 1.129
1.35.2 Frequency response of second-order system . . . . . . . 1.131
1.35.3 Transfer function . . . . . . . . . . . . . . . . . . . . . . 1.133
1.36 Steady State and Transient Response . . . . . . . . . . . . . . . . . 1.138
1.37 Phase and Group delays . . . . . . . . . . . . . . . . . . . . . . . . 1.140
1.38 Ideal filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.141
1.39 Zero phase and linear phase transfer functions . . . . . . . . . . . . 1.144
1.40 Analog to Digital Conversion . . . . . . . . . . . . . . . . . . . . 1.146
1.40.1 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . 1.146
1.40.2 Aliasing effect . . . . . . . . . . . . . . . . . . . . . . . 1.148
1.40.3 Sampling theorem . . . . . . . . . . . . . . . . . . . . . 1.149
1.40.4 Anti-Aliasing Filter . . . . . . . . . . . . . . . . . . . . 1.150
1.40.5 Sample-and-hold circuit . . . . . . . . . . . . . . . . . . 1.150
1.40.6 Quantization . . . . . . . . . . . . . . . . . . . . . . . . 1.151
1.41 Reconstruction of Analog signal . . . . . . . . . . . . . . . . . . . 1.152
1.42 Sampling of Bandpass signals . . . . . . . . . . . . . . . . . . . . 1.156
Questions and Answers . . . . . . . . . . . . . . . . . . . . . . . 1.201
Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . 1.214
Multiple Choice Questions . . . . . . . . . . . . . . . . . . . . . 1.216
Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.221
MATLAB Programs . . . . . . . . . . . . . . . . . . . . . . . . . 1.225
Chapter 2 The Z-Transform 2.1 - 2.112
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1
2.2 Definition of the z-Transform . . . . . . . . . . . . . . . . . . . . . 2.1
2.3 z-Transform and ROC of Finite Duration Sequences . . . . . . . . . 2.2
2.3.1 Right-hand sequence . . . . . . . . . . . . . . . . . . . . 2.2
2.3.2 Left hand sequence . . . . . . . . . . . . . . . . . . . . 2.2
2.3.3 Two-sided sequence . . . . . . . . . . . . . . . . . . . . 2.3
2.4 z-Transform and ROC of Infinite Duration Sequence . . . . . . . . 2.4
2.5 ROC of Two-sided Sequence . . . . . . . . . . . . . . . . . . . . . 2.5
2.6 Stability and ROC . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7
2.7 Properties of Region of Convergence . . . . . . . . . . . . . . . . . 2.8
2.8 Properties of the z-Transform . . . . . . . . . . . . . . . . . . . . . 2.8
2.9 The system Function . . . . . . . . . . . . . . . . . . . . . . . . . 2.23
2.10 Poles and Zeros of a System Function . . . . . . . . . . . . . . . . 2.25
2.11 Stability Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.26
2.12 Relationship between the Fourier Transform and the z-Transform . . 2.27
2.13 Relationship between s-Plane and z-Plane . . . . . . . . . . . . . . 2.29
2.14 Inverse z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 2.30
2.14.1 Long division method . . . . . . . . . . . . . . . . . . . 2.30
2.14.2 Partial fraction expansion method . . . . . . . . . . . . . 2.32
2.14.3 Residue method . . . . . . . . . . . . . . . . . . . . . . 2.40
2.14.4 Convolution Method . . . . . . . . . . . . . . . . . . . . 2.44
2.15 Solution of difference equations using one sided z-transform . . . . 2.48
2.16 Deconvolution Using z-transform . . . . . . . . . . . . . . . . . . 2.54
2.17 Simple Digital Filters . . . . . . . . . . . . . . . . . . . . . . . . . 2.66
2.17.1 Simple IIR digital Filters . . . . . . . . . . . . . . . . . 2.66
2.17.2 Highpass IIR digital filter . . . . . . . . . . . . . . . . . 2.68
2.18 Simple FIR Digital filters . . . . . . . . . . . . . . . . . . . . . . . 2.72
2.18.1 Lowpass FIR digital filters . . . . . . . . . . . . . . . . 2.72
2.18.2 Highpass FIR digital filter . . . . . . . . . . . . . . . . . 2.74
2.19 Comb Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.75
2.20 Allpass Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.76
2.21 Minimum Phase, Maximum Phase and Non-minimum Phase Systems 2.79
Questions and Answers . . . . . . . . . . . . . . . . . . . . . . . 2.87
Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . 2.92
Multiple Choice Questions . . . . . . . . . . . . . . . . . . . . . 2.92
Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.96
MATLAB Programs . . . . . . . . . . . . . . . . . . . . . . . . . 2.99
Chapter 3 The Discrete Fourier Transform 3.1 - 3.118
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1
3.2 Discrete Fourier series . . . . . . . . . . . . . . . . . . . . . . . . 3.1
3.3 Properties of the Discrete Fourier Series . . . . . . . . . . . . . . . 3.2
3.3.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2
3.3.2 Time shifting . . . . . . . . . . . . . . . . . . . . . . . . 3.3
3.3.3 Symmetry property . . . . . . . . . . . . . . . . . . . . 3.3
3.3.4 Periodic convolution . . . . . . . . . . . . . . . . . . . . 3.4
3.4 The Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . 3.5
3.4.1 Time-domain aliasing due to frequency sampling . . . . . 3.5
3.5 Relationship of The DFT to Other Transforms . . . . . . . . . . . . 3.22
3.5.1 Relationship to the Fourier transform . . . . . . . . . . . 3.22
3.5.2 Relationship to the z-transform . . . . . . . . . . . . . . 3.23
3.6 Properties of the Discrete Fourier transform . . . . . . . . . . . . . 3.25
3.6.1 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . 3.26
3.6.2 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . 3.26
3.6.3 Circular shift of a sequence . . . . . . . . . . . . . . . . 3.26
3.6.4 Time reversal of the sequence . . . . . . . . . . . . . . . 3.31
3.6.5 Circular frequency shift . . . . . . . . . . . . . . . . . . 3.31
3.6.6 Complex conjugate property . . . . . . . . . . . . . . . . 3.32
3.6.7 Circular convolution . . . . . . . . . . . . . . . . . . . . 3.34
3.6.8 Circular correlation . . . . . . . . . . . . . . . . . . . . 3.35
3.6.9 Multiplication of two sequences . . . . . . . . . . . . . . 3.36
3.6.10 Parseval’s theorem . . . . . . . . . . . . . . . . . . . . . 3.36
3.7 Comparison between Circular Convolution and Linear Convolution . 3.41
3.8 Methods to Evaluate Circular Convolution of Two Sequences . . . . 3.42
3.8.1 Concentric circle method . . . . . . . . . . . . . . . . . 3.42
3.8.2 Matrix multiplication method . . . . . . . . . . . . . . . 3.42
3.9 Linear Convolution From Circular Convolution . . . . . . . . . . . 3.48
3.10 Filtering Long Duration Sequences . . . . . . . . . . . . . . . . . . 3.52
3.10.1 Overlap-save method . . . . . . . . .
Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.60
MATLAB Programs . . . . . . . . . . . . . . . . . . . . . . . . . 4.61
Chapter 5 Infinite Impulse Response Filters 5.1 - 5.141
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1
5.2 Frequency Selective Filters . . . . . . . . . . . . . . . . . . . . . . 5.1
5.3 Design of Digital Filters from Analog Filters . . . . . . . . . . . . . 5.3
5.3.1 Digital versus analog filters . . . . . . . . . . . . . . . . 5.5
5.3.2 Advantages and disadvantages of digital filters . . . . . . 5.5
5.4 Analog Lowpass Filter Design . . . . . . . . . . . . . . . . . . . . 5.6
5.5 Analog lowpass Butterworth Filter . . . . . . . . . . . . . . . . . . 5.6
5.6 Steps to Design an Analog Butterworth Lowpass Filter . . . . . . . 5.14
5.7 Analog Lowpass Chebyshev Filters . . . . . . . . . . . . . . . . . . 5.17
5.7.1 Pole locations for Chebyshev filter . . . . . . . . . . . . 5.19
5.7.2 Chebyshev Type - 2 filter . . . . . . . . . . . . . . . . . 5.22
5.8 Comparison between Butterworth Filter and Chebyshev Filter . . . 5.23
5.9 Steps to Design an Analog Chebyshev Lowpass Filter . . . . . . . . 5.23
5.10 Frequency Transformation in Analog Domain . . . . . . . . . . . . 5.29
5.10.1 Lowpass to lowpass filter . . . . . . . . . . . . . . . . . 5.29
5.10.2 Lowpass to highpass . . . . . . . . . . . . . . . . . . . . 5.29
5.10.3 Lowpass to bandpass . . . . . . . . . . . . . . . . . . . . 5.30
5.10.4 Lowpass to bandstop . . . . . . . . . . . . . . . . . . . . 5.31
5.11 Design of Highpass, Bandpass and Bandstop Filters . . . . . . . . . 5.31
5.12 Design of IIR Filters from Analog Filters . . . . . . . . . . . . . . 5.33
5.12.1 Approximation of derivatives . . . . . . . . . . . . . . . 5.33
5.12.2 Design of IIR filter using impulse invariance technique . . 5.35
5.12.3 Design of IIR filter using bilinear transformation . . . . . 5.44
5.12.4 The matched z-transform . . . . . . . . . . . . . . . . . 5.48
5.13 Frequency Transformation in Digital Domain . . . . . . . . . . . . 5.52
5.13.1 Lowpass to lowpass . . . . . . . . . . . . . . . . . . . . 5.52
5.13.2 Lowpass to highpass . . . . . . . . . . . . . . . . . . . . 5.52
5.13.3 Lowpass to bandpass . . . . . . . . . . . . . . . . . . . . 5.52
5.13.4 Lowpass to bandstop . . . . . . . . . . . . . . . . . . . . 5.53
5.14 Realization of Digital Filters . . . . . . . . . . . . . . . . . . . . . 5.54
5.14.1 Direct form I realization . . . . . . . . . . . . . . . . . . 5.55
5.14.2 Direct form II realization . . . . . . . . . . . . . . . . . 5.58
5.14.3 Signal flowgraph . . . . . . . . . . . . . . . . . . . . . . 5.62
5.14.4 Transposition theorem and transposed structure . . . . . . 5.63
5.14.5 Cascade form . . . . . . . . . . . . . . . . . . . . . . . 5.65
5.14.6 Parallel form structure . . . . . . . . . . . . . . . . . . . 5.67
5.14.7 Lattice structure of IIR system . . . . . . . . . . . . . . . 5.72
5.14.8 Conversion from Lattice structure to direct form . . . . . 5.74
5.14.9 Conversion from direct form to Lattice structure . . . . . 5.75
5.14.10 Lattice-Ladder structure . . . . . . . . . . . . . . . . . . 5.76
Questions and Answers . . . . . . . . . . . . . . . . . . . . . . . 5.109
Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . 5.124
Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.125
MATLAB Programs . . . . . . . . . . . . . . . . . . . . . . . . . 5.127
Chapter 6 Finite Impulse Response Filters 6.1 - 6.157
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1
6.2 Linear Phase FIR Filters . . . . . . . . . . . . . . . . . . . . . . . 6.1
6.3 Frequency Response of Linear Phase FIR Filters . . . . . . . . . . . 6.5
6.4 Location of the Zeros of Linear Phase FIR Filters . . . . . . . . . . 6.13
6.5 The Fourier Series Method of Designing FIR Filters . . . . . . . . . 6.16
6.6 Design of FIR Filters using Windows . . . . . . . . . . . . . . . . 6.29
6.6.1 Rectangular window . . . . . . . . . . . . . . . . . . . 6.32
6.6.2 The Triangular or Bartlett window . . . . . . . . . . . . 6.35
6.6.3 Raised cosine window . . . . . . . . . . . . . . . . . . . 6.38
6.6.4 Hanning window . . . . . . . . . . . . . . . . . . . . . . 6.39
6.6.5 Hamming window . . . . . . . . . . . . . . . . . . . . . 6.40
6.6.6 Blackman window . . . . . . . . . . . . . . . . . . . . . 6.43
6.6.7 Kaiser window . . . . . . . . . . . . . . . . . . . . . . . 6.53
6.6.8 Summary of windows . . . . . . . . . . . . . . . . . . . 6.64
6.7 Digital Differentiator . . . . . . . . . . . . . . . . . . . . . . . . . 6.67
6.8 Hilbert Transformers . . . . . . . . . . . . . . . . . . . . . . . . . 6.74
6.9 Frequency Sampling Method of Designing FIR Filters . . . . . . . 6.80
6.9.1 Frequency sampling realization . . . . . . . . . . . . . . 6.81
6.9.2 Frequency response . . . . . . . . . . . . . . . . . . . . 6.81
6.9.3 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.82
6.10 Optimum Equiripple Approximation of FIR Filters . . . . . . . . . 6.99
6.11 Realization of FIR Filters . . . . . . . . . . . . . . . . . . . . . . . 6.102
6.11.1 Transversal structure . . . . . . . . . . . . . . . . . . . . 6.102
6.11.2 Linear phase realization . . . . . . . . . . . . . . . . . . 6.105
6.11.3 Lattice structure of an FIR filter . . . . . . . . . . . . . . 6.109
6.11.4 Polyphase realization of FIR filter . . . . . . . . . . . . . 6.115
Questions and Answers . . . . . . . . . . . . . . . . . . . . . . . 6.129
Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . 6.141
Multiple Choice Questions . . . . . . . . . . . . . . . . . . . . . 6.142
Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.144
MATLAB Programs . . . . . . . . . . . . . . . . . . . . . . . . . 6.146
Chapter 7 Finite Word Length Effects in Digital Filters 7.1 - 7.72
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1
7.2 Number Representation . . . . . . . . . . . . . . . . . . . . . . . . 7.2
7.3 Types of Number Representation . . . . . . . . . . . . . . . . . . . 7.3
7.3.1 Fixed point representation . . . . . . . . . . . . . . . . . 7.3
7.3.2 Sign-magnitude form . . . . . . . . . . . . . . . . . . . 7.3
7.3.3 One’s-complement form . . . . . . . . . . . . . . . . . . 7.3
7.3.4 Two’s-complement form . . . . . . . . . . . . . . . . . . 7.4
7.3.5 Addition of two fixed point numbers . . . . . . . . . . . 7.5
7.3.6 Multiplication in fixed point arithmetic . . . . . . . . . . 7.7
7.4 Floating Point Numbers . . . . . . . . . . . . . . . . . . . . . . . . 7.7
7.4.1 Comparison of fixed point and floating point arithmetic . 7.8
7.5 Block Floating Point Numbers . . . . . . . . . . . . . . . . . . . . 7.9
7.6 Quantization Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9
7.6.1 Truncation . . . . . . . . . . . . . . . . . . . . . . . . . 7.9
7.6.2 Rounding . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10
7.6.3 Error due to truncation and rounding . . . . . . . . . . . 7.10
7.7 Input Quantization Error . . . . . . . . . . . . . . . . . . . . . . . 7.15
7.7.1 Steady state input noise power . . . . . . . . . . . . . . . 7.16
7.7.2 Steady state output noise power . . . . . . . . . . . . . . 7.18
7.8 Product Quantization Error . . . . . . . . . . . . . . . . . . . . . . 7.21
7.9 Coefficient Quantization error . . . . . . . . . . . . . . . . . . . . 7.29
7.10 Zero-input Limit Cycle Oscillations . . . . . . . . . . . . . . . . . 7.30
7.10.1 Dead band . . . . . . . . . . . . . . . . . . . . . . . . . 7.31
7.11 Overflow Limit Cycle Oscillations . . . . . . . . . . . . . . . . . . 7.32
7.12 Signal Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.33
7.13 Quantization in Floating Point Realization of IIR Digital Filters . . . 7.37
7.14 Finite Word Length Effects in FIR Digital Filters . . . . . . . . . . 7.41
7.15 Quantization Effects in the Computation of the DFT . . . . . . . . . 7.43
7.16 Quantization Errors in FFT Algorithms . . . . . . . . . . . . . . . 7.45
Questions and Answers . . . . . . . . . . . . . . . . . . . . . . . 7.59
Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . 7.68
Multiple Choice Questions . . . . . . . . . . . . . . . . . . . . . 7.68
Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.71
Chapter 8 Multirate Signal Processing 8.1 - 8.54
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1
8.2 Down Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1
8.3 Spectrum of the Down Sampled Signal . . . . . . . . . . . . . . . . 8.2
8.4 Upsampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9
8.5 Spectrum of the Up-sampled Signal . . . . . . . . . . . . . . . . . 8.11
8.6 Anti-imaging Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 8.12
8.7 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.15
8.7.1 First identity . . . . . . . . . . . . . . . . . . . . . . . . 8.15
8.7.2 Second identity . . . . . . . . . . . . . . . . . . . . . . 8.15
8.7.3 Third identity . . . . . . . . . . . . . . . . . . . . . . . 8.16
8.7.4 Fourth identity . . . . . . . . . . . . . . . . . . . . . . . 8.17
8.7.5 Fifth identity . . . . . . . . . . . . . . . . . . . . . . . . 8.17
8.7.6 Sixth identity . . . . . . . . . . . . . . . . . . . . . . . . 8.18
8.8 Cascading Sample Rate Converters . . . . . . . . . . . . . . . . . . 8.19
8.9 Efficient Transversal Structure for Decimator . . . . . . . . . . . . 8.24
8.10 Efficient Transversal Structure for Interpolator . . . . . . . . . . . . 8.26
8.11 Polyphase Structure of Decimator . . . . . . . . . . . . . . . . . . 8.28
8.12 Polyphase Decimation using the z-transform . . . . . . . . . . . . . 8.31
8.13 Polyphase Structure of Interpolator . . . . . . . . . . . . . . . . . . 8.31
8.14 Polyphase Interpolation using the z-transform . . . . . . . . . . . . 8.33
8.15 Multistage Implementation of Sampling Rate Conversion . . . . . . 8.34
8.16 Implementation of Narrow Band Lowpass Filter . . . . . . . . . . . 8.35
8.17 Filter Banks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.35
8.17.1 Analysis filter bank . . . . . . . . . . . . . . . . . . . . 8.36
8.17.2 Synthesis filter bank . . . . . . . . . . . . . . . . . . . . 8.36
8.17.3 Subband coding filter bank . . . . . . . . . . . . . . . . 8.36
8.18 Quadrature - Mirror Filter (QMF) Bank . . . . . . . . . . . . . . . 8.37
8.18.1 Alias free filter bank . . . . . . . . . . . . . . . . . . . . 8.40
Questions and Answers . . . . . . . . . . . . . . . . . . . . . . . 8.44
Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . 8.46
Multiple Choice Questions . . . . . . . . . . . . . . . . . . . . . 8.46
MATLAB Programs . . . . . . . . . . . . . . . . . . . . . . . . . 8.48
Chapter 9 Statistical Digital Signal Processing 9.1 - 9.45
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1
9.2 Random processes . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2
9.3 Random Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2
9.4 Random variable . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3
9.5 Discrete-time Random Signals . . . . . . . . . . . . . . . . . . . . 9.3
9.6 Statistical Properties of Random Signal . . . . . . . . . . . . . . . 9.4
9.6.1 Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4
9.6.2 Mean square . . . . . . . . . . . . . . . . . . . . . . . . 9.6
9.6.3 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6
9.6.4 Autocorrelation of random process . . . . . . . . . . . . 9.7
9.6.5 Autocovariance of random process . . . . . . . . . . . . 9.7
9.6.6 Crosscorrelation of random processes . . . . . . . . . . . 9.7
9.6.7 Crosscovariance of random processes . . . . . . . . . . . 9.7
9.7 Wide-sense Stationary Random Process . . . . . . . . . . . . . . . 9.8
9.7.1 Power in a random signal . . . . . . . . . . . . . . . . . 9.11
9.7.2 Ergodic process . . . . . . . . . . . . . . . . . . . . . . 9.12
9.8 Power Density Spectrum . . . . . . . . . . . . . . . . . . . . . . . 9.12
9.9 The DTFT of the Crosscorrelation Sequence . . . . . . . . . . . . . 9.13
9.10 White Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.14
9.11 Response of LTI System Due to Random Signals . . . . . . . . . . 9.14
9.12 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.20
9.13 Estimation of Autocorrelation . . . . . . . . . . . . . . . . . . . . 9.23
9.14 The Periodogram . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.24
9.14.1 Average value of IN (ω) . . . . . . . . . . . . . . . . . . 9.25
9.14.2 Variance of the periodogram . . . . . . . . . . . . . . . . 9.26
9.15 The Use of DFT in Power Spectrum Estimation . . . . . . . . . . . 9.31
9.16 Performance Characteristics of Nonparametric Power Spectrum
Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.32
9.16.1 Periodogram . . . . . . . . . . . . . . . . . . . . . . . . 9.32
9.16.2 Bartlett’s power spectrum estimate . . . . . . . . . . . . 9.33
9.16.3 Welch power spectrum estimate . . . . . . . . . . . . . . 9.34
9.17 Computational Requirements of Nonparametric Power Spectrum
Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.36
9.17.1 Bartlett’s method . . . . . . . . . . . . . . . . . . . . . . 9.37
9.17.2 Welch’s method (50% overlap) . . . . . . . . . . . . . . 9.37
9.17.3 Blackman-Tukey method . . . . . . . . . . . . . . . . . 9.38
9.18 Advantages and Disadvantages of Nonparametric Power Spectrum
Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.39
Questions and Answers . . . . . . . . . . . . . . . . . . . . . . . 9.42
Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . 9.43
MATLAB Programs . . . . . . . . . . . . . . . . . . . . . . . . . 9.44
Chapter 10 Applications of Digital Signal Processing 10.1 - 10.12
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1
10.2 Speech Processing . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1
10.3 Speech Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2
10.4 Speech Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5
10.5 Subband coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6
10.6 Channel Vocoder . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7
10.7 Homomorphic Vocoder . . . . . . . . . . . . . . . . . . . . . . . . 10.8
10.8 Digital Processing of Audio Signals . . . . . . . . . . . . . . . . . 10.9
10.9 Radar Signal Processing . . . . . . . . . . . . . . . . . . . . . . . 10.10
10.10DSP Based Measurement System . . . . . . . . . . . . . . . . . . . 10.11
Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . 10.12
Chapter 11 Digital Signal Processors 11.1 - 11.76
11.1 Overview of Digital Signal Processors . . . . . . . . . . . . . . . . 11.1
11.2 Selecting Digital Signal Processors . . . . . . . . . . . . . . . . . . 11.5
11.3 Applications of PDSPs . . . . . . . . . . . . . . . . . . . . . . . . 11.6
11.3.1 Communications systems . . . . . . . . . . . . . . . . . 11.6
11.3.2 Audio signal processing . . . . . . . . . . . . . . . . . . 11.7
11.3.3 Control and data acquisition . . . . . . . . . . . . . . . . 11.7
11.3.4 Biometric Information Processing . . . . . . . . . . . . . 11.8
11.3.5 Image/video processing . . . . . . . . . . . . . . . . . . 11.8
11.4 Von Neumann Architecture . . . . . . . . . . . . . . . . . . . . . . 11.8
11.5 Harvard Architecture . . . . . . . . . . . . . . . . . . . . . . . . . 11.9
11.6 VLIW Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . 11.10
11.7 Multiply Accumulate Unit (MAC) . . . . . . . . . . . . . . . . . . 11.11
11.8 Pipelining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.13
11.9 Architecture of TMS320C50 . . . . . . . . . . . . . . . . . . . . . 11.15
11.9.1 Bus structure . . . . . . . . . . . . . . . . . . . . . . . . 11.16
11.9.2 Central processing unit . . . . . . . . . . . . . . . . . . 11.17
11.9.3 On-chip memory . . . . . . . . . . . . . . . . . . . . . . 11.21
11.9.4 On-chip peripherals . . . . . . . . . . . . . . . . . . . . 11.23
11.10 Addressing modes . . . . . . . . . . . . . . . . . . . . . . . . . . 11.25
11.10.1 Immediate addressing . . . . . . . . . . . . . . . . . . . 11.26
11.10.2 Indirect addressing . . . . . . . . . . . . . . . . . . . . . 11.26
11.10.3 Register addressing . . . . . . . . . . . . . . . . . . . . 11.26
11.10.4 Memory mapped register addressing . . . . . . . . . . . 11.27
11.10.5 Direct addressing mode . . . . . . . . . . . . . . . . . . 11.27
11.10.6 Circular addressing mode . . . . . . . . . . . . . . . . . 11.28
11.11 Instruction Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.28
11.11.1 Summary of instructions . . . . . . . . . . . . . . . . . . 11.28
11.11.2 Classification of instructions . . . . . . . . . . . . . . . . 11.34
11.12 Simple Assembly Language Programs . . . . . . . . . . . . . . . . 11.48
11.13 Architecture of TMS320C54x . . . . . . . . . . . . . . . . . . . . 11.60
11.13.1 Internal memory organization . . . . . . . . . . . . . . . 11.60
11.13.2 Overflow handling . . . . . . . . . . . . . . . . . . . . . 11.63
11.13.3 The carry bit . . . . . . . . . . . . . . . . . . . . . . . . 11.64
11.13.4 Dual 16-bit mode . . . . . . . . . . . . . . . . . . . . . 11.64
11.14 Accumulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.64
Questions and Answers . . . . . . . . . . . . . . . . . . . . . . . 11.72
Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . 11.75
Appendix A Mathematical Identities A.1 - A.2
A.1 Infinite Summation Formulas . . . . . . . . . . . . . . . . . . . . . A.1
A.2 Finite Summation Formulas . . . . . . . . . . . . . . . . . . . . . A.1
A.3 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2
A.4 Complex Variables . . . . . . . . . . . . . . . . . . . . . . . . . . A.2
A.5 Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . A.2