Part I The Analysis of Discrete-time Signals
1 Understanding Sampling ....................................... 3
1.1 The Sample-and-hold Operation ........................... 3
1.2 The Ideal Sampler in the Frequency Domain ............... 4
1.2.1 Representing the Ideal Sampler Using Complex
Exponentials: A Simple Approach .................. 4
1.2.2 Representing the Ideal Sampler Using Complex
Exponentials: A More Careful Approach ............ 5
1.2.3 The Action of the Ideal Sampler in the
Frequency Domain ................................. 9
1.3 Necessity of the Condition ............................. 10
1.4 An Interesting Example ................................. 11
1.5 Aliasing ............................................... 11
1.6 The Net Effect ......................................... 11
1.7 Undersampling .......................................... 14
1.8 The Experiment ......................................... 14
1.9 The Report ............................................. 15
1.10 Exercises .............................................. 15
2 Signal Reconstruction ....................................... 17
2.1 Reconstruction ......................................... 17
2.2 The Experiment ......................................... 18
2.3 The Report ............................................. 18
2.4 Exercises .............................................. 19
3 Time-limited Functions Are Not Band-limited ................. 21
3.1 A Condition for Analyticity ............................ 21
3.2 Analyticity Implies Lack of Compact Support ............ 23
3.3 The Uncertainty Principle .............................. 23
3.4 An Example ............................................. 24
3.5 The Best Function ...................................... 25
3.6 Exercises .............................................. 26
4 Fourier Analysis and the Discrete Fourier Transform ......... 29
4.1 An Introduction to the Discrete Fourier Transform ...... 29
4.2 Two Sample Calculations ................................ 30
4.3 Some Properties of the DFT ............................. 31
4.4 The Fast Fourier Transform ............................. 35
4.5 A Hand Calculation ..................................... 36
4.6 Fast Convolution ....................................... 37
4.7 MATLAB, the DFT, and You ............................... 37
4.8 Zero-padding and Calculating the Convolution ........... 39
4.9 Other Perspectives on Zero-padding ..................... 41
4.10 MATLAB and the Serial Port ............................. 42
4.11 The Experiment ......................................... 42
4.12 Exercises .............................................. 42
5 Windowing ................................................... 45
5.1 The Problems ........................................... 45
5.2 The Solutions .......................................... 47
5.3 Some Standard Window Functions ......................... 47
5.3.1 The Rectangular Window .......................... 48
5.3.2 The Triangular Window ........................... 48
5.4 The Raised Cosine Window ............................... 48
5.5 A Remark on Widths ..................................... 49
5.6 Applying a Window Function ............................. 49
5.7 A Simple Comparison .................................... 50
5.8 MATLAB's Window Visualization Tool ..................... 51
5.9 The Experiment ......................................... 51
5.10 Exercises .............................................. 51
6 Signal Generation with the Help of MATLAB ................... 53
6.1 Introduction ........................................... 53
6.2 A Simple Sinewave Generator ............................ 53
6.3 A Simple White Noise Generator ......................... 54
6.4 The Experiment ......................................... 54
6.5 Exercises .............................................. 55
7 The Spectral Analysis of Random Signals ..................... 57
7.1 The Problem ............................................ 57
7.2 The Solution ........................................... 58
7.3 Warm-up Experiment ..................................... 59
7.4 The Experiment ......................................... 61
7.5 Exercises .............................................. 63
Part II Analog to Digital and Digital to Analog Converters
8 The General Structure of Sampled-data Systems ............... 67
8.1 Systems for Spectral Analysis .......................... 67
8.2 Systems for Implementing Digital Filters ............... 68
9 The Operational Amplifier: An Overview ...................... 69
9.1 Introduction ........................................... 69
9.2 The Unity-gain Buffer .................................. 69
9.3 Why the Signal is Fed Back to V- ....................... 71
9.4 The "Golden Rules" ..................................... 71
9.5 The Inverting Amplifier ................................ 72
9.6 Exercises .............................................. 72
10 A Simple Digital to Analog Converter ........................ 75
10.1 The Digital to Analog Converter ....................... 75
10.2 Practical Difficulties ................................ 77
10.3 The Experiment ........................................ 78
10.4 Exercises ............................................. 79
11 The Binary Weighted DAC ..................................... 81
11.1 The General Theory .................................... 81
11.2 Exercises ............................................. 83
12 The R-2R Ladder DAC ......................................... 85
12.1 Introduction .......................................... 85
12.2 The Derivation ........................................ 85
12.3 Exercises ............................................. 87
13 The Successive Approximation Analog to Digital Converter .... 89
13.1 General Theory ........................................ 89
13.2 An Example ............................................ 90
13.3 The Sample-and-hold Subsystem ......................... 91
14 The Single- and Dual-slope Analog to Digital Converters ..... 93
14.1 The Single-slope Converter ............................ 93
14.2 Problems with the Single-slope A/D .................... 94
14.3 The Dual-slope A/D .................................... 95
14.4 A Simple Example ...................................... 95
14.5 Exercises ............................................. 96
15 The Pipelined A/D ........................................... 99
15.1 Introduction .......................................... 99
15.2 The Fully Pipelined A/D .............................. 100
15.3 The Experiment ....................................... 102
15.4 Exercises ............................................ 102
16 Resistor-chain Converters .................................. 103
16.1 Properties of the Resistor Chain ..................... 103
16.2 The Resistor-chain DAC ............................... 103
16.3 The Flash A/D ........................................ 104
16.4 Exercises ............................................ 105
17 Sigma—Delta Converters ..................................... 109
17.1 Introduction ......................................... 109
17.2 The Sigma-Delta A/D .................................. 110
17.3 Sigma-Delta A/Ds, Oversampling, and the Nyquist
Criterion ............................................ 112
17.4 Sigma-Delta DACs ..................................... 113
17.5 The Experiment ....................................... 113
Part III Digital Filters
18 Discrete-time Systems and the Z-transform .................. 117
18.1 The Definition of the Z-transform ..................... 117
18.2 Properties of the Z-transform ......................... 117
18.2.1 The Region of Convergence (ROC) ................ 117
18.2.2 Linearity ...................................... 119
18.2.3 Shifts ......................................... 119
18.2.4 Multiplication by k ............................ 119
18.3 Sample Transforms ..................................... 120
18.3.1 The Transform of the Discrete-time Unit Step
Function ....................................... 120
18.3.2 A Very Similar Transform ....................... 120
18.3.3 The Z-transforms of Two Important Sequences .... 121
18.3.4 A Two-sided Sequence ........................... 122
18.4 Linear Time-invariant Systems ......................... 122
18.5 The Impulse Response and the Transfer Function ........ 123
18.6 A Simple Example ...................................... 124
18.7 The Inverse Z-transform ............................... 124
18.7.1 Inversion by Contour Integration ............... 124
18.7.2 Inversion by Partial Fractions Expansion ....... 125
18.7.3 Using MATLAB to Help ........................... 126
18.8 Stability of Discrete-time Systems .................... 127
18.9 From Transfer Function to Recurrence Relation ......... 128
18.10 The Sinusoidal Steady-state Response of Discrete-
time Systems .......................................... 130
18.11 MATLAB and Linear Time-invariant Systems ............. 132
18.11.1 Individual Commands ........................... 132
18.11.2 The ltiview Command ........................... 134
18.12 Exercises ............................................ 134
19 Filter Types ............................................... 139
19.1 Finite Impulse Response Filters ....................... 139
19.2 Infinite Impulse Response Filters ..................... 140
19.3 Exercises ............................................. 140
20 When to Use С (Rather than Assembly Language) .............. 143
20.1 Introduction .......................................... 143
20.2 A Simple Low-pass Filter .............................. 143
20.3 A Comparison with an RC Filter ........................ 144
20.4 The Experiment ........................................ 145
20.5 Exercises ............................................. 145
21 Two Simple FIR Filters ..................................... 147
21.1 Introduction .......................................... 147
21.2 The Experiment ........................................ 149
21.3 Exercises ............................................. 149
22 Very-narrow-band Filters ................................... 151
22.1 A Very Simple Notch Filter ............................ 151
22.2 From Simple Notch to Effective Bandpass ............... 151
22.3 The Transfer Function ................................. 152
22.4 The Experiment ........................................ 152
22.5 Exercises ............................................. 153
23 Design of IIR Digital Filters: The Old-fashioned Way ....... 155
23.1 Analog Filter Design .................................. 155
23.2 Two Simple Design Examples ............................ 157
23.3 Why We Always Succeed in Our Attempts at Factoring .... 159
23.4 The Bilinear Transform ................................ 160
23.5 The Passage from Analog Filter to Digital Filter ...... 161
23.6 MATLAB and the Bilinear Transform ..................... 162
23.7 The Experiment ........................................ 164
23.8 Exercises ............................................. 164
24 New Filters from Old ....................................... 165
24.1 Transforming Filters .................................. 165
24.2 Functions that Take the Unit Circle into Itself ....... 165
24.3 Converting a Low-pass Filter into a High-pass
Filter ................................................ 167
24.4 Changing the Cut-off Frequency of an Existing Low-
pass Filter ........................................... 168
24.5 Going from a Low-pass Filter to a Bandpass Filter ..... 170
24.6 The Experiment ........................................ 171
24.7 The Report ............................................ 171
24.8 Exercises ............................................. 172
25 Implementing an IIR Digital Filter ......................... 173
25.1 Introduction .......................................... 173
25.2 The Direct Form I Realization ......................... 174
25.3 The Direct Form II Realization ........................ 175
25.4 Trouble in Paradise ................................... 175
25.5 The Solution: Biquads ................................. 177
25.6 Exercises ............................................. 178
26 IIR Filter Design Using MATLAB ............................. 181
26.1 Individual Commands ................................... 181
26.2 The Experiment: Part I ................................ 183
26.3 Fully Automatic Filter Design ......................... 183
26.4 The Experiment: Part II ............................... 183
26.5 Exercises ............................................. 184
27 Group Delay and Phase Delay in Filters ..................... 185
27.1 Group and Phase Delay in Continuous-time Filters ...... 185
27.2 A Simple Example ...................................... 187
27.3 A MATLAB Experiment ................................... 187
27.4 Group Delay in Discrete-time Systems .................. 188
27.5 Exercises ............................................. 189
28 Design of FIR Filters ...................................... 191
28.1 FIR Filter Design ..................................... 191
28.2 Symmetric FIR Filters ................................. 194
28.3 A Comparison of FIR and IIR Filters ................... 197
28.4 The Experiment ........................................ 197
28.5 Exercises ............................................. 198
29 Implementing a Hilbert Filter .............................. 199
29.1 An Introduction to the Hilbert Filter ................. 199
29.2 Problems and Solutions ................................ 200
29.3 The Experiment ........................................ 200
29.4 Exercises ............................................. 201
30 The Goertzel Algorithm ..................................... 203
30.1 Introduction .......................................... 203
30.2 First-order Filters ................................... 203
30.3 The DFT as the Output of a Filter ..................... 204
30.4 Comparing the Two Methods ............................. 205
30.5 The Experiment ........................................ 206
30.6 Exercises ............................................. 206
References .................................................... 207
Index ......................................................... 209
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