Verified Signal Processing Algorithms in Matlab and C

Verified Signal Processing Algorithms in Matlab and C
اسم المؤلف
Arie Dickman
التاريخ
23 أبريل 2022
المشاهدات
108
التقييم
(لا توجد تقييمات)
Loading...

Verified Signal Processing Algorithms in Matlab and C
Arie Dickman
Advised by Israel Greiss
1 Overview of Signal Detection . 1
1.1 Introduction . 1
1.2 About Signal Detection 1
1.3 Conclusions . 8
2 Measures for Verification and Implementation of Algorithms 9
2.1 Introduction . 9
2.2 Measures Details . 9
2.2.1 Algorithm Verification in Frequency 9
2.2.2 Preparing Coefficients File for C Software . 11
2.2.3 Algorithm Verification in Time 12
2.2.4 Algorithm Transient Time Measurement . 14
2.2.5 Algorithm Processing Time Measurement . 15
2.3 Conclusions . 15
3 Narrow Band & Wide Band Processing Basics 17
3.1 Narrow Band Processing . 18
3.1.1 Introduction . 18
3.1.2 Numerically Controlled Oscillator 18
3.1.3 Finite Impulse Response Decimators 21
3.1.4 Low Decimation Ratio Finite Impulse Response
Filters 22
3.1.5 Higher Decimation Ratio Finite Impulse Response
Filters 29
3.1.6 Conclusions . 36
3.2 Wide Band Processing . 37
3.2.1 Introduction . 37
3.2.2 Shaping Windows 37
3.2.3 Designing Self Made Windows 44
3.2.4 Fast Fourier Transform . 46
Contentsxii
3.2.5 Fast Fourier Transform Implementation . 46
3.2.6 Implementing Large Fast Fourier Transform by Smaller
Ones . 54
3.2.7 Conclusions . 57
References . 58
4 Complex Finite Impulse Response Filters 59
4.1 Introduction . 59
4.2 Complex Finite Impulse Response Filter Design 59
4.2.1 Modified Low Pass Filter Design Method 60
4.2.2 Matlab Direct Complex Filter Design Method 61
4.2.3 Preferred Design Method . 62
4.2.4 Property of Complex Filters . 62
4.2.5 Implementation of Complex Filters . 63
4.3 Conclusions . 63
5 Infinite Impulse Response Filters . 65
5.1 Introduction . 65
5.2 Infinite Impulse Response Filter Design . 65
5.2.1 Low Pass Filter Design 67
5.2.2 High Pass Filter Design 75
5.2.3 Band Stop Filter Design 78
5.2.4 All Pass Filter Design . 81
5.2.5 Band Pass Filter Design 83
5.3 Conclusions . 87
6 Non-Linear and Batch Filters 89
6.1 Introduction . 89
6.2 Median Filter 89
6.3 Batch Filter with Comparison to Median Filter 92
6.4 Conclusions . 93
7 Interpolators 95
7.1 Introduction . 95
7.2 Interpolator Design . 95
7.2.1 Fractional Interpolation/Decimation 98
7.3 Conclusions . 106
8 Direct Digital Synthesis . 107
8.1 Introduction . 107
8.2 Direct Digital Synthesizer Details 107
8.2.1 Frequency Modulation of a Direct Digital Synthesizer 109
8.2.2 Receiver—Frequency Demodulator for Verification 111
8.2.3 Amplitude Modulation of a Direct Digital Synthesizer 113
8.2.4 Receiver—Amplitude Demodulator for Verification 115
8.3 Conclusions . 117
Contentsxiii
9 Inverse Fast Fourier Transform . 119
9.1 Introduction . 119
9.2 Inverse Fast Fourier Transform Details 119
9.3 Inverse Fast Fourier Transform Implementation . 121
9.4 Frequency Modulation of Inverse Fast Fourier Transform
Output . 122
9.5 Conclusions . 124
Reference . 124
10 Hilbert Transform 125
11 Channelizer . 127
11.1 Introduction 127
11.2 Channelizer Fundamentals . 127
11.2.1 Comparing Channelizer and Narrow Band
Processing Times 129
11.2.2 Channelizer Design Considerations 130
11.3 Conclusions 134
12 Correlation 135
12.1 Introduction 135
12.2 Correlation Details . 135
12.2.1 Fast Correlation by Using Fast Fourier Transform 138
12.3 Pattern Recognition 140
12.4 Conclusions 144
13 Adaptive Filters 145
13.1 Introduction 145
13.2 Equalizers 145
13.2.1 Least Mean Squares Algorithm . 146
13.2.2 Normalized Least Mean Squares Algorithm . 150
13.2.3 Multi-input Equalizer . 151
13.3 Interference Cancellation 154
13.3.1 Least Mean Squares Cancelling Algorithm 156
13.3.2 Recursive Least Squares Cancelling Algorithm . 157
13.4 Conclusions 158
Reference . 159
14 Unequally Spaced Sampling 161
14.1 Introduction 161
14.2 Modified Lomb–Scargle Algorithm 161
14.3 Algorithm Performance . 165
14.4 Algorithm Quality Factor 166
14.5 Algorithm Sensitivity to Interference 167
14.6 Algorithm Implementation in C . 169
14.7 Conclusions 171
References . 171
Contentsxiv
15 Encountered Problems in Signal Processing & Solutions 173
15.1 Introduction 173
15.2 When Input Signal Includes Spikes 173
15.3 When Input Signal Includes Fast Transitions 178
15.4 Algorithms Processing Time Improvement 179
15.5 Mean Center Frequency Estimation of Signals . 184
15.5.1 Mean Center Frequency Estimation for Multiple
Signals 184
15.5.2 Mean Center Frequency Estimation for a Single
Channel . 186
15.6 Power Level Estimation . 189
15.7 Noise Floor Estimation . 191
15.8 Bandwidth Estimation 195
15.9 Automatic Level Control 200
15.10 Frequency Response Estimation 203
15.11 Group Delay Estimation 207
15.12 Conclusions 209
Related Software . 211
Index . 213
Contents
A
A2d, 2, 9, 17, 18, 20, 22, 29, 37, 189, 200
Adaptive, 145, 159
Adaptive filters, 145–159
Algorithms, 9, 12–15, 46, 57, 58, 99, 102, 104,
105, 119–121, 127, 131, 145–151, 154,
156–159, 161–171, 179, 180, 182, 186,
187, 191, 194, 198, 200, 202, 203,
205–207, 209
Alias, 18, 19, 102
All pass, 81–83
AM, 108, 115, 116, 190, 197
Amplitude, 107, 113–117, 120, 121, 129, 153,
167, 175, 176, 203, 204
Analog, 17
Antipodal, 137
Ap, 22, 23, 29, 33, 45, 59, 60, 68, 84, 85, 97,
99, 100, 103, 112, 116
Arrays, 12, 13, 21, 37, 46, 47, 51, 52, 63,
91, 95, 99, 108, 120, 135–140,
169, 206
Ast, 22, 29, 99
Attenuation, 22, 26, 34, 71, 93, 101, 103,
157, 158
Automatic level control (ALC), 189,
200–203, 209
Autoscaling, 46, 121
Average, 3, 90, 200
B
Band, 2, 5, 17–58, 78, 83, 85, 89, 186,
195, 198
Band pass, 83–87
Band stop, 78–81
Bandwidth, 1, 2, 9, 17, 22, 29, 38,
42, 79, 128, 131, 134, 191,
195–200, 209
Baseband, 17, 18
Batch filters, 89–93
Benchmark, 37, 63
Bin, 2, 3, 5, 18, 120, 121, 124, 130, 131, 184,
189, 194
Binary, 12, 13
Biquads, 68, 76, 79, 85
Bit, 20, 46, 47, 121
Bit reversed order, 47, 49
Bro, 47
Butterflies, 46, 121
Butterworth, 65
C
Cancellation, 145
Cascade, 31, 33, 34
Cascaded integrator comb (CIC), 21, 22,
29–33, 36, 89
Center, 144
Center frequency, 184–188, 209
cfirpm, 61, 62
Channelizer, 127–134
Channels, 2, 15, 17, 18, 36–38, 42–44, 57,
127, 129–134, 145, 146, 148, 151, 154,
158, 186–188
Chebwin, 39, 42–45, 130, 131
Clip, 175
Clocks_per_sec, 15
Clock_t, 15
Index214
Codes, 2, 3, 5, 14, 15, 18, 20–22, 30, 34, 35,
37, 38, 40, 43, 44, 47, 51, 52, 54–56,
59–61, 67, 69, 70, 73, 75–80, 83, 86,
87, 89–91, 98, 99, 102–104, 106–111,
113–115, 121–123, 125–127, 129, 131,
132, 135, 137, 139, 140, 142, 144, 147,
150, 154, 156–158, 162, 163, 169–171,
175–177, 180, 182, 184, 187–189, 191,
192, 194, 196, 198, 200, 202–209
Coefficients, 11–12, 20, 22, 24, 26, 27, 29, 32,
33, 35–37, 39, 40, 43–45, 60–63, 66,
68–70, 73, 74, 76, 77, 79–81, 85–87, 97,
99, 101–103, 108, 134, 147, 148, 156
Coefficients file, 11–12, 20, 27, 35, 36, 60, 62,
63, 66, 69, 71, 73, 76, 99
Coherent, 5–8
Compensator, 29, 34–36, 67
Complex, 5, 18, 20, 21, 37, 46, 47, 51, 52,
58–63, 99, 119, 120, 125–128, 137,
139, 203, 205, 209
Conj, 63, 121, 139, 205, 209
Conjugate, 62
Constraint, 21
Convergence, 146, 147, 153, 156, 158
Convolution, 33
Correlations, 135–144
Cos, 20
CPU, 9, 15, 26, 29, 32, 33, 35–37, 40, 43, 44,
51, 54, 67, 73, 76, 80, 83, 86, 91, 98,
102, 104, 106, 129, 130, 134, 138–140,
144, 148, 151, 171, 177, 180, 184, 188,
194, 203, 207
Cramer – Rao bound, 5
D
db, 1, 2, 4, 6, 22, 27, 29, 30, 34, 35, 39, 42, 43,
57, 65, 81, 92, 105, 106, 115, 130, 131,
146, 153, 155, 156, 166, 174, 175, 182,
191, 192, 194, 196, 199, 200
Dbm, 1, 179, 182, 184, 189, 191–194,
196, 202
Decimal, 179, 196
Decimation, 9, 18, 21–23, 29–31, 33, 36, 43,
44, 77, 81, 87, 95, 98–106
Decimation in frequency, 47, 51
Decimators, 18, 21–22, 29–34, 36, 130, 131
Delay, 73, 74, 79, 89, 90, 92, 93, 135, 140, 144
Demodulate, 107, 117, 128
Demodulation, 157, 203
Demodulator, 63, 111–113, 115–117, 187, 203
Designs, 18, 21–24, 29–36, 44, 57, 59–63,
65–87, 95–107, 117, 130–134
Detections, 1–8, 140, 191
Difference equations, 65, 81
Digital, 17, 18, 37, 107–117, 127
Digital direct synthesizer (DDS), 107–109,
111, 113, 115, 117
Digital down converter (DDC), 2
Digitally controlled attenuator
(DCA), 200
Digit reversed order, 51
Direct form, 65, 67, 81
Distort, 90, 176, 200
Doppler, 144
Doubles, 22, 63, 92, 137, 171
Dro, 52, 54
Droop, 22, 29
DSP, vii
Dynamic, 17, 46, 121, 173, 176, 200
E
Ellip, 68, 76, 79, 85
ENOB, 17
Equalizers, 145–154, 158
Estimation, 5, 146, 153, 158, 162, 165–167,
184–200, 203–209
Exponent, 46, 120, 205, 209
F
False alarms, 1, 3, 8
False detections, 2
Fast Fourier transform (FFT), 2, 5, 18, 37, 38,
41, 44, 46–58, 96, 119–121, 124–126,
129–132, 144, 161, 162, 171, 173, 184,
186, 191, 195, 200
fdesign, 22, 44, 59, 60, 67
fftshift, 51
Figure of merit, 137
Files, 12, 13
Filters, 17, 21–37, 44, 59–63, 65–87, 89–93,
95–97, 99–105, 126, 145, 156, 157,
186, 196, 203, 204, 207
Filtfilt, 92
Finite impulse response (FIR), 22, 24, 27, 29,
32–36, 44, 45, 59, 63, 67–69, 73–81,
83, 85–87, 89, 92, 99, 134, 146, 148,
156, 157
Firls, 22
Firpm, 22
Fixed point, 26, 27, 31, 36, 39, 40, 42–44, 66,
67, 70, 73, 75–77, 79, 82, 86
Flattopwin, 44, 130–132
Float, 12, 39, 169
Index215
Floating point, 24, 26, 27, 29, 40, 43, 44, 46,
67, 69, 73, 76, 79, 80, 86, 98, 102, 104,
106, 121, 182
Floor, 20, 100, 173, 174, 191–195
Fp, 21–23, 29, 30, 33, 34, 45, 60, 68, 75, 92,
97, 99–103, 112, 116
FPGA, 134, 179
Fractional, 95, 98–106
Frames, 5–7
Frequencies, 2, 9, 17–24, 29, 46, 60, 65, 73,
74, 78, 79, 83, 89, 90, 92, 95–99, 103,
105–109, 111–113, 117, 120, 122–130,
134, 135, 144, 161, 162, 165–167, 169,
171, 184–187, 189, 191, 195, 203, 207
Frequency modulation (FM), 108–112, 122,
124, 127–129, 154, 184, 186, 187, 190
Frequency response, 11, 23, 24, 27, 30, 34, 38,
39, 41, 42, 44, 45, 60–62, 69, 76, 79,
82, 85, 86, 96, 98, 111, 115, 124, 134,
203–207, 209
Fs, 2, 6, 17, 19–21, 41, 59–61, 63, 68, 72, 74,
78, 81, 84, 85, 92, 95–100, 103,
105–110, 113, 114, 116, 120, 122, 123,
125, 127–130, 133, 155, 156, 163, 164,
174, 179, 184, 185, 187–189, 192, 197,
200, 201, 204, 208
Fstop, 22, 23, 30, 34
G
Gain, 2, 22, 27, 67, 81, 82
Geometric, 32
Group delay, 73–75, 77, 80–81, 87, 207–209
H
H file, 11, 20, 29, 39, 40, 47, 49, 52, 66,
69–71, 76, 79, 86, 108, 169, 180
High pass, 75–77
Higher decimation, 29–36
Hilbert transform, 125–126
HPF, 77
I
IF, 1, 9, 17, 18, 22, 29, 127, 191, 195,
196, 204
imag, 21, 47, 51, 63, 137, 169
Images, 18, 19, 21, 96
Implements, 20, 37, 63, 66, 69, 76, 79, 86, 95,
103, 104, 106–109, 113, 117, 119, 121,
124, 127, 131, 138, 144, 159, 170, 179,
187, 203, 207
Include, 61, 62, 67, 75, 78, 83, 96, 144, 162,
173–180, 187, 196
Increment, 21, 108
Infinite impulse response (IIR), 65–87, 89,
140, 148
In-place, 51
Input, 1, 12, 21, 24, 27, 29–32, 35–37, 43, 44,
46, 47, 63, 65, 73, 89, 91, 92, 95,
97–99, 101, 102, 120, 126, 127, 130,
145, 148, 151, 153, 161, 162, 173–179,
189, 200, 202, 206, 207, 209
int, 29, 71, 75, 77, 169, 202
Integration, 35
Interference, 145, 154, 156–158, 167–169
Interference cancellation, 145, 154–158
Interleave, 139, 169, 202
Interp1, 104–106
Interpolation, 95, 96, 98–106
Interpolator, 95–106
Inverse discrete Fourier transform (IDFT),
119, 120
Inverse fast Fourier transform (IFFT), 119–126
Iteration, 148, 151
J
Jitter, 17
L
Lag, 67
Latency, 13
Leads, 36, 67
Least mean square (LMS), 146, 147,
149–151, 156–159
Linear, 65, 74, 89, 104, 106
Lomb–Scargle, 161–165
Loop, 26, 40, 67, 78, 79
Low decimation, 22–29, 32
Low pass, 44, 60–62, 67–75
Low pass filter (LPF), 18, 21, 60, 76, 95, 98,
126, 184
LSB, 63
M
Main, 163
Margins, 67, 79
Mean, 22, 74, 137, 146–150, 156–157,
175, 184–188
Mean square error, 145, 146, 153
Measures, 9–15, 23, 166, 175, 195, 196, 198
Median, 89–93
Index216
Memory, 9, 15, 179
Modulation, 108, 109, 111, 113–115,
122–124, 184
Multi input equalizer, 153, 154
Multi path, 154
Multiple, 31, 99, 101, 103, 127, 184–186,
189, 190
Multirate, 21, 22, 29
N
Narrow, 78, 83
Narrow band, 2, 17–58, 127, 129–130, 134
Noise, 1–4, 6, 8, 67, 89, 92, 93, 108, 137, 154,
157, 191, 195
Noise, 1–4, 7, 20, 67, 89, 92, 93, 108, 137,
154, 155, 157, 164, 173–174, 191–197,
199, 209
Noise floor, 20, 173, 174, 191–196, 198, 209
Non coherent, 5–7
Non linear, 89–93
Normal distribution, 2
Normalization, 153, 189
Normalized, 65, 150
Normalized Least Mean Squares (NLMS),
150–151, 153, 158
Notch, 78–81
Numerically controlled oscillator (NCO),
18–22, 107, 126, 130, 131, 184, 186,
205, 209
Nyquist, 9, 17, 21, 101
O
Offline, 93, 163, 169
Online, 163, 170
Optimal, 35
Order, 1, 2, 5, 18, 22, 29, 31–33, 36, 37,
40–42, 44, 47, 51–52, 65–68, 73, 76,
77, 79, 80, 85, 86, 90, 92, 101, 111,
115, 130, 134, 146–148, 150, 151, 153,
158, 162, 178, 184, 187, 190, 191, 195
Output, 13, 14, 17, 20, 21, 26, 29, 30, 32, 36,
37, 40, 46, 47, 51, 63, 65, 71, 73, 74,
92, 95, 99, 102, 105, 107, 108,
110–112, 114–116, 120–124, 126, 136,
139, 151, 189, 202, 204
P
Parabolic approximation, 192
Parseval, 189
Passband, 18, 21, 22, 29, 33–35, 44, 59–62,
65, 130, 195
Pattern recognition, 135, 140–144
Peaks, 121, 144, 192, 200
Peak to average, 121
Periodogram, 161, 162, 166, 171
Phase, 21, 47, 51, 67, 74, 79, 81, 82, 108, 121,
125, 129, 184, 186, 203, 204
Plomb, 161, 165, 166, 171
Polyphase, 99–104
Power, 1–3, 5, 6, 47, 51, 96, 137, 178, 179,
182, 184–186, 189–192, 194, 196, 198,
200, 202, 203, 209
Probability, 1–3, 5, 8
Processing, 2, 5–8, 17–58, 127, 130, 131, 134,
138, 173–209
Processing time, 5, 9, 15, 21, 22, 26, 27, 29,
32, 36, 37, 40, 43, 44, 51, 54–56, 63,
67, 73, 74, 76, 77, 80, 81, 83, 86, 87,
91, 98, 102, 104, 106, 110, 114, 121,
123, 126, 127, 129–131, 134, 135,
137–140, 144, 148, 151, 158, 159, 171,
177, 179–184, 188, 194, 203, 207, 209
Q
Quality factor, 166–167
Quantization, 2, 66, 67, 73, 74, 77, 80,
81, 86, 87
R
Radix, 46–49, 51, 120, 121
Radix 2, 46–51, 54, 57, 120, 121
Radix 4, 46, 51–54, 57, 120, 121
Range, 2, 17, 23, 27, 46, 51, 81, 121, 173, 176,
184, 186, 187, 191, 200
Real, 21, 37, 46, 47, 51, 58–60, 63, 99, 107,
120, 125, 126, 134, 137, 145, 156, 157,
169, 182, 189
Receivers, 1, 2, 9, 17, 22, 29, 111–113,
115–117, 184, 189, 191, 195, 200, 204
Recursive, 157–158
Recursive Least Squares (RLS), 157–158
References, 135, 137, 138, 140, 144, 203
Rejection, 27, 29, 37–39, 42, 44, 57, 79, 92,
93, 130, 131, 134
Remez, 22
Resamples, 95, 98, 99, 102, 105, 106
Resolution, 15, 20, 23, 46, 108, 121, 161, 179,
184, 192
Resonances, 78, 79
Resources, 9, 15, 58, 77, 131, 134, 169
Responses, 9, 14, 21–37, 44, 59–63, 67, 68,
76, 79, 81, 85, 92, 96, 97, 101, 103,
112, 116, 119, 120, 126, 129, 130, 133,
Index217
142, 145, 146, 148, 151, 153, 154, 156,
157, 185, 202–204, 207, 208
RF, 1, 2, 17, 187
Ripple, 22, 34, 35, 65, 101, 103
Round, 21, 40, 43, 66, 70, 73, 76, 77, 79, 80,
82, 86, 108
Rules of thumb, 23
S
Samples, 5, 14, 17, 18, 21, 22, 26, 29, 35–37,
40, 43, 44, 67, 73, 74, 76, 80, 83, 86,
89–92, 95, 96, 98, 102, 104–106, 108,
110, 114, 120, 123, 130, 131, 134–142,
144, 161, 162, 171, 174–177, 188, 202,
203, 207
Sampling, 2, 17, 20–22, 26, 29, 32, 36, 38, 41,
65, 95, 96, 98, 99, 102, 106, 108,
127–131, 134, 161–171, 173, 184, 187
Second order section (SOS), 67–69, 76,
79, 85, 86
Sections, 2, 3, 5, 9, 11, 14, 15, 67–69, 71, 73,
76, 77, 79, 80, 85, 86, 202
Self made windows, 44–46, 57
Sensitive, 17, 77
Sequence, 32, 46, 47, 66, 120, 146
Shape factor, 22, 29
Shaping windows, 2, 18, 37–44, 57, 130,
131, 190
Shift, 18, 60, 71, 76, 79, 86, 126, 140,
141, 144
Short, 20–22, 108, 140, 161, 187
Signal to noise ratio (SNR), 1–6, 8, 17, 106,
157, 165–167, 191, 194, 195, 198
Signals, 1–8, 17, 22, 37, 46, 89, 90, 96, 97, 99,
101, 102, 109, 113, 119, 125, 126, 128,
135, 144, 156, 157, 162, 163, 173–209
Sin, 20, 107, 109, 113, 122
Sine, 9, 12, 37, 107, 126
Snapshot, 5
Software, vii, 11–13, 21, 36, 63, 108, 131,
134, 179
Specifications, 9, 15, 44, 67, 70, 73–81,
83, 86, 87
Spectral, 2, 37, 57, 130, 178
Spectrum, 20, 46, 96, 97, 99, 110, 114, 119,
123, 146, 191–193, 195, 196, 198, 200
Spikes, 89, 173–177, 209
Spurs, 17, 20, 108
Stability, 67
Stages, 9, 21, 29–31, 33, 36
State, 32, 69, 76, 79, 148, 153
std, 111, 115, 121, 166, 167, 169
Step, 3, 14, 26, 40, 47, 129, 146
Stopband, 18, 22, 26, 27, 29, 101, 103
Streams, 17, 18, 127, 128, 186
Sweeps, 9
Switch, 71, 76, 79, 86
Synthesizer, 107–117
T
Target, 12, 15, 37, 63
Thermal noise, 1, 2
Thresholds, 2–5, 8, 140, 176, 197
tic, 55, 56, 158
Times, 3, 5, 9, 14–15, 18, 22, 26, 36, 37, 44,
47, 55, 56, 63, 67, 74, 77, 78, 81, 87,
89, 92, 95, 98, 102, 106, 112, 116,
119–121, 125–127, 129–131, 134,
140–142, 144–146, 153, 158, 161, 162,
169, 175, 179, 184, 189, 200, 202
toc, 55, 56, 158
Toolbox, 22
Transfer function, 65
Transient, 9, 14–15, 89, 92, 112, 116, 140
Transitions, 178–179, 209
Trial and error, 23, 33
Twiddle factors, 46–49, 52, 120
U
Uncertainty, 79
Unequal, 161
Unequally spaced, 161–171
Uniform distribution, 3, 4
Unwrap, 205–207, 209
USB, 63
V
Variables, 9, 32, 69, 76, 79, 144, 145, 169, 200
Variance, 147, 150
Verification, 9–15, 22–24, 33, 107,
111–113, 115–117
W
Wave, 9, 12, 37, 107
Wide, 2, 5, 17–58, 146, 186, 195, 198
Windows, 37–45, 129–132, 162, 173, 178,
184, 185, 190
WOLA, 40–44, 129–132, 134, 173, 178, 184,
185, 191, 195, 200
Z
Zero order hold (ZOH), 96
Zone, 9, 17, 89, 130, 195–197
Z transform, 22, 29, 32
Index

كلمة سر فك الضغط : books-world.net
The Unzip Password : books-world.net

تحميل

يجب عليك التسجيل في الموقع لكي تتمكن من التحميل
تسجيل | تسجيل الدخول

التعليقات

اترك تعليقاً