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i
Statistical Signal Processing
An Introduction to
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September 10, 2008
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ii
An Introduction to
Statistical Signal Processing
Robert M. Gray
and
Lee D. Davisson
Information Systems Laboratory
Department of Electrical Engineering
Stanford University
and
Department of Electrical Engineering and Computer Science
University of Maryland
c 2004 by Cambridge University Press. Copies of the pdf file may be
downloaded for individual use, but multiple copies cannot be made or printed
without permission.
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to our Families
Contents
page ix
xii
xiii
1
10
2.1
Introduction
10
2.2
Spinning pointers and flipping coins
14
2.3
Probability spaces
22
2.4
Discrete probability spaces
45
2.5
Continuous probability spaces
54
2.6
Independence
69
2.7
Elementary conditional probability
70
2.8
Problems
74
83
3.1
Introduction
83
3.2
Random variables
94
3.3
Distributions of random variables
103
3.4
Random vectors and random processes
113
3.5
Distributions of random vectors
116
3.6
Independent random variables
125
3.7
Conditional distributions
128
3.8
Statistical detection and classification
133
3.9
Additive noise
136
143
145
146
152
153
v
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vi
Contents
157
159
168
169
183
4.1
Averages
183
4.2
Expectation
186
4.3
Functions of random variables
189
4.4
Functions of several random variables
196
4.5
Properties of expectation
196
4.6
Examples
198
4.7
Conditional expectation
207
4.8
⋆Jointly Gaussian vectors
210
4.9
Expectation as estimation
212
219
222
229
232
235
237
244
246
251
256
259
277
5.1
Linear filtering of random processes
278
5.2
Linear systems I/O relations
280
5.3
Power spectral densities
286
5.4
Linearly filtered uncorrelated processes
288
5.5
Linear modulation
295
5.6
White noise
298
5.7
⋆Time averages
302
5.8
⋆Mean square calculus
305
5.9
⋆Linear estimation and filtering
333
351
365
6.1
Discrete time linear models
366
6.2
Sums of iid random variables
371
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