statistical digital signal processing
TRANSCRIPT
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Statistical
SignalProcessing
ECE 5615 Lecture Notes
Spring 2015
© 2007–2015Mark A. Wickert
Equalizer
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Chapter 1Course Introduction/Overview
Contents1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1-3
1.2 Where are we in the Comm/DSP Curriculum? . . . . 1-4
1.3 Instructor Policies . . . . . . . . . . . . . . . . . . . . 1-6
1.4 The Role of Computer Analysis/Simulation Tools . . . 1-7
1.5 Required Background . . . . . . . . . . . . . . . . . . 1-8
1.6 Statistical Signal Processing? . . . . . . . . . . . . . . 1-9
1.7 Course Syllabus . . . . . . . . . . . . . . . . . . . . . 1-10
1.8 Mathematical Models . . . . . . . . . . . . . . . . . . 1-11
1.9 Engineering Applications . . . . . . . . . . . . . . . . 1-13
1.10 Random Signals and Statistical Signal Processing in
Practice . . . . . . . . . . . . . . . . . . . . . . . . . . 1-14
1-1
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CHAPTER 1. COURSE INTRODUCTION/OVERVIEW
.
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1.1. INTRODUCTION
1.1 Introduction
Course perspective
Course syllabus
Instructor policies
Software tools for this course
Required background
Statistical signal processing overview
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1.3. INSTRUCTOR POLICIES
1.3 Instructor Policies
Working homework problems will be a very important aspect
of this course Each student is to his/her own work and be diligent in keeping
up with problems assignments
Homework papers are due at the start of class
If work travel keeps you from attending class on some evening,
please inform me ahead of time so I can plan accordingly, andyou can make arrangements for turning in papers
The course web site
will serve as an information source between weekly class meet-
ings Please check the web site updated course notes, assignments,
hints pages, and other important course news; particularly ondays when weather may result in a late afternoon closing of thecampus
Grading is done on a straight 90, 80, 70, ... scale with curving
below these thresholds if needed
Homework solutions will be placed on the course Web site inPDF format with security password required; hints pages mayalso be provided
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CHAPTER 1. COURSE INTRODUCTION/OVERVIEW
1.4 The Role of Computer Analysis/
Simulation Tools
In working homework problems pencil and paper type solu-tions are mostly all that is needed
– It may be that problems will be worked at the board bystudents
– In any case pencil and paper solutions are still required tobe turned in later
Occasionally an analytical expression may need to be plotted,here a visualization tool such as Python (via IPython consoleor notebook), MATLAB or Mathematica will be very helpful
Simple simulations can be useful in enhancing your under-standing of mathematical concepts
The use of Python for computer work is encouraged since it isfast and efficient at evaluating mathematical models and run-ning Monte-Carlo system simulations
There will be one or more Python or MATLAB based simula-tion projects, and the Hayes text supports this with MATLABbased exercises at the end of each chapter
– I intend to rework MATLAB examples in Python, includ-ing converting code from the text to freely available Python
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1.5. REQUIRED BACKGROUND
1.5 Required Background
Undergraduate probability and random variables (ECE 3610 or
similar) Linear systems theory
– A course in discrete-time linear systems such as, ECE5650 Modern DSP, is highly recommend (a review is pro-vided in Chapter 2)
Familiarity with linear algebra and matrix theory, as matrix no-tation will be used throughout the course (a review is providedin Chapter 2)
A desire to use the IPython Notebook, but falling back to MATLABor Mathematica if needed. Homework problems will requireits use of some vector/matrix computational tool, and a rea-sonable degree of proficiency will allow your work to proceed
quickly
A desire to dig in!
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CHAPTER 1. COURSE INTRODUCTION/OVERVIEW
1.6 Statistical Signal Processing?
The author points out that the text title is not unique, in fact
A Second Course in Discrete-Time Signal Processing is alsoappropriate
The Hayes text covers:
– Review of discrete-time signal processing and matrix the-ory for statistical signal processing
– Discrete-time random processes
– Signal modeling
– The Levinson and Related Recursions
– Wiener and Kalman filtering
– Spectrum estimation
– Adaptive filters
The intent of this course is not entirely aligned with the texttopics, as this course is also attempting to fill the void leftby Random Signals, Detection and Extraction of Signals from
Noise, and Estimation Theory and Adaptive Filtering
A second edition to the classic detection and estimation theorytext by Van Trees is another optional text for 2015; This is the
text I first learned from
Two Steven Kay books on detection and estimation are nowoptional texts, and may take the place of the Hayes book in thefuture
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1.7. COURSE SYLLABUS
1.7 Course Syllabus
ECE 5615/4615
Statistical Signal ProcessingSpring Semester 2015
Instructor: Dr. Mark Wickert Office: EN292 Phone: [email protected] Fax: 255-3589http://www.eas.uccs.edu/wickert/ece5615/
Office Hrs: Wednesday 10:40–11:45 AM, other times by appointment.
RequiredTexts:
Monson H. Hayes, Statistical Digital Signal Processing and Modeling, John
Wiley, 1996.
OptionalTexts:
H.L. Van Trees, K.L. Bell, and Z. Tian, Detection, Estimation, and Modulation
Theory, Part I , 2nd. ed., Wiley, 2013. Steven Kay, Fundamentals of Statistical
Signal Processing, Vol I: Estimation Theory, Vol II: Detection Theory, Prentice
Hall, 1993/1998. ISBN-978-0133457117/978-0135041352
OptionalSoftware:
Scientific Python (PyLab) via the IPython command line or notebook (http://
ipython.org/ install.html). IPython Notebook is highly recommended. A Linux
Virtual machine is available with all the needed tools. The ECE PC Lab also
has IPython and the notebook installed. If needed the full version of MATLAB
for windows (release 2014b) is also in the lab.
Grading: 1.) Graded homework assignments and computer projects worth 55%.2.) Midterm exam worth 20%.
3.) Final exam worth 25%.
Topics Text Sections
1. Introduction and course overview Notes ch 1
2. Background: discrete-time signal processing, linear algebra 2.1–2.4 (N ch2)
3. Random variables & discrete-time random processes 3.1–3.7 (N ch3)
4. Classical detection and estimation theory N ch4
(optional texts)
5. Signal modeling 4.1–4.8 (N ch5)
6. The Levinson recursion 5.1–5.5 (N ch6)
7. Optimal filters including the Kalman filter 7.1–7.5 (N ch8)
8. Spectrum estimation 8.1–8.8 (N ch9)
9. Adaptive filtering 9.1–9.4
(N ch10)
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CHAPTER 1. COURSE INTRODUCTION/OVERVIEW
1.8 Mathematical Models
Mathematical models serve as tools in the analysis and design
of complex systems
A mathematical model is used to represent, in an approximateway, a physical process or system where measurable quantitiesare involved
Typically a computer program is written to evaluate the math-ematical model of the system and plot performance curves
– The model can more rapidly answer questions about sys-tem performance than building expensive hardware pro-totypes
Mathematical models may be developed with differing degreesof fidelity
A system prototype is ultimately needed, but a computer sim-ulation model may be the first step in this process
A computer simulation model tries to accurately represent allrelevant aspects of the system under study
Digital signal processing (DSP) often plays an important rolein the implementation of the simulation model
If the system being simulated is to be DSP based itself, the sim-ulation model may share code with the actual hardware proto-type
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1.8. MATHEMATICAL MODELS
The mathematical model may employ both deterministic andrandom signal models
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CHAPTER 1. COURSE INTRODUCTION/OVERVIEW
1.9 Engineering Applications
Communications, Computer networks, Decision theory and decision
making, Estimation and filtering, Information processing, Power en-gineering, Quality control, Reliability, Signal detection, Signal anddata processing, Stochastic systems, and others.
Relation to Other Subjects2
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1.10. RANDOM SIGNALS AND STATISTICAL SIGNAL PROCESSING IN PRACTICE
1.10 Random Signals and Statistical Sig-
nal Processing in Practice
A typical application of random signals concepts involves oneor more of the following:
– Probability
– Random variables
– Random (stochastic) processes
Example 1.1: Modeling with Probability
Consider a digital communication system with a binary sym-metric channel and a coder and decoder
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CHAPTER 1. COURSE INTRODUCTION/OVERVIEW
The channel introduces bit errors with probability P e.bit/ D
A simple code scheme to combat channel errors is to repetitioncode the input bits by say sending each bit three times
The decoder then decides which bit was sent by using a major-ity vote decision rule
The system can tolerate one channel bit error without the de-coder making an error
The probability of a symbol error is given by
P e.symbol/ D P .2 bit errors/ C P .3 bit errors/
Assuming bit errors are statistically independent we can write
P .2 bit errors/ D .1 / C .1 /
C .1 / D 32.1 /
P .3 errors/ D D 3
The symbol error probability is thus
P e.symbol/ D 32 23
Suppose P e.bit/ D D 103, then P e.symbol/ D 2:998
106
The error probability is reduced by three orders of magnitude,but the coding reduces the throughput by a factor of three
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1.10. RANDOM SIGNALS AND STATISTICAL SIGNAL PROCESSING IN PRACTICE
Example 1.2: Separate Queues vs A Common Queue
A well known queuing theory result3 is that multiple servers, with
a common queue for all servers, gives better performance than mul-tiple servers each having their own queue. It is interesting to seeprobability theory in action modeling a scenario we all deal with inour lives.
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CHAPTER 1. COURSE INTRODUCTION/OVERVIEW
Common Queue Analysis
The number of servers is defined to be m, the mean customerarrival rate is per unit of time, and the mean customer servicetime is T s units of time
In the single queue case we let u D T s D traffic intensity
Let D u=m D server utilization
For stability we must have u < m and < 1
As a customer we are usually interested in the average time inthe queue, which is defined as the waiting time plus the servicetime (Tanner)
T CQ
Q D T w C T s D Ec.m; u/T s
m.1 / C T s
D
Ec.m; u/ C m.1 /
T s
m.1 /
where
Ec.m; u/ D um=mŠ
um=mŠ C .1 /Pm1
kD0 uk=kŠ
is known as the Erlang-C formula
To keep this problem in terms of normalized time units, we
will plot T Q=T s versus the traffic intensity u D T s
The normalized queuing time is
T CQ
Q
T sD
Ec.m; u/ C .m u/
m u D
Ec.m; u/
m u C 1
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1.10. RANDOM SIGNALS AND STATISTICAL SIGNAL PROCESSING IN PRACTICE
Separate Queue Analysis
Since the customers randomly choose a queue, arrival rate into
each queue is just =m
The server utilization is D .=m/ T s which is the same asthe single queue case
The average queuing time is (Tanner)
T SQ
Q D T s
1
D T s
1 T sm
D mT s
m T s
The normalized queuing time is
T SQ
Q
T sD
m
m T sD
m
m u
Create the Erlang-C function in MATLAB:
We will plot T Q=T s versus u D T s for m D 2 and 4
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CHAPTER 1. COURSE INTRODUCTION/OVERVIEW
The plots are shown below:
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Queuing time of common queue and separate queues for m D 2 servers
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1.10. RANDOM SIGNALS AND STATISTICAL SIGNAL PROCESSING IN PRACTICE
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Example 1.3: Power Spectrum Estimation
The second generation wireless system Global System for Mo-bile Communications (GSM), uses the Gaussian minimum shift-keying (GMSK) modulation scheme
xc.t/ Dp
2P c cos
2f 0t C 2f d
1XnD1
ang.t nT b/
where
g.t/ D 1
2
erf
r 2
ln 2BT b
t
T b
1
2
C erf
r 2
ln 2BT b
t
T bC
1
2
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CHAPTER 1. COURSE INTRODUCTION/OVERVIEW
The GMSK shaping factor is BT b D 0:3 and the bit rate isRb D 1=T b D 270:833 kbps
We can model the baseband GSM signal as a complex randomprocess
Suppose we would like to obtain the fraction of GSM signalpower contained in an RF bandwidth of B Hz centered aboutthe carrier frequency
There is no closed form expression for the power spectrum of a
GMSK signal, but a simulation, constructed in MATLAB, canbe used to produce a complex baseband version of the GSMsignal
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1.10. RANDOM SIGNALS AND STATISTICAL SIGNAL PROCESSING IN PRACTICE
Using averaged periodogram spectral estimation we can esti-mate S GMSK.f / and then find the fractional power in any RFbandwidth, B , centered on the carrier
P fraction D
R B=2B=2
S GMSK.f / df R 11
S GMSK.f / df
– The integrals become finite sums in the MATLAB calcu-lation
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B 100 kHz! 67.8%!
B 50 kHz! 38.0%!
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GSM Power Containment vs. RF Bandwidth
Fractional GSM signal power in a centered B Hz RF bandwidth
An expected result is that most of the signal power (95%) iscontained in a 200 kHz bandwidth, since the GSM channelspacing is 200 kHz
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CHAPTER 1. COURSE INTRODUCTION/OVERVIEW
Example 1.4: A Simple Binary Detection Problem
Signal x is measured as noise alone, or noise plus signal
x D
(n; only noise for hypothesis H 0V C n; noise + signal for hypothesis H 1
We model x as a random variable with a probability density function dependent upon which hypothesis is present
0 V V 1 ! 1 V 1"1 !
X
p x X H 0! " p x X H 1! "
V T
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We decide that the hypothesis H 1 is present if x > V T , whereV
T is the so-called decision threshold
The probability of detection is given by
P D D Pr.x > V T jH 1/ DZ 1
V T
px.X jH 1/ dX
0 V V 1 ! 1 V 1"1 !
X
p x
X H 1
! "
V T
P D
Pr x vT
H 1
#! "#
Area corresponding to P D
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1.10. RANDOM SIGNALS AND STATISTICAL SIGNAL PROCESSING IN PRACTICE
We may choose V T such that the probability of false alarm,defined as
P F D Pr.x < V T jH 0/ DZ 1
V T px.X jH 0/ dX
is some desired value (typically small)
0 V V 1 ! 1 V 1"1 !
X
p x
X H 0
! "
V T
P F
Pr x vT
H 0
#! "#
Area corresponding to P F
Example 1.5: A Waveform Estimation Theory Problem
Consider a received phase modulated signal of the form
r.t/ D s.t/ C n.t/ D Ac cos.2f ct C .t// C n.t/
where .t/ D kpm.t/, kp is the modulator phase deviationconstant, m.t/ the modulation, and n.t/ is additive white Gaus-sian noise
The signal we wish to estimate is m.t/, information phasemodulated onto the carrier
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CHAPTER 1. COURSE INTRODUCTION/OVERVIEW
We may choose to develop an estimation procedure that ob-tains Om.t/, the estimate of m.t/ such that the mean square erroris minimized, i.e.,
MSE D Efjm.t/ Om.t/j2g
Another approach is maximum likelihood estimation
A practical implementation of a maximum likelihood basedscheme is a phase-locked loop (PLL)
As a specific example here we consider the MATLAB Simulinkblock diagram shown below:
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Complex baseband Simulink block diagram
Results from the simulation using a squarewave input for twodifferent signal-to-noise power ratios (20 dB and 0 dB) areshown below
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1.10. RANDOM SIGNALS AND STATISTICAL SIGNAL PROCESSING IN PRACTICE
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Input/output waveforms
Example 1.6: Nonrandom Parameter in White Gaussian Noise
We are given observations
ri D A C ni ; i D 1 ; : : : ; N
with the ni iid of the form N.0; 2n /
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CHAPTER 1. COURSE INTRODUCTION/OVERVIEW
The joint pdf is
prja.RjA/ D
N
Yi D11
p 2 2n exp.Ri A/
2
2 2n
The log likelihood function is
ln prja.RjA/ç D N
2 ln
1
2 2n
1
2 2n
N XiD1
.Ri A/2
Solve the likelihood equation
@ lnŒprja.RjA/ç
@A
ˇ̌̌ˇ
AD Oaml
D 1
2n
N Xi D1
.Ri A/
ˇ̌̌ˇ
AD Oaml
D 0
H) Oaml D 1
N
N
Xi D1Ri
Check the bias:
Ef Oaml.r/ Ag D E
1
N
N Xi D1
ri A
D 1
N
N
Xi D1Efri g„ƒ‚…A
A D 0
No Bias!
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1.10. RANDOM SIGNALS AND STATISTICAL SIGNAL PROCESSING IN PRACTICE
Example 1.7: Sat-Com Adaptive Equalizer
Wideband satellite communication channels are subject to bothlinear and non-linear distortion
Uplink
Channel
Transmitter
Transponder
Receiver
PSK
Mod
HPA
(TWTA)
HPA
(TWTA)
Modulation
Impairments
Bandpass
Filtering
Bandpass
Filtering
Bandpass
Filtering
PSK Demod
(bit true with
full synch)
Adaptive
Equalizer
Other Signals
M o d .
M o d .
WGNNoise
(off)
Data
Source
Recovered
Data
Downlink
Channel
WGN
Noise
(on)
Other
Signals
M o d .
Other
Signals
Spurious PM Incidental AM
Clock jitter
IQ amplitude imbalance IQ phase imbalance
Waveform asymmetryand rise/fall time
Phase noise Spurious PM Incidental AM
Spurious outputs
Phase noise Spurious PM Incidental AM
Spurious outputs
BPSK QPSK
OQPSK
Wideband Sat-Comm simulation model
An adaptive filter can be used to estimate the channel dis-tortion, for example a technique known as decision feedback equalization
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CHAPTER 1. COURSE INTRODUCTION/OVERVIEW
M1 Tap
ComplexFIR
Re
z-1M
1 Tap
Complex
FIR
Im 2
2
M2 Tap
Real
FIR
M2 Tap
Real
FIR
CM Error/
LMS Update
DD Error/
LMS Update
CM Error/
LMS Update
DD Error/
LMS Update
Tap
Weight
UpdateSoft I/Q outputs
from demod at
sample rate = 2Rs
Recovered
I Data
Recovered
Q Data
Stagger for
OQPSK, omit
for QPSK
+
+
-
-
-
-
+
+ AdaptMode
Decision
Feedback
Decision
Feedback
µCM
, µ
DD
µDF
, γ
An adaptive baseband equalizer4
Since the distortion is both linear (bandlimiting) and nonlin-ear (amplifiers and other interference), the distortion cannot becompletely eliminated
The following two figures show first the modulation 4-phasesignal points with and with out the equalizer, and then the bit
error probability (BEP) versus received energy per bit to noisepower spectral density ratio (Eb=N 0)
4Mark Wickert, Shaheen Samad, and Bryan Butler. “An Adaptive Baseband Equalizer for HighData Rate Bandlimited Channels,Ó Proceedings 2006 International Telemetry Conference, Session5, paper 06–5-03.
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1.10. RANDOM SIGNALS AND STATISTICAL SIGNAL PROCESSING IN PRACTICE
−1.5 −1 −0.5 0 0.5 1 1.5−1.5
−1
−0.5
0
0.5
1
1.5
In−phase
Q u a d r a t u r e
−1.5 −1 −0.5 0 0.5 1 1.5−1.5
−1
−0.5
0
0.5
1
1.5
In−phase
Q u a d r a t u r e
Before Equalization: R b = 300 Mbps After Equalization: R
b = 300 Mbps
OQPSK scatter plots with and without the equalizer
6 8 10 12 14 16 18 20 22 2410
−7
10−6
10−5
10−4
10−3
10−2
Eb/N
0 (dB)
P r o b a b i l i t y o f B i t E r r o r
Theory EQ NO EQ
4.0 dB 8.1 dB
Semi-Analytic Simulation
300 MBPS BER Performance with a 40/0 Equalizer
BEP versus Eb=N 0 in dB