Here are the key steps to estimate the Doppler frequency from radar returns to determine vehicle speed: 1. Demodulate the received radar signal to baseband to extract the complex envelope. This will produce a signal of the form s[n] = (A/2)exp(j2πFDnΔ + φ), where FD is the Doppler frequency related to vehicle speed. 2. Sample the complex envelope at a rate Fs higher than twice the maximum expected Doppler frequency to avoid aliasing. 3. Estimate the Doppler frequency FD from the sampled complex envelope using techniques like periodogram or autoregressive spectral estimation. The Doppler estimate allows calculating the vehicle speed as v = cFD/2F0. 4.

1. Professional Development Short Course On:
Practical Statistical Signal Processing — using MATLAB
Instructor:
Dr. Steven Kay
ATI Course Schedule: http://www.ATIcourses.com/schedule.htm
ATI's Practical Statistical Signal Processing: http://www.aticourses.com/practical_statistical_signal.htm
349 Berkshire Drive • Riva, Maryland 21140
888-501-2100 • 410-956-8805
Website: www.ATIcourses.com • Email: ATI@ATIcourses.com

2. Practical Statistical Signal Processing Using MATLAB
with Radar, Sonar, Communications, Speech & Imaging Applications
June 22-25, 2009
Middletown, Rhode Island
$1895 (8:30am - 4:00pm)
"Register 3 or More & Receive $10000 each
Off The Course Tuition."
Course Outline
1. MATLAB Basics. M-files, logical flow, graphing,
debugging, special characters, array manipulation,
Summary vectorizing computations, useful toolboxes.
This 4-day course covers signal processing 2. Computer Data Generation. Signals, Gaussian
systems for radar, sonar, communications, speech, noise, nonGaussian noise, colored and white noise,
imaging and other applications based on state-of- AR/ARMA time series, real vs. complex data, linear
the-art computer algorithms. These algorithms models, complex envelopes and demodulation.
include important tasks such as data simulation,
3. Parameter Estimation. Maximum likelihood,
parameter estimation, filtering, interpolation, best linear unbiased, linear and nonlinear least
detection, spectral analysis, beamforming, squares, recursive and sequential least squares,
classification, and tracking. Until now these minimum mean square error, maximum a posteriori,
algorithms could only be learned by reading the general linear model, performance evaluation via Taylor
latest technical journals. This course will take the series and computer simulation methods.
mystery out of these designs by introducing the 4. Filtering/Interpolation/Extrapolation. Wiener,
algorithms with a minimum of mathematics and linear Kalman approaches, time series methods.
illustrating the key ideas via numerous examples
using MATLAB. 5. Detection. Matched filters, generalized matched
filters, estimator-correlators, energy detectors,
Designed for engineers, scientists, and other detection of abrupt changes, min probability of error
professionals who wish to study the practice of receivers, communication receivers, nonGaussian
statistical signal processing without the headaches, approaches, likelihood and generalized likelihood
this course will make extensive use of hands-on detectors, receiver operating characteristics, CFAR
MATLAB implementations and demonstrations. receivers, performance evaluation by computer
Attendees will receive a suite of software source simulation.
code and are encouraged to bring their own laptops 6. Spectral Analysis. Periodogram, Blackman-
to follow along with the demonstrations. Tukey, autoregressive and other high resolution
Each participant will receive two books methods, eigenanalysis methods for sinusoids in noise.
Fundamentals of Statistical Signal Processing: Vol. I 7. Array Processing. Beamforming, narrowband
and Vol. 2 by instructor Dr. Kay. A complete set of vs. wideband considerations, space-time processing,
notes and a suite of MATLAB m-files will be interference suppression.
distributed in source format for direct use or 8. Signal Processing Systems. Image processing,
modification by the user. active sonar receiver, passive sonar receiver, adaptive
noise canceler, time difference of arrival localization,
channel identification and tracking, adaptive
Instructor beamforming, data analysis.
Dr. Steven Kay is a Professor of Electrical 9. Case Studies. Fault detection in bearings,
Engineering at the University of acoustic imaging, active sonar detection, passive sonar
Rhode Island and the President of detection, infrared surveillance, radar Doppler
estimation, speaker separation, stock market data
Signal Processing Systems, a analysis.
consulting firm to industry and the
government. He has over 25 years
of research and development What You Will Learn
experience in designing optimal • To translate system requirements into algorithms
statistical signal processing algorithms for radar, that work.
sonar, speech, image, communications, vibration, • To simulate and assess performance of key
and financial data analysis. Much of his work has algorithms.
been published in over 100 technical papers and • To tradeoff algorithm performance for
the three textbooks, Modern Spectral Estimation: computational complexity.
Theory and Application, Fundamentals of • The limitations to signal processing performance.
Statistical Signal Processing: Estimation Theory, • To recognize and avoid common pitfalls and traps
and Fundamentals of Statistical Signal in algorithmic development.
Processing: Detection Theory. Dr. Kay is a Fellow • To generalize and solve practical problems using
of the IEEE. the provided suite of MATLAB code.
Register online at www.ATIcourses.com or call ATI at 888.501.2100 or 410.956.8805 Vol. 97 – 41

3. www.ATIcourses.com
Boost Your Skills 349 Berkshire Drive
Riva, Maryland 21140
with On-Site Courses Telephone 1-888-501-2100 / (410) 965-8805
Tailored to Your Needs
Fax (410) 956-5785
Email: ATI@ATIcourses.com
The Applied Technology Institute specializes in training programs for technical professionals. Our courses keep you
current in the state-of-the-art technology that is essential to keep your company on the cutting edge in today’s highly
competitive marketplace. Since 1984, ATI has earned the trust of training departments nationwide, and has presented
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For a Free On-Site Quote Visit Us At: http://www.ATIcourses.com/free_onsite_quote.asp
For Our Current Public Course Schedule Go To: http://www.ATIcourses.com/schedule.htm

4. References
1∗. S. Kay, Fundamentals of Statistical Signal
Processing: Estimation Theory, Prentice-Hall, 1993
2∗. S. Kay, Fundamentals of Statistical Signal
Processing: Detection Theory, Prentice-Hall, 1998
3. L. Scharf, Statistical Signal Processing, Addison-
Wesley,
Reading, MA, 1991 (more advanced treatment)
4. R.N. McDonough, A.D. Whalen, Detection of
Signals in Noise, Academic Press, New York, 1995
5. H.L. Van Trees, Detection, Estimation, and
Modulation Theory, Vol. I, J. Wiley, New York, 1968
(fairly involved but a classic)
6. G.M. Jenkins, D.G. Watts, Spectral Analysis and its
Applications, Holden-Day, 1968
7. S. Kay, Modern Spectral Estimation: Theory and
Application, Prentice-Hall, 1988
8. M.B. Priestley, Spectral Analysis and Time Series,
Academic Press, 1981
9. R.A. Monzingo, T.W. Miller, Adaptive Arrays, J.
Wiley, 1980
10. D.H. Johnson, D.E. Dudgeon, Array Signal
Processing, Prentice-Hall, 1993
∗
Provided as part of course materials

5. Summary of Slides
Slide number
1. Matlab Basics 4 -16
2. Computer Data Generation 17 - 49
3. Parameter Estimation 50 - 107
4. Detection 108 - 182
5. Spectral Analysis 183 - 208
6. Array Processing 209 - 222
7. Case Studies 223 - 247
8. Description of MATLAB Programs 248 - 253

6. MATLAB Basics
Version: 5.2 for Windows
Useful toolboxes: signal processing, statistics, symbolic
m files: script files
Fortran vs. MATLAB example:
Signal generation
Math: s[n ] = cos(2π f 0n ) n = 0,1, K, N − 1
Fortran: pi=3.14159
f0=0.25
N=25
do 10 I=1,N
10 s(I)=cos(2*pi*f0*(I-1))
MATLAB: f0=0.25;N=25;
s=cos(2*pi*f0*[0:N-1]’);

7. Notes: pi already defined, [0:N-1]’ is a column vector,
cosine of vector of samples produces a vector output,
MATLAB treats vectors and matrices as elements

8. Noise Generation
Simplest model for observation noise is white Gaussian
noise (WGN)
Definition: zero mean, all samples are uncorrelated,
power
spectral density (PSD) is flat, and first order
probability density function (PDF) is
Gaussian
exp ⎛ − 2 x 2 ⎞
1 1
PDF: p( x ) = ⎜ ⎟
⎝ 2σ
p(x )
2πσ 2
⎠
where σ 2 = variance x
MATLAB Example: σ2 = 1
4
2
] 0
n
[
x
-2
-4
0 20 40 60 80 100
n
wgn.m
0 25
0
1
f
o 20
t
u
o
s 15
e
m
o
c 10
t
u
o
f 5
o
r
e
b 0
m
u -3 -2 -1 0 1 2 3
n x

9. wgn.m
1
0.9
0.8
0.7
0.6
)
x
(
p 0.5
,
F
D
P 0.4
0.3
0.2
0.1
0
-3 -2 -1 0 1 2 3
x
Note: randn(‘state’,0) sets random number generator to
default seed and thus generates the same set of
random numbers each time the program is run.
MATLAB code:
% wgn.m
%
% This program generates and plots
the time series, histogram, and

10. % estimated PDF for real white
Gaussian noise.
randn('state',0)
x=randn(100,1);
subplot(2,1,1)
plot(x)
xlabel('n')
ylabel('x[n]')
grid
subplot(2,1,2)
hist(x)
xlabel('x')
ylabel('number of outcomes out of
100')
title('wgn.m')
figure
pdf(x,100,10,-3,3,1)
xlabel('x')
ylabel('PDF, p(x)')
title('wgn.m')
% pdf.m
%
function
pdf(x,N,nbins,xmin,xmax,ymax)
%

11. % This function subprogram computes
and plots the
% PDF of a set of data.
%
% Input parameters:
%
% x - Nx1 data array
% N - number of data points
% nbins - number of bins (<N/10)
% xmin,xmax,ymax - axis scaling
%
[y,xx]=hist(x(1:N),nbins);
delx=xx(2)-xx(1);
bar(xx,y/(N*delx))
grid
axis([xmin xmax 0 ymax]);

12. Complex White Gaussian Noise
Definition: x [n ] = w1[n ] + jw2 [n ]
where w1[n ] and w2 [n ] are independent of each other
and
each one is real WGN with variance of σ 2 / 2
Mean: E (x [n ]) = 0
Variance: var(x [n ]) = var(w1[n ]) + var(w 2 [n ]) = σ 2
MATLAB code:
% cwgn.m
%
% This program generates complex
white Gaussian noise and
% then estimates its mean and
variance.
%
N=100;
varw=1;
x=sqrt(varw/2)*randn(N,1)+j*sqrt(varw
/2)*randn(N,1);
muest=mean(x)
varest=cov(x)

13. NonGaussian Noise
Generation: transform WGN using a nonlinear
memoryless
transformation
Example: Laplacian noise
1 ⎛ 2 ⎞
p( x ) = exp ⎜ − 2 x ⎟
2σ 2 ⎝ σ ⎠
Use the transformation
x = F −1 (w )
where w is uniform random variable on the interval
[0,1]
and F is the cumulative distribution function of the
Laplacian
PDF.
F (x )
1
x

14. Example: σ 2 = 1
5
] 0
n
[
x
-5
0 200 400 600 800 1000
n
laplaciannoise.m
1
0.8
)
x 0.6
(
p
,
F 0.4
D
P
0.2
0
-5 0 5
x

15. MATLAB Code:
% laplaciannoise.m
%
% This program uses a memoryless
transformation of a uniform
% random variable to generate a set
of independent Laplacian
% noise samples.
%
rand('state',0)
varx=1;N=1000;
u=rand(N,1);
for i=1:N
if u(i)>0.5
x(i,1)=sqrt(varx)*(1/sqrt(2))*log(1/(
2*(1-u(i))));
else
x(i,1)=sqrt(varx)*(1/sqrt(2))*log(2*u
(i));
end
end
subplot(2,1,1)
plot(x)
xlabel('n')
ylabel('x[n]')

16. axis([0 1000 -5 5]);
subplot(2,1,2)
pdf(x,N,50,-5,5,1)
title('laplaciannoise.m')

17. Solving Parameter Estimation Problems
Approach:
1. Translate problem into manageable estimation
problem
2. Evaluate best possible performance (bounds)
3. Choose optimal or suboptimal procedure
4. Evaluate actual performance
a. Analytically – exact or approximate
b. By computer simulation

18. Radar Doppler Estimation
(Step 1)
Problem: Given radar returns from automobile,
determine speed to within 0.5 mph
transmit
Physical basis: Doppler effect
receive – receive-
approaching moving away

19. Received frequency is
2v
F = F0 + F0
c
{
FD
where v = velocity, c= speed of light, F0 = sinusoidal
transmit
frequency
To measure the velocity use
c F − F0
v =
2 F0
and estimate the frequency to yield
c F − F0
ˆ
v =
ˆ
2 F0

20. Modeling and Best Possible Performance
(Step 2)
Preprocessing: first demodulate to baseband to produce
the
sampled complex envelope or
s [n ] = (A / 2) exp( j 2π FD n ∆ + ϕ )
%
⎛ F = 2v F ⎞
⎜ D ⎟
⎝ c 0⎠
Fs = 1/ ∆ > 2FD = 2 ⎛ max F0 ⎞
2v
Must sample at ⎜ ⎟
⎝ c ⎠
Example: v max =300 mph, F0 =10.5 Ghz (X-band),
c = 3x108 m/s
2v max
FD −max = F0 ≈ 9388 Hz
c
⇒ Fs > 18, 776 complex samples/sec
How many samples do we need?
Spec: error must be less than 0.5 mph for

21. (A / 2)2
SNR = 10 log10 > −10 dB
σ 2
Cramer-Rao Lower Bound for Frequency
• tells us the minimum possible variance for estimator
– very useful for feasibility studies
6
var( fD ) ≥
ˆ (*) (see [Kay 1988])
(2π )2ηN (N 2 − 1)
where fD = FD / Fs , N = number of complex samples,
η =linear SNR
cFs
Since FD = (2v / c )F0 ⇒ v = fD
2F0
and we can show that
2
⎛ cFs ⎞
var(v ) = ⎜
ˆ ˆ
⎟ var( fD )
⎝ 2F0 ⎠
For an error of 0.5 mph (0.22 m/s) set

22. 3 var(v ) = 0.22 ⇒ var( fD ) = 7.47x10 −8
ˆ ˆ
99.8%
ˆ
v
v − 0.5 v v + 0.5
and finally we have from (*) that
1/ 3
⎡ 6 ⎤
N >⎢ ≈ 272 samples
ˆ )⎥
⎢ (2π ) η var( fD ⎥
2
⎣ ⎦

23. Descriptions of MATLAB Programs
1. analogsim – simulates the action of an RC filter on a
pulse
2. arcov - estimates the AR power spectral density
using he covariance method for AR parameter
estimation for real data.
3. arexamples - gives examples of the time series and
corresponding power spectral density for various AR
models. It requires the function subprograms:
gendata.m and armapsd.m.
4. armapsd - computes a set of PSD values, given the
parameters of a complex or real AR or MA or ARMA
model.
5. arpsd - plots the AR power spectral density for some
simple cases. The external subprogram armapsd.m is
required.
6. arpsdexample - estimates the power spectral density
of two real sinusoids in white Gaussian noise using the
periodogram and AR spectral estimators.
It requires the functions subprograms: per.m and
arcov.m.

24. 7. arrivaltimeest - simulates the performance of an
arrival time estimator for a DC pulse. The estimator is
a running correlator which is the MLE for white
Gaussian noise.
8. avper - illustrates the effect of block averaging on
the periodogram for white Gaussian noise.
9. classicalbayesian - demonstrates the difference
between the classical approach and the Bayesian
approaches to parameter modeling.
10. cwgn - generates complex white Gaussian noise and
then estimates its mean and variance.
11. DClevelhist - generates Figures 1.4, 1.5 in
"Fundamentals of Statistical Signal Processing:
Detection Theory", S. Kay
12. DCleveltime - generates a data set of white
Gaussian noise only and also a DC level A in white
Gaussian noise
13. discretesinc – plots the graph in linear and dB
quantities of a discrete sinc pulse in frequency

25. 14. estperform - compares the frequency estimation
performance for a single complex sinusoid in complex
white Gaussian using the peak location of the
periodogram and an AR(1) estimator.
15. Fig35new - computes Figure 3.5 (same as Figure
4.5) in "Fundamentals of Statistical Signal Processing:
Detection Theory", S. Kay. The function subprograms
Q.m and Qinv.m are required.
16. Fig39new - computes Figure 3.9 in "Fundamentals
of Statistical Signal Processing: Detection Theory", S.
Kay. The function subprograms Q.m and Qinv.m are
required.
17. Fig77new - computes Figure 7.7 in "Fundamentals
of Statistical Signal Processing: Detection Theory", S.
Kay.
18. gendata - generates a complex or real AR, MA, or
ARMA time series given the filter parameters and
excitation noise variance.
19. kalman - implementation of the vector state-scalar
observation linear Kalman filter. See (13.50)-(13.54) of
"Fundamentals of Statistical Signal Processing:
Estimation Theory" by S. Kay for more details.

26. 20. kalmanexample - uses the linear Kalman filter to
estimate the tap weights for a random TDL channel. It
generates Figures 13.16-13.18 in "Fundamentals of
Statistical Signal Processing: Estimation Theory", S.
Kay. It requires the function subprogram kalman.m.
21. laplaciannoise - uses a memoryless transformation
of a uniform random variable to generate a set of
independent Laplacian noise samples.
22. linearmodel - computes the optimal estimator of
the parameters of a real or complex linear model.
Alternatively, it is just the least squares estimator.
23. linearmodelexample - implements a line fit to a
noise corrupted line. The linear model or least squares
estimator is used. The function subprogram
linearmodel.m is required.
24. MAexample – plots out the PDF of an MA process
25. mlevar - computes the mean, variance, PDF of the
MLE for the power of a WGN process and compares it
to the CRLB.
26. montecarloroc - uses a Monte Carlo approach to
determine the detection performance of a Neyman-
Pearson detector for a DC level in WGN. The true

27. performance is shown in "Fundamentals of Statistical
Signal Processing: Detection Theory", S. Kay, in Figure
3.9 for d^2=1. The function subprogram roccurve.m is
required.
27. pcar - estimates the frequencies of real sinusoids
by using the principal component AR approach. Futher
details can be found in "Modern Spectral Estimation:
Theory and Application", by S. Kay.
28. pdf - computes and plots the PDF of a set of data.
29. per - computes the periodogram spectral estimator.
Futher details can be found in "Modern Spectral
Estimation: Theory and Application", by S. Kay.
30. perdetectexample - illustrates the detection
performance of a periodogram, which is an incoherent
matched filter.
31. perexamples - illustrates the capability of the
periodogram for resolving spectral lines.
32. plot1 – plots a sinusoid
33. psk - implements a matched filter receiver for the
detection of a PSK signal. The data are assumed real.

28. 34. pskexample - illustrates the optimal
detection/decoding of a PSK encoded digital sequence.
The bits are decoded and the probability of error is
computed and compared to the number of actual errors.
The external function subprogram psk.m is required.
35. Q - computes the right-tail probability
(complementary cumulative distribution function) for a
N(0,1) random variable.
36. Qinv - computes the inverse Q function or the
value which is exceeded by a N(0,1) random variable
with a probability of x.
37. repcorr - implements a replica correlator for either
real or complex data.
38. repcorrexample - illustrates the replica-correlator.
It requires the subprogram repcorr.m.
39. roccurve - determines the ROCs for a given set of
detector outputs under H0 and H1.
40. sampling – plots out an analog sinusoid and the
samples taken
41. seqls - implements a sequential least squares
estimator for a DC level

29. in WGN of constant variance.
42. shift - shifts the given sequence by a specified
number of samples. Zeros are shifted in either from
the left or right.
43. signdetexample - implements a sign detector for a
DC level in Gaussian-mixture noise. A comparison is
made to a replica correlator, which is just the sample
mean.
44. sinusoid - generates a sinusoid
45. stepdown - implements the step-down procedure to
find the coefficients and prediction error powers for all
the lower order predictors given the filter parameters
and white noise variance of a pth order AR model. See
(6.51) and (6.52). This program has been translated
from Fortran into Matlab. See "Modern Spectral
Estimation" by S. Kay for further details.
46. timedelaybfr - implements a time delay
beamformer for a line array of 3 sensors. The emitted
signal is sinusoidal and is assumed to be at
broadside or at 90 degrees (perpendicular to line array).

30. 47. wgn - generates and plots the time series,
histogram, and estimated PDF for real white Gaussian
noise.
48. wiener - implements a Wiener smoother for
extracting an AR(1) signal in white Gaussian noise and
also for predicting an AR(1) signal for no observation
noise present.

31. Boost Your Skills
with On-Site Courses
Tailored to Your Needs
The Applied Technology Institute specializes in training programs for technical
professionals. Our courses keep you current in the state-of-the-art technology that is
essential to keep your company on the cutting edge in today’s highly competitive
marketplace. For 20 years, we have earned the trust of training departments nationwide,
and have presented on-site training at the major Navy, Air Force and NASA centers, and for a
large number of contractors. Our training increases effectiveness and productivity. Learn
from the proven best.
ATI’s on-site courses offer these cost-effective advantages:
• You design, control, and schedule the course.
• Since the program involves only your personnel, confidentiality is maintained. You can
freely discuss company issues and programs. Classified programs can also be arranged.
• Your employees may attend all or only the most relevant part of the course.
• Our instructors are the best in the business, averaging 25 to 35 years of practical, real-
world experience. Carefully selected for both technical expertise and teaching ability, they
provide information that is practical and ready to use immediately.
• Our on-site programs can save your facility 30% to 50%, plus additional savings by
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• The ATI Satisfaction Guarantee: You must be completely satisfied with our program.
We suggest you look at ATI course descriptions in this catalog and on the ATI website.
Visit and bookmark ATI’s website at http://www.ATIcourses.com for descriptions of all
of our courses in these areas:
• Communications & Computer Programming
• Radar/EW/Combat Systems
• Signal Processing & Information Technology
• Sonar & Acoustic Engineering
• Spacecraft & Satellite Engineering
I suggest that you read through these course descriptions and then call me personally, Jim
Jenkins, at (410) 531-6034, and I’ll explain what we can do for you, what it will cost, and what
you can expect in results and future capabilities.
Our training helps you and your organization
remain competitive in this changing world.
Register online at www.aticourses.com or call ATI at 888.501.2100 or 410.531.6034