Elec4621:
Advanced Digital Signal Processing
Chapter 7: Introduction to Linear
Prediction
Dr. D. S. Taubman
April 18, 2011
1
1.1
Linear Prediction
Statistical Formulation
Let X [n] denote the elements of a random process, having outcomes x [n]. The
random
Elec4621:
Advanced Digital Signal Processing
Chapter 2: Z-Transforms, Filters and
Oscillators
Dr. D. S. Taubman
March 4, 2011
1
Introduction to the Z-Transform
The Z-transform of a sequence, x [n], is a formal power series, dened by
X (z) =
4
[
x [n] z n
Elec4621:
Advanced Digital Signal Processing
Chapter 8: Wiener and Adaptive
Filtering
Dr. D. S. Taubman
May 2, 2011
1
Wiener Filtering
A Wiener lter is one which provides the Minimum Mean Squared Error
(MMSE) prediction, y [n], of the outcomes, y [n], fro
Elec4621 Topics, S1 2014, Week 13
David Taubman
June 2, 2014
1
Quiz-2 Revision
2
Introduction to Time-Frequency Analysis
2.1
Block-wise DFT as a Filter Bank
Filter bank equivalent
easiest to see by taking the inner product perspective
block size N give
Elec4621:
Signal Processing 2
Chapter 4: The Discrete Fourier
Transform
Dr. D. S. Taubman
March 18, 2011
1
Denition
Recall that the Discrete Time Fourier Transform (DTFT) is dened on sequences
of innite extent according to
x($)
=
4
[
x[n]ejn$ ,
for $
n=4
Elec4621 Topics, S1 2014, Week 7
David Taubman
April 14, 2014
1
Statistics
1.1
Random Vectors (ctd)
Gaussian PDF
fX (x) =
1
e
(2 )m CX
1
2 (x
X )t CX1 (x X )
then CX diagonal means that the elements of X are mutually independent.
1.2
Random Processes
Se
Elec4621 Topics, S1 2014, Week 3
David Taubman
March 17, 2014
1
Spillover from Week 2
All-pass
Minimum phase
Oscillators
2
Filter Structures
2.1
Canonical Form
M
y [n] =
N
ak x [n
k] +
k=0
H (z) =
a0 + a1 z
1 b1 z
bk y [n
1
1
H 2 ( z)
+ + aM z
bN z
a0
Elec4621 Topics, S1 2014, Week 5
David Taubman
March 31, 2014
1
Digital Filter Design
1.1
Introduction to FIR Design Methods
Windowing
Not generally optimal for any application, but simple
Frequency sampling
Essentially take inverse DFT of desired res
Elec4621 Topics, S1 2014, Week 9
David Taubman
May 5, 2014
1
Wiener Filters
1.1
Mathematics
Want to predict next output of one random process, Y [n], based on current and past values of another, X [n].
Minimize
2
E
J=
=E
Y [n]
Y [n]
2
N
2
=E
Y [n]
ak X
Elec4621 Topics, S1 2014, Week 1
David Taubman
March 3, 2014
1
Linear Time Invariant Operators
1.1
Linear Spaces
Closure under addition and scalar multiplication
Inner products and Cauchy-Schwartz inequality
a, a b, b
a, b
Norms and Triangle inequality
Elec4621 Topics, S1 2014, Week 4
David Taubman
March 24, 2014
1
Finite Word Length E ects
1.1
Fixed-point realization of multiplication and accumulation
Typical W -bit binary adder circuit
complexity is W 1-bit adder cells
delay is W 1-bit adder propag
Elec4621 Topics, S1 2014, Week 12
David Taubman
May 26, 2014
1
Filter Banks
General M-channel analysis/synthesis lter bank:
With ideal lters, we saw that subbands can be combined to reconstruct original signal exactly.
With practical lters, we are inte
Elec4621 Topics, S1 2014, Week 11
David Taubman
May 26, 2014
1
Multi-rate Systems
1.1
Decimation
We write
M for the operation y [n] = x [M n]
In the Fourier domain: y( ) =
1
M
M 1
m=0
x
2 m
M
That is, we stretch the spectrum by M , add all translates by
Elec4621:
Advanced Digital Signal Processing
Chapter 3: Filter Implementation
Techniques
Dr. D. S. Taubman
March 11, 2011
1
1.1
Filter Structures
Canonical Form
In the previous chapter, we gave the following dierence equation as a general
form for LTI lte
Elec4621 Topics, S1 2014, Week 10
David Taubman
May 15, 2014
1
Wiener Filtering Applications
A little more on this
2
Linear Prediction Revisited
Recall the normal equations
RXX [0]
RXX [1]
.
.
RXX [N
RXX [1]
RXX [0]
.
.
1] RXX [N
.
.
RXX [N
RXX [N
.
.
2]
Elec4621:
Advanced Digital Signal Processing
Chapter 5: Digital Filter Design
Dr. D. S. Taubman
April 1, 2011
Over the years many dierent strategies have been developed to design digital
lters. In this chapter, we will cover some of these strategies and h
Elec4621:
Advanced Digital Signal Processing
Chapter 6: Random Processes and
Power Spectrum Estimation
Dr. D. S. Taubman
April 8, 2011
1
Random Variables and Vectors
We will consistently use upper case (as in X) to refer to random variable (RV),
and lower
Elec4621:
Advanced Digital Signal Processing
Chapter 1: Foundations of Digital
Signal Processing
Dr. D. S. Taubman
March 3, 2011
1
Introduction
For some, the material presented in these notes may seem excessively mathematical: Why do we need to go back to
Elec4621:
Advanced Digital Signal Processing
Chapter 10: Subband Transforms and the
Discrete Wavelet Transform
Dr. D. S. Taubman
May 16, 2011
1
Tree-Structured Subband Transforms
We have already seen two-channel lter banks used to represent a signal,
x[n]
Elec4621 Topics, S1 2014, Week 8
David Taubman
May 5, 2014
1
Power Spectrum Estimation (ctd)
1.1
Bartlett Method
Divide N samples into P blocks and average periodograms from each of the P blocks.
Variance of the estimates reduces by P , as does the spec
Elec4621 Topics, S1 2014, Week 2
David Taubman
March 10, 2014
1
Z-transform and Discrete LTI Filters
Connection between polynomial multiplication and convolution
Relationship to Fourier transform
Connection between rational polynomials and realizable d
Elec4621
Advanced Digital Signal Processing
Chapter 11: Time-Frequency Analysis
Dr. D. S. Taubman
May 23, 2011
In this last chapter of your notes, we are interested in the problem of nding the instantaneous power spectrum of a signal. Earlier we encounter
Elec4621 Topics, S1 2014, Week 6
David Taubman
April 7, 2014
1
IIR Filter Design
1.1
Impulse Invariant Method
Have analog lter, Ha (s). Want h [n] = ha (t)|t=n
Partial fraction expansion
N
N
i
Ha (s) =
i=1
s
=
pi
i=1
N
N
pi n
ie
h [n] =
n
i ai
=
i=1
ai
Elec4621:
Advanced Digital Signal Processing
Chapter 9: Multi-Rate Signal Processing
Dr. D. S. Taubman
May 16, 2011
1
Decimation
Let x [n] be a sequence which has been sampled at rate 1. That is, in an
appropriate time scale, x [n] is obtained by sampling
ELEC4042 Signal Processing 2
MATLAB Review (prepared by A/Prof Ambikairajah)
Introduction
MATLAB is a powerful mathematical language that is used in most engineering companies today. Its
strength lies in its numerical analysis. MATLAB is not the fastest l
934
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 8, NO. 7, JULY 1999
Image Registration and Object Recognition
Using Affine Invariants and Convex Hulls
Zhengwei Yang and Fernand S. Cohen, Senior Member, IEEE
Abstract This paper is concerned with the proble
2012 International Conference on Communication Systems and Network Technologies
A Novel Technique for Robust Image Registration Using Log Polar Transform
Yagnesh N. Makwana
Research Scholer , Dr.K.N.Modi University
Ajay K. Somkuwar
Department of Electroni
1422
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 14, NO. 10, OCTOBER 2005
Image Registration Using Log-Polar Mappings
for Recovery of Large-Scale Similarity and
Projective Transformations
Siavash Zokai and George Wolberg, Senior Member, IEEE
AbstractThis
A Survey of Image Registration
LISA GOTTESFELD
Department
of Computer
Techniques
BROWN
Sctence,
Colunzbza
Unzl,ersity,
New
York,
NY
10027
Registration M a fundamental task in image processing used to match two or more
pictures taken, for example, at diffe
www.ietdl.org
Published in IET Image Processing
Received on 11th August 2013
Revised on 6th December 2013
Accepted on 11th December 2013
doi: 10.1049/iet-ipr.2013.0575
ISSN 1751-9659
Sparse-induced similarity measure: mono-modal
image registration via spa