Phil Schniter
ECE-6202
Review of Linear Algebra:
In this class, a vector will be an element of CN , i.e., an array of
complex-valued numbers of functions. We will assume column
vectors and denote vectors using boldface font when typset, or
an underline w
Phil Schniter
ECE-6202
Stochastic Models:
We now focus on stochastic modeling, where we model a given
random process x(n) using a designed random process x(n).
Motivation: Describing a class of signals rather than a specic
instance of a signal. Examples
SPECTRAL FACTORIZATION
Polynomial root nding and the Leja ordering
Spectral factorization is an important ingredient in the design of
minimum-phase lters, and has many other applications. First we
need to learn about polynomial root nding, and the problem
Phil Schniter
ECE-6202
Wiener Filtering [Ch.7]:
Filtering refers to the processing of an input x(n) by an LSI system
W (z) in an attempt to match a desired output d(n).
x(n)
W (z)
d(n)
+
d(n)
e(n)
In this chapter, we assume that x(n) and d(n) are jointl
Phil Schniter
ECE-6202
The Least-Squares (LS) Approach:
Until now weve focused on minimizing MSE Ecfw_|e(n)|2 . The
Wiener lter is optimal but requires knowledge of Rx (n) and
rxd (n). The LMS algorithm bypasses the need to know these
statistics but take
Phil Schniter
ECE-6202
Adaptive Filtering [Ch.9]:
Previously, we considered the design of W (z) in
x(n)
W (z)
d(n)
+
d(n)
e(n)
to minimize the MSE Ecfw_|e(n)|2 under the assumption that x(n)
and d(n) are jointly WSS RPs with known statistics.
Now we re
Phil Schniter
ECE-6202
Spectral Estimation [Ch. 8]:
We now take a more in-depth look at estimating the power
spectrum of an ergodic WSS RP x(n):
rx (k)ejk , rx (k) = Ecfw_x(n + k)x (n)
j
Px (e ) =
k=
from a nite-sample realization cfw_x(n)N 1
n=0
The re
Phil Schniter
ECE-6202
Because is diagonal, the GD update decouples into
(i+1)
ql
(i)
= 1 l ql
for l = 0, . . . , p 1,
implying that the initialization q(0) yields the time-i error
(i)
ql
= 1 l
Thus, limi q
(i)
i (0)
ql
for l = 0, . . . , p 1.
q1
= 0 i |
Phil Schniter
ECE-6202
Adaptive Filtering [Ch.9]:
Previously, we considered the design of W (z) in
x(n)
W (z)
d(n)
+
d(n)
e(n)
to minimize the MSE Ecfw_|e(n)|2 under the assumption that x(n)
and d(n) are jointly WSS RPs with known statistics.
Now we re
176 Problem Solutions
7.2 In this problem we consider the design of a threestep predictor using a rst-order lter
we) 2 mm) + w(1)z_l
In other words7 with min) the input to the predictor W the output
+ = w(0):c(n) + w(1):r(n - 1)
is the minimum meansquare
ECE-6202
Adaptive Filtering
Sp15
Homework #6
Apr. 23, 2015
HOMEWORK SOLUTIONS #6
1. (a) From Wiener theory, we know
w
=
R1 r xd for
x
Rx
=
Ecfw_x(n)xH (n) = Ecfw_d(n)dH (n)
r xd
=
Ecfw_x(n)d (n) = Ecfw_d(n)d (n)
and so
=
R
r (p 1)
=
r xd
r(0)
r (1)
.
.
.
Phil Schniter
ECE-6202
Levinson Recursion [Ch. 5] / Lattice Filters [Ch. 6]:
Many signal-processing applications (e.g., all-pole models) require
the solution of a set of p Hermitian Toeplitz equations.
We will see that this task requires O(p2 ) multipli
Phil Schniter
ECE-6202
Signal Modeling [Ch. 4]:
Having a parametric model for a signal x(n) allows us to
store and/or communicate it using only a few numbers,
estimate it from a noisy observation,
extrapolate it to an unobserved time interval (e.g., t
The Ohio State University
Department of Electrical and Computer Engineering
ECE 6202
STOCHASTIC SIGNAL PROCESSING
Instructor:
Prof. Phil Schniter, 616 Dreese Labs, [email protected]
E-Access:
http:/carmen.osu.edu
Oce Hours:
TBD. (Outside of oce hours, pl
Phil Schniter
ECE-6202
Random Variables [Ch. 3]:
Imagine an innite bag of marbles, each one labeled with a real
number. We draw a marble from the bag and record the number.
number if m4 drawn
Bag of
marbles
x
m1
m4
number if m3 drawn
m2
number if m2 draw
Phil Schniter
ECE-6202
Review of Linear Algebra:
In this class, a vector will be an element of CN , i.e., an array of
complex-valued numbers of functions. We will assume column
vectors and denote vectors using boldface font when typset, or
an underline w
Phil Schniter
ECE-6202
Periodogram Averaging:
Neither the periodogram nor the modied periodogram were
consistent estimators of the power spectrum. How do we build a
consistent estimator?
Suppose we had access to K uncorrelated realizations cfw_xi (n)K
i
Phil Schniter
ECE-6202
Modied Periodogram:
For the periodogram, we saw that
1
1
Pper (e ) = |XN (ej )|2 =
N
N
2
j
E Pper (ej ) = Px (ej )
wR (n)x(n)e
jn
n=
WB (ej ) for
wR (n) rectangular window
wB (n) Bartlett window
The modied periodogram Pmod (ej ) u
Phil Schniter
ECE-6202
Spectral Estimation [Ch. 8]:
We now take a more in-depth look at estimating the power
spectrum of an ergodic WSS RP x(n):
rx (k)ejk , rx (k) = Ecfw_x(n + k)x (n)
j
Px (e ) =
k=
from a nite-sample realization cfw_x(n)N 1
n=0
The re
Phil Schniter
ECE-6202
ARMA Spectral Estimation:
If we believe that x(n) is well modeled using an ARMA(p, q)
process, then we might rst try to estimate the coecients a(k)
and b(k) of the model and then estimate the power spectrum as
Px (ej ) = |H(ej )|2