EE567 – ADVANCED DIGITAL SIGNAL PROCESSING - FINAL
Date : 10thJune Monday, 2013
.
Due:14th June Fridayday, 16:00am
Instructor : Hüseyin ÖZKARAMANLI
Answer All Questions
(Group work will be penalized)
Q1. Consider the noise cancellation system shown below. The useful signal is a sinusoid
s(n)  cos(w0 n   ),
w0 = /12 where the phase  is a random variable
uniformly distributed from 0 to 2. The noise signals are given by
v1 (n)  0.95v1 (n  1)  w(n)
and
v2 (n)  0.7v2 (n  1)  w(n)
where the sequence w(n) is WGN(0,1).
a) Design an optimum filter of order M and choose a reasonable value M0 for M by
plotting the MMSE as a function of M.
b) Design an LMS filter with M0 coefficients and choose the step size  to achieve a
10% misadjustment.
c) Plot the signals s(n), s(n)+v1(n), v2(n), the clean signal e0(n) using the optimum
filter, and the clean signal elms(n) using the LMS filter and comment upon the
obtained results.
d) Re-solve the above noise cancellation problem using the RLS and comment on
the differences between RLS and LMS in terms of speed of convergence and
excess error.
Signal
source
s(n) + v1(n)
+
primary input
_
^
Noise
source
v2(n)
Reference
input
y (n)
Filter
e(n)
Q2.
Consider the process x(n) generated using AR(1) model
x(n)  0.729 x(n  1)  w(n)
where w(n) is WGN(0,1).
We want to design a linear predictor of x(n) using the Steepest Descent Algorithm
(SDA). Let
yˆ (n)  xˆ(n)  co,1 x(n 1)  co,2 x(n  2)  co,3 x(n  3)
a) Determine the 3x3 autocorrelation matrix R of x(n) and compute its eigenvalues
 
3
.
b) Determine the 3x1 crosscorrelation vector d .
c) Choose the step size  so that the resulting response is overdamped. Now
implement the SDA
i
i 1
ck  ck ,1 ck ,2
ck ,3   ck 1  2   d  Rck 1 
d) Repeat part (c) by choosing  so that the response is underdamped.
Q3. Problem 10.5 from Manolakis (Refer to the text for figures and equations)
Q4. Figure 4 shows a 2 band filterbank with analysis filters H 0 ( z ), and H1 ( z ) and
synthesis filters F0 ( z ), and F1 ( z ) . It is well known that filterbanks cannot be orthogonal
and symmetric at the same time. One possible solution to obtaining symmetric and
orthogonal filterbanks is what is known as biorthogonal filterbank constructions.
Biorthogonal filterbanks have the perfect reconstruction property
xˆ[n]  Ax[n  D]
where A and D are constants (D an integer).
In this problem we would like to design a 2 band bi-orthogonal filter bank where the
filters have the bi-orthogonality property
 h [ n ] f [ n  2k ]   [ k ] .
0
0
n
H0(z)
2
2
F0(z)
xˆ[n]  Ax[nD]
x[n]
+
H1(z)
2
2
F1(z)
Figure 4: Biorthogonal Filterbank
Assume that the lowpass synthesis filter in the time domain have impulse response
1
f0(n) = [ 1 4 6 4 1 ] / 8=  (n)  4 ( n  1)  6 ( n  2)  4 ( n  3)  1 ( n  4)  .
8
We desire the dual filter h0(n) to be able to interpolate polynomials upto degree 3. This
condition implies that the z-transform of h0(n), H0(z) should be of the following form
H 0 ( z )  B( z )Q( z )
where B(z) is a polynomial which has 3 zeros at z=-1 and Q(z) is an arbitrary polynomial
chosen in such a way that biorthogonality condition is satisfied.
It is pointed out that all filters H 0 ( z ) , H1 ( z ) , F0 ( z ) and F1 ( z ) are symmetric polynomials
possibly with a shift.
a) Derive the dual filter h0(n).
b) Complete the filter-bank by finding all the other filters. You should use alias
cancellation conditions for calculating the filters h1(n) and f1(n).
Note that perfect reconstruction condition is given by:
F0(z)H 0(z)  F1(z)H1(z)  2z l
F0(z)H 0(z)  F1(z)H1(z)  0
Where
Alias cancellation condition:
F0(z)H0(z)  F1(z)H1(z)  0
No Distortion condition:
F0(z)H0(z)  F1(z)H1(z)  2zl
For alias cancellation one can choose
F0(z)=H1(-z) and
F1(z) = -H0(-z)
c) Verify that the system has the perfect reconstruction property by letting x(n) be a linear
sequence of length 1000 (i.e., x(n)= 0 1 2 3 ……1000) and tracing it through the two
channels and finally reconstructing xhat. That is to say show that x(n) = A xhat (n - D) .
(verification of perfect reconstruction should be done in MATLAB)