Solutions for Assignment #4
Problem 1 Part (a)
The Leaky-LMS recursion is
b n+1 = (1 )w
b n rn (rH
b n dn )
w
nw
b n + rn dn
= [(1 )I rn rH
n ]w
(15)
b n is independent
where I is the M M identity matrix. Taking the expectation of both sides (w
of rn ),
b
Due Date: February 11, 2014 (In class)
Assignment #1
1. Consider the most trivial of all linear array processors that has beamformer weight values
wi = 1 i = 1, 2, . . . , M . Assume that a unit-value signal (m(t) = 1) on a carrier
frequency fc is arrivin
Solutions for assignment #3
Problem 1
By the Matrix Inversion Lemma, if A = B1 + CD1 CH , where A CM M , B CM M ,
b
C CM L , D CLL , then A1 = B BC(D + CH BC)1 CH B. Set A = R(k),
B =
k b
1
[R(k 1)] , C = rk , D = k. Then,
k1
k b
k b
k b
[R(k 1)]1
[R(k 1
Solution for assignment #2
We list the expressions of the max-SINR, MMSE, MVDR, and ML beamformers:
wmaxSIN R = R1
i+n s , C,
1
wM M SE = R s , C,
R1 s
wM V DR = H 1 ,
s R s
R1 s
.
wM L = H i+n
s R1
i+n s
(1)
(2)
(3)
(4)
First, wmaxSIN R and wM L are dire
Assignment #6
Dead line: April, 17, Thursday (In classs)
1. You have an antenna array with M = 7 elements. Generate one BPSK signal with SNR
= 10 dB and random, unknown but fixed, angle of arrival 1 .
(a) Plot PbM F () and PbM V DR () using a data record
Deadline: 13th Mar., (Thursday) in class
Assignment #4
1. It can be shown that if the learning gain is such that 0 < < max1 + where max is
the maximum eigenvalue of the input autocorrelation matrix and is the norm-square
weight, Leaky-LMS converges in the
Solutions for Assignment #5
10
LMS
RLS
CMA
0
10
|y()|2 (dB)
20
30
40
50
60
70
80
60
40
20
0
20
40
60
80
Fig. 13. Assignment 4, Problem 3: LMS, RLS and CM algorithms after 1000 iterations.
Comments:
As in Problem 3 of Assignment # 3, we have four BPSK sign
Solutions for Assignment #5
10
LMS
RLS
CMA
0
10
|y()|2 (dB)
20
30
40
50
60
70
80
60
40
20
0
20
40
60
80
Fig. 13. Assignment 4, Problem 3: LMS, RLS and CM algorithms after 1000 iterations.
Comments:
As in Problem 3 of Assignment # 3, we have four BPSK sign
Due date: April 1, (in class)
Assignment #5
1. Consider again the antenna-array and signal model of Problem 3 of Assignment #3.
Fix the SNR of the user of interest at 12dB and the SNRs of the interferers at 10dB, 12dB,
and 14dB. Run the normalized LMS, RL
Dead line: 4th march, tue. (in class)
Assignment #3
1. Use the Matrix Inversion Lemma (Woodburys identity) to derive a recursion for the
1
b
inverse of the estimated input autocorrelation matrix [R(k)]
based on the sampleaverage recursion
b 1) + rk rH
(k
Solutions for Assignment #6
Problem 1, Part (a)
25
MF
MVDR
20
15
P() (dB)
10
5
0
5
10
15
20
25
80
60
40
20
0
20
40
60
80
Fig. 14. Assignment 5, Problem 1, Part (a): PbM F () and PbM V DR () using a data record size of
N = 20 samples.
25
MF
MVDR
20
15
P()