This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: University of Minnesota
Dept. of Electrical and Computer Engineering _ EE 8581
DETECTION AND ESTIMATION THEORY
Spring 2010 March 4, 2010 Student ii): ' Solve the three problems that appear on the next pages. Problem 1: Mark has built a new powermeasuring device. The device outputs a reading that is equal
to samples of the square of the current i(k), 1 S k S 10 ﬂowing through a 1 £2 resistor. Mark is not sure whether his new device is dominated by thermal white noise or whether
the noise is a colored Gaussian noise. If the noise is white, the readings can be modeled
as the sum of the square of the current i(k) and a zeromean white Gaussian noise process
w(k), 1 S k S 10, of variance 1. If the noise is colored, the readings can be modeled as the sum of the square of the current i(k) and a zero—mean Gaussian process n(k),
1 S k S 10 , with a 10x10 covariance matrix equal to Kn = In the above equation the (if) element of matrix Kn is the covariance of 110') and n0),
i.e., Kn = [E(n(i)n(i))], and the vectors {$2} are known orthonormal 10x1 vectors. In particular, if Mark applies a current i(k) at the input he will measure:
H1: r(k) = i2(k) +w(k), 1S k S 10,
H0: r(k) = i2(k)+n(k), 1S k S10.
1. Find a 10x10 coordinate transformation matrix U such that 7 = U F: where 7" = [r(k)], 1 S k S 10, 7 = [r'(k)], 1 S k S 10, and the cOvariance matrices
of 7' are diagonal under H] and Ho. (The covariance matrices of ?’ under H1 and
Ho are not equal.) W: m a; aim/afar mamch gleamkt
J is eMd/é Kn ;., 4
M i/fﬂaiétﬁ 7%“ 2. Formulate the binary hypothesis testing problem in the new coordinate system
and devise a test to help Mark determine which of the two noise sources is
affecting his measurement. Assume that you can use a signal generator that
produces any desired current i(k) and that you can record both i(k) and the output measur:;;4(kc)lfor ’17:; 105mm F) Wu M’hj U I? H': 5001‘; PHD/4) l‘Lé‘a 3. Is the test singular, that is, can you achieve perfect detection or rejection? Problem 2:
1. Let x and y be two random variables and let 2mg (y) be the Bayes minimum mean
squared estimate of x given y. Using the properties of iterated expectation and/or
the orthogonality principle show that the following three results hold: a. EOC) = E( fms b. E(xy) = E(y @1520»
c. E ((x — 2mg (y)) ) = 50:2) — Ecx 2mm). m flit %Jg)=£ [x \Q
EM: 53(E(x\$))= £39?"le
£605); : E3 XN} EM— 53:43)] ‘ (03%; Margie > A. l).
c. {(QX’MM
m Emma) ,— 5, gm] Hm ' E ( ((X"£"5U»2> ’ E581 ’ ffxiknslslj 2. Using the above results or otherwise, answer the following questions. Suppose w
and z are scalar random variables, and that 1
132(2) = 5' ’Z' < 1'
0, otherwise. You are told that the Bayes leastsquares estimate of w given an observation 2 is 1
1 “.220
A _ . __ 2 A
Wms — —§szgn(z) —
E, and its associated meansquare estimation error is Ams = 1/12. However, you would
prefer to, use the following adhoc estimator WAH = —Z. (a) Is it possible to determine bCWAH) = E (w — wAH ), the bias of your new
estimator, from the information given? If your answer is no, brieﬂy
explain your reasoning. If your answer is yes, calculate b(wAH). E[U“Wna]: £2 [ mate] +2]
New {Bole}; )Q 9M?) l
5") £3[E[Wl?))= 4/; ‘ y), ogJZ’l' YZ 'YLA an 51%} 0 (b) Is it possible to determine RAH = E ((w — WAH)2), the meansquare
estimation error obtained using this estimator, from the information given? If your answer is no, brieﬂy explain your reasoning. If your answer is yes,
calculate AAH. Problem 3:
Let N
y = 2 xi.
i=1 Here the xi are independent zero mean Gaussian random variables with variance 1. We
observe y. In parts 1 to 4, treat N as a continuous deterministic variable. 1. Find the maximum likelihood estimate IVMLE of N. 2. Is IVMLE unbiased?
A , g 1: 2]
E[Nme *— N ' 7 3
. Wh
at
is
th
e
v
arian
c
e
of
1V
M
LE
? \[M
[M
: E
[7*]
 N 7’ 3W
kI2
2N2 .— 4. Is IVMLE efﬁcient? £[ 32' gnﬂgllv W: V ...
View
Full
Document
This note was uploaded on 06/04/2010 for the course EE 8581 taught by Professor Staff during the Spring '08 term at Minnesota.
 Spring '08
 Staff

Click to edit the document details