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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 ...
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 Spring '08
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