ELEC 533: Second Exam
Professor Behnaam Aazhang
Instructions: Read carefully before beginning ! ! !
* This exam is a take home exam.
* The time for this exam is 2 hours. The exam must be taken in a contiguous 2-hour slot beginning
when you start reading the first problem.
* The completed exam should be uploaded by Midnight on Thursday 3rd of December.
* This exam is a closed books, notes, old exams, internet, homeworks and solutions. You are
allowed to use three cheat sheets 8.5 by 11 inches and two sided (you may write whatever you think is
necessary on these cheat sheets, as in definitions, formulas, Fourier transform formulas, densities formula,
etc.) For example. three sheets of notes: one for this exam, the one from the quiz, and the one from Exam
1. No other references are allowed.
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* All problems are weighted equally.
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1
1. Suppose {Nt; t ∈ R} is a zero-mean white Gaussian process which is the input to a linear time-varying
system with impulse response
h(t, τ) =

N∑
n=1
fn(t)fn(τ) : for 0 ≤ t ≤ 1, 0 ≤ τ ≤ 1,
0 : otherwise
where f1, f2, . . . , fN are functions satisfying
∫ 1
0
fn(t)fm(t)dt = δnm. Here δnm is zero when n 6= m and
it is 1 when n = m. Let {Xt; t ∈ R} denote the resulting output process (i.e., the output process when
the input, Nt, goes through the linear system h(t, τ)).
(a) Find the Karhunen-Loeve expansion of {Xt; t ∈ [0, 1]}. That is, find an expansion that would
write the process as a sum product of sequence of functions of time and random variables.
Xt =
∑
n
σn(t)Zn,
(b) What is the joint distribution of any two of the coefficients (i.e., random variables) in the expan-
sion? That is, fZnZm(u, v).
2. Recall that the spectral representation of the process Xt is defined by Xˆν as
Xˆb − Xˆa m.s.==
∫ +∞
−∞
∫ b
a
e−i2piνtdνXtdt ∀a ≤ b,
or alternatively as
Xt
m.s.
==
∫ +∞
−∞
e+i2piνtdXˆν ∀t
where i =
√−1.
(a) What is the spectral representation of X ′t, defined as the m.s. derivative of the process Xt? Write
it in terms of Xˆν .
(b) What is the spectral representation of Yt
m.s.
==
∫∞
−∞ h(t− τ)Xτdτ defined as the output of a linear
system when Xt is the input? Write it in terms of Xˆν and H(ν) = F{h(t)} where F{.} denotes
the Fourier transform.
3. Suppose {Yt; t ≥ 0} is a random process with mean function µY (t) = µt and autocorrelation function
RY (t, s) = σ
2 min (t, s) + µ2ts,
for t ≥ 0 and s ≥ 0 and where µ > 0 and σ2 > 0 are constants.
(a) Find the linear minimum mean square estimator (MMSE) of Yt+γ based on the observation of
{Yτ ; 0 ≤ τ ≤ t}. Also find the corresponding mean square error. Note γ is a known parameter.
(b) Suppose now that {Yt; t ≥ 0} is a homogeneous Poisson counting process with rate λ. Show that
the estimator from part (a) with µ = λ is actually the overall MMSE estimate of Yt+γ (over all
estimates, including non-linear ones).
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