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Matlab assignment and exercises
Estimation and Detection (ET 4386)
Matlab assignment and exercises
Outline
Matlab assignment
Homework 1, problem 3: best linear unbiased estimator (BLUE)
The maximum likelihood estimator (MLE)
Exercise 1: exponential pdf family
Exercise 2: a non-Gaussian example
Exercise 3: the BLUE and MLE for a Laplacian pdf
The least squares estimator (LSE)
Exercise 4: parameter estimation by LSE
1
Application to Communications
s[m℄
x[m℄
n[m℄
channel
s[m℄
x[m] = h[m] ⋆ s[m] + n[m] =
x[m] = s[m] ⋆ h[m] + n[m] =
x[m℄
h[m℄
L−1
X
h[l]s[m − l] + n[m]
l=0
K−1
X
k=0
s[k]h[m − k] + n[m]
2
h[l] has length L
s[k] has length K
Application to Communications
Defining x = [x[0], . . . , x[K + L − 1]]T and n = [n[0], . . . , n[K + L − 1]]T , we obtain
Symbol estimation model:

h[0]

..
..

.
.

H=
h[L − 1]
..

.

Channel estimation model:

s[0]

..
..

.
.

S=
s[K − 1]
..

.

x = Hs + n




h[0] 

..

.

h[L − 1]
s = [s[0], . . . , s[K − 1]]T
x = Sh + n




s[0] 

..

.

S[K − 1]
3
h = [h[0], . . . , h[L − 1]]T
Application to Communications
Most communications systems (GSM, UMTS, WLAN, ...) consist of two periods:
Training period: During this period we try to estimate the channel by transmitting
some known symbols, also known as training symbols or pilots.
Data period: During this period we use the estimated channel to recover the
unknown data symbols that convey useful information.
What kind of processing do we use in each of these periods?
During the training period we use one of the previously developed estimation
techniques on the channel estimation model, x = Sh + n, assuming that S is
known.
During the data period we use one of the previously developed estimation techniques on the symbol estimation model, x = Hs + n, assuming that H is known.
4
Application to Communications
Channel estimation
Let us assume that cov(n) = C = σ 2 I
BLUE, LSE (or when the noise is Gaussian also the MVU and MLE):
ĥ = (SH S)−1 SH x
LMMSE (or when the noise and channel are Gaussian also the MMSE):
−1 H
ĥ = (SH S + σ 2 C−1
h ) S x
Remark: Note that the LMMSE estimator requires the knowledge of Ch = E{hhH }
which is generally not available.
5
Application to Communications
Symbol estimation
Let us assume that cov(n) = C = σ 2 I
BLUE, LSE (or when the noise is Gaussian also the MVU and MLE):
ŝ = (HH H)−1 HH x
LMMSE (or when the noise and symbols are Gaussian also the MMSE):
−1 H
ŝ = (HH H + σ 2 C−1
s ) H x
Remark: Note that the LMMSE estimator requires the knowledge of Cs = E{ssH }
which can be set to σs2 I if the data symbols have energy σs2 and are uncorrelated.
6
Field estimation using Wireless Sensor Network (WSN)
Static field estimation
Linear data model: y = Mu + n. Measurements: y of size M ×1 (given with
location of sensors). Temperature over the line: u of size (N ×1 with N = 20).M
is an M ×N measurement or observation matrix. n is an M ×1 WGN vector with
zero mean and covariance matrix σ 2 I.
Covariance matrix Cu of size N ×N with [Cu ]ij = e−(0.3kxi −xj k) with i, j = 1 . . . N .
Here xi = [xi , yi ]T and xj = [xj , yj ]T are the coordinates of the i-th and j-th points
on the line.
Estimate u using LMMSE estimator ( σ = 0.1, E(u) = µI, µ = 16.4)
7
Homework 1, problem 3
Best linear unbiased estimator
If a parameter vector θ is linearly related to a parameter vector a,
a = Bθ + b
where B is a known p × p invertible matrix and b is a known p × 1 vector, their
BLUEs maintain the same linear relation, i.e.
â = Bθ̂ + b
Assume that x = Hθ + w, where E(w) = 0 and E(wwT ) = C, then BLUE is given
by
T
θ̂ = H C
−1
H
8
−1
HT C−1 x
The maximum likelihood estimator (MLE)
The maximum likelihood estimator finds the θ that maximizes p(x; θ) over θ for a
given x,
θ̂ = arg max p(x; θ)
θ
or,
θ̂ = arg max ln p(x; θ)
θ
Taking the derivative of the log likelihood function we obtain the following equation
to find θ
∂ ln p(x; θ)
=0
∂θ
or
9
∂p(x; θ)
=0
∂θ
Exercise 1
The MLE for the exponential family of pdfs
Consider N IID observations from the exponential family of pdfs,
p(x; θ) = exp [A(θ)B(x) + C(x) + D(θ)]
where A, B, C and D are functions of their respective arguments. Find an equation to be solved for the MLE.
Apply now the results to the following pdfs:
(a) Gaussian
1
1
p(x; µ) = √ exp − (x − µ)2
2
2π
(b) Exponential

 λ exp(−λx), x > 0
p(x; λ) =
 0,
x<0
10
Exercise 2
The MLE for a non-Gaussian example
For N IID observations from a µ[0, θ] pdf, find the MLE for θ.

 1, 0 < x < θ
θ
p(x; θ) =
 0, otherwise.
11
Exercise 3
The BLUE, MLE and CRB for a Laplacian pdf
Consider N IID observations x[0], x[1], . . . , x[N − 1] from the Laplacian pdf,
p(x; θ) =
1
exp (−|x − θ|)
2
(a) Find the BLUE for θ.
(b) Find the MLE for θ by using a single observation.
12
The least squares estimator (LSE)
Consider the general model,
x = H(θ, w)
where w is some noise vector, the LSE of θ is given by
θ̂ =arg min ||x − H(θ, 0)||2
θ
For a linear model, the LSE of θ is given by
T
θ̂ = H H
−1
HT x
No assumption about the noise.
The orthogonality condition
0T = (x − Hθ)T H
13
Exercise 4
The LSE for parameter estimation
For the signal model
s[n] =
p
X
Ai cos [2πfi n] ,
i=1
n = 0, 1, . . . , N − 1
where the frequencies fi are known and the amplitudes Ai are to be estimated,
find the LSE normal equations (do not try to solve them). The observation is
x[n] = s[n] + w[n] where w[n] is independent noise.
If the frequencies are specifically known to be fi =
i
N,
explicitly find the LSE and
the minimum LS error.
Finally, if x[n] = s[n] + w[n], where w[n] is WGN with variance σ 2 , determine the
pdf of the LSE, assuming the given frequencies.
14
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