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SGN-21006 Advanced Signal Processing
Examination on Wednesday 13.12.2018

Examiner Ioan Tabus, room TF 419
Calculators can not be used in the exam

The sheet with exam guestions can remain at the student

1. (5 points) Draw the diagram of an adaptive filter block and specify the involved signals. Draw

K

D.

the block diagram of adaptive noise cancelled (or channel egualization, or echo cancellation)
and specify the meaning in this application of each signal.

(a) (4 points) Find the least sguares estimate of wo in the very simple filter model y(n) = wo

when the desired data d(n) is given for n = 1,...,N (therefore the model is the particular
case of d(n) = y(n) + e(n) = wou(n) + e(n), where input is assumed constant u(n) = 1
for n = 1,...,N). What is the intuitive meaning of the estimate formula that you have
obtained?

(b) (3 points) Find the recursive least sguares solution wo(N) for the model at (a) as a
simple eguation connecting wo(N) to wo(N — 1) (by elementary derivations, no need to use
the general RLS eguations). Try to find also the exponentially weighted solution (using the
weighting function B(n,i) = A).

(4 points) Prove the linearity property of the expectation operator: given two discrete ran-
dom variables, X and Y, with joint pmf g(x, y) = Prob(X = 1; Y = y) and two constants a
and b, then F[aX + bY] = aE[X] + bEIY].

(3 points) State the ”Recursive least sguare” problem.

(6 points)

(a) (2 points) The first definition of the power spectrum density (PSD) is

[0.0]

w)= YZ rike iv

k=- 0

What modification to the definition is needed for transforming it into a method for
estimating the PSD, given the data y(1),...,y(N)?

(b) (2 points) What are the two main ways of estimating the autocovariance function
T(7) from a set of data y(1),...,y(N)? Which of the two methods is preferred as an
intermediate step for estimating the PSD by the correlogram method?

(c) (2 points) A second definition of the power spectrum density (PSD) is

2
: L
a) = EAN

Y ylt)e

t=1

 

 

What modifications to the definition are needed in order to transforming it into a
method for estimating the PSD, given the data y(1),...,y(N)?