Impulse Response Measurement and Equalization Digital Signal Processing LPP Erasmus Program Aveiro 2012 Digital Signal Processing LPP Erasmus Program Aveiro

Impulse Response Measurement and Equalization

Digital Signal Processing LPP Erasmus Program

Aveiro 2012

Impulse response measurement and equalization LPP Erasmus Visit, Aveiro 2012

Impulse response measurement

Measurement of impulse responses is a common task in different areas (specially important in audio processing)

Typically, PC hardware is used to perform this task using the following setup:

The following assumptions are made:The whole system between x[n] and y[n] shows time-invariant behaviour.All involved components are sufficiently linearThe impulse response can be considered finite (h[n]≈0 for n > M)

h(t)DACADC

x[n]

y[n]

x(t)

y(t)PC



In theory, any input signal can be used to measure the impulse response of a linear system.The output signal can be expressed in terms of this impulse response by:

Assuming that the impulse response only has N significant samples, the former equation can be rewritten in matrix form as:

0

][][][][][k

knxkhnxnhny

]1[

]1[]0[

]1[00

0]0[]1[00]0[

]2[

]1[]0[

Nh

hh

Lx

xxx

LNy

yy

where L is the length of the input signal



Then, the impulse response can be theoretically obtained as:

]2[

]1[]0[

]1[00

0]0[]1[00]0[

]1[

]1[]0[ *1

LNy

yy

Lx

xxx

Nh

hh

where X-1* represents the pseudo-inverse of matrix X: (XT·X)-1·XT or XT·(X·XT)-1

However, in practice, very long input signals would be needed to attain high suppression of measurement noise, resulting in extremely large problems which may turn out to become computationally intractable.



An atractive approach consist in using as input signal one whose auto-correlation function is similar to a delta sequence:

In this case, the impulse response of the system can be obtained by cross-correlating the output and the input signals:



Assuming that the length of the input signal is K and that the impulse response has only N significant samples, we have:

][·][][1

0

knxkhnyNK

k

][]·[1][1

0

nxnyK

RK

nyx

][]·[1][][]·[]·[1)(1

0

1

0

1

0

1

0

nxknxK

khnxknxkhK

RNK

k

K

n

K

n

NK

kyx

kk

nxknxK

K

n

01

][]·[1 1

0

][][ hRyx



Using this indirect approach, a great inmunity against noise is achieved:

The energy of the input signal can be as large as desired, since we have just to extend its duration.

As noise corrupting the output signal is uncorrelated with the input signal x, its contribution in the measurement of the impulse response is very small:

][][]·[][1

0

nrknxkhnyNK

k

][][]·[1][][

0

1

0

hnxnrK

hRK

nyx



Typical test signals:

Chirp(Continuous-

time frequency varying

sinusoid)

PR sequence(13-bit Barker

Code)

White Gaussian Noise

x(t) Rxx()

0 20 40 60 80 100 120-1.5

-1

-0.5

0

0.5

1

1.5

0 50 100 150 200 250-40

-20

0

20

40

60

80

0 20 40 60 80 100 120-1.5

-1

-0.5

0

0.5

1

1.5

0 50 100 150 200 250

-50

0

50

100

150

0 20 40 60 80 100 120-2

-1

0

1

2

0 50 100 150 200 250

0

50

100

150


Impulse response equalization

An Equalizer is a system whose main objective is to undo the effect of a previous stage, typically a transmission channel:

Ideally, the composite impulse response of the channel and the equalizer is a delta (flat frequency response), so that the output of the equalizer is a delayed version of the input signal.

1. Fixed equalizers (its coefficients do not change with time)2. Adaptive equalizers (its coefficients are updated periodically based on the

current channel characteristics)

We can distinguish between two types of equalizers:

+Channelc[n]

Equalizerg[n]][ns ][̂ns

][nn


Fixed Equalizer (Zero-Forcing)


In zero-forcing equalization, the equalizer response g[n] attempts to completely inverse the channel by forcing :

][][][ 0nnngnc

0

1

0

]1[

]1[]0[

]1[00

]1[]1[0]0[]1[0]1[

]0[]1[00]0[

Ng

gg

Lc

cLccLc

ccc

c

Or using matrix representation:

1~·~~ *1Cg



Adaptive Equalizer

When the channel is time-varying (LTV), it is necessary to update the equalizer coefficients in order to track the channel changes.

Define the input signal to the equalizer as a vector ck where:

and an equalizer weight vector gk, where

Then, the output signal is given by:

And the error signal ek is given by:

Tk Nkckckckcc ]1[]2[]1[][

TNk kgkgkgkgg ][][][][ 1210

kTkkkkk grssse ˆ

kTkk grs ˆ



An adaptive channel equalizer has the following structure.

Adaptive Equalizer



The least mean-square (LMS) algorithm carries out the mean squared error by recursively updating the coefficients using the following rules:

kkkk

kkk

kTkk

reggssegrs

·ˆ

ˆ

1

The step parameter , controls the adaptation rate and must be chosen carefully to guarantee convergence.

The equalizer is converged if the error ek becomes steady.

Adaptive Equalizer

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Impulse Response Measurement and Equalization Digital Signal Processing LPP Erasmus Program Aveiro 2012 Digital Signal Processing LPP Erasmus Program Aveiro