fractional delay filters presentation

8/9/2019 Fractional Delay Filters Presentation

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Fractional Delay Filte

Aksh


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Introduction

What do we look for in a fractional delay filter :

Flat magnitude response.

Continuous control of the delay.

Efficient such that fast coefficient update takes place.

Some applications :

Synchronization in digital modems

Speech coding

Tuning of music


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Ideal Fractional Delay

)()( txty cc

A delay element, whose purpose is to delay an incoming continuo

signalxc(t) by (in seconds).

j

c

id

c

j

eX

H

Xe

)(

)(Y)(

)()(Y

c

c

The transfer function Hid() of the delay element can be expresse

the below equation.


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Discrete-Time System

Consider a discrete-time delay system where Dis the desire

y(n) = x(n - D) The spectrum of a discrete-time signal can be expressed by

the discrete-time Fourier transform (DTFT).

where D = Dint+ d and 0


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Characteristics of the Ideal FractionalDelay Element

D

D

DeH

eH

eH

idg

idp

j

id

j

id

Djid

)(

)(

)}(arg{

1|)(|

||,)(

,

,

Phase delay

Group delay

Ideal phase

response

The delay element is a linear-phase all pass system.


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Inferences from ideal impulse response

The impulse response hid(n) of the delay element is a shifte

sampled version of the sinc function which is infinitely long.

The filter is not BIBO stable.

This kind of ideal filter is nonrealizable.

Some finite-length approximation for the sinc function has t


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Design of fractional delay filters

Lagrange interpolation design

Make the error function maximally flat at a certain frequency by sderivative to 0.

This leads to the Lagrange interpolation formula

for n= 0,1,2N

N

k kn

kDnh

0

)(

Advantages

This method gives excellent approximation at low frequencies

Easy formula for finding out the coefficients.

Ideal method where good accuracy at high frequencies is not r


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Lagrange interpolation method

Odd vs. ev

filters.

Tradeoff b

magnitude

phase delay

[1]


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Implementation in matlab :


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Fractional delay in a FIR filter :


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Change in the magnitude response


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Least Squared Integral Error Design

One of the most attractive method for designing realizable FD filters a

obtained by the truncation of the ideal impulse response.

This is for 0


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Windowing Method

A method basically continuation of the Least squares method.

A method to reduce the Gibbs phenomenon in FIR filter design is to us

function.

Here the ideal impulse response is truncated and shaped which is of l

N+1 with the help of a window function.

The obtained impulse response does not impact any least squared me

it is a modification of the least square method.


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Windowing Method

The coefficients are given by :

The windowing method is suitable for real-time systems where the fra

delay is changed since the coefficients can be updated quickly.

Also this method is fast and easy.


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Windowing Method in matlab


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REFERENCES

[1] Laakso, Timo I.; Vlimki, Vesa; Karjalainen, Matti; Laine, Unto K. (

1996), Splitting the unit delay - tools for fractional delay filter design,

Processing Magazine.

INTERPOLATED ALLPASS FRACTIONAL-DELAY FILTERS USING ROOT

DISPLACEMENT. Huseyin Hachabiboglu, Banu G unel, Ahmet M. Kond

Principles of fractional delay filters; V Valimaki, TI Laakso.

Fractional Delay Digital Filters ;Javier Diaz-Carmona and Gordana Jova

Dolecek Institute ITC Celaya, Institute INAOE Puebla, Mexico.

fractional delay filters presentation

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