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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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Lagrange interpolation method
Implementation in matlab :
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Lagrange interpolation method
Fractional delay in a FIR filter :
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Lagrange interpolation method
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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Windowing Method in matlab
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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.