Variable fractional delay filter

=Introduction=

One important advantage of the variable digital filter is that the frequency characteristics, including the magnitude response and the phase response are tunable without re-implementation of the filter. Variable fractional delay (VFD) filters pay attention to the phase response, i.e., the phase is adjustable without redesigning the filter. They are very useful in the applications such as the time delay estimation, speech coding, discrete-time signal interpolation, speech-assisted video processing, sampling rate conversion system, etc. Because of the wide range of the application, the precise design of VFD filter is an important topic. Several topics on VFD filter design have been studied for a long time, and there has been various designing methods proposed including Lagrange-type VFD filter, weighted least-square (WLS) approach, maximally-flat methods, etc. WLS approach can achieve higher accuracy with independent variable magnitude and fractional-delay responses. Moreover, the complexity reduction by coefficient symmetry for variable digital filter is also proved. Considering the predominant advantage of WLS method, we will introduce the designing approach proposed by Tian-Bo Deng and Yong Lian in 2006 which is named as Weighted-Least-Square using coefficient symmetry.

Fundemantal Concept

A continuous signal "x(t)" which is delayed by an amount "t_d" can be represented as a linear operation: $y(t)\; =\; mathcal\{L\}\{x(t)\}\; =\; x(t-t\_d)$.

To discuss the discrete case, we define the sampling period "T_s" such that discrete signal "x [n] " can be related with its continuous counterpart as $x\; [n]\; equiv\; x(nT\_s),\; quad\; ninmathbb\{Z\},$.

Then, the equation for delay can be showed as: $y\; [n]\; equiv\; y(nT\_s)=mathcal\{L\}\{x(nT\_s)\}=x(nT\_s-t\_d)=x(nT\_s-frac\{t\_d\}\{T\_s\}T\_s)=x((n-frac\{t\_d\}\{T\_s\})T\_s)=x\; [n-D]$ (see the figure)
By defining $D=frac\{t\_d\}\{T\_s\}=x$ where D is the delay amount. It can be splitted into the integer part and the fractional part as: $D\; =\; n+p,\; quad\; ninmathbb\{Z\},$ and $pin\; [-0.5,0.5]$, where n is the integer part,i.e., $n=round(frac\{t\_d\}\{T\_s\})$ and p is the fractional part of the delay,i.e., $p=t\_d-n$.

Thus, the frequency response is: :$H(w)=\; frac\{Y(omega)\}\{X(omega)\}\; =\; e^\{-jomega\; D\}$
and the transfer function in Z-domain ais: :$H(z)=\; frac\{Y(z)\}\{X(z)\}\; =\; z^\{-D\}=z^\{-n-p\}$

However, from the viewpoint of the discrete-time signal processing, the delay amount "D" is just meaningful for the integer part. In other words, the output value will be the several previous point for the integer delay "D"; for the noninteger "D", the output will lie between two sampling points. The noninteger delay problem will be solved by resampling method for which the task is the multirate filter design technique; however, it requires a lot of computation and costs are high.

Variable Fractional Delay(VFD) FIR Filter Design: WLS using coefficient symmetry

There are so many methods studied for the VFD FIR filter design. Here, we will introduce a method which was proposed by Tian-Bo Deng and Yong Lian in 2006 named as weighted-least-square(WLS) design using coefficient symmetry. Firstly, we let the desired frequency response is $H\_d(omega,p)=exp^\{-j\; omega\; p\}$, and assume the variable frequency response is :$H(omega,p)=sum\_\{n=-N\}^\{N\}h\_n\; [p]\; e^\{-jomega\; n\}$, where $h\_n\; [p]\; =\; sum\_\{m=0\}^\{M\}h\; [n,m]\; p^\{m\}$, in which the FIR filter coefficients are represented by the polynomial expandsion with variable "p". Thus, the frequency response can be formulated as :$H(omega,p)=sum\_\{n=-N\}^\{N\}sum\_\{m=0\}^\{M\}h\; [n,m]\; p^\{m\}e^\{-jomega\; n\}=sum\_\{m=0\}^\{M\}(sum\_\{n=-N\}^\{N\}h\; [n,m]\; e^\{-jomega\; n\})p^\{m\}=sum\_\{m=0\}^\{M\}H\_m(omega)p^\{m\}$

This form can be implemented as the Farrow structure. Farrow structure is very useful for the variable digital filter since we can adjust the variables instantaneously. For the VFD case, the variables are the fractional part of the delay "p". The critical point is the coefficient symmetry; that is: $h\; [-n,m]\; =(\{-1\})^\{m\}h\; [n,m]$. In other words, :In addition, there is another constraint to be opposed:$h\; [n,0]\; =delta\; [n]$

The goal is to find the coefficients "h [n,m] " to approximate the disired frequency respose "H_d". By the weighted least-square(WLS) method, we can assume an error function:$ext\{e\}(omega,p)=\; H(omega,p)-H\_d(omega,p)$, and the objective function is :$J(omega,p)=\; int\_\{0\}^\{alphapi\}int\_\{-0.5\}^\{0.5\}W(omega,p)left|e(omega,p)\; ight|^2dpdomega$where "W($omega$,p)" is the weighted function. In order to find the optimal coefficients "h [n,m] " to minimize the objective function "J", we do the partial deriviation to the objective function with respect to the error function "e"::$\{part\; J\; over\; part\; e\}=0$By very complicate matrix operation(see the ), we can find the optimal "e" which will imply the optimal coefficients "h [n,m] " we want.