Fir digital filters  (Page 3/7)

To develop the theory for linear phase FIR filters, a careful definition of phase shift is necessary. If the real and imaginary parts of $H\left(\omega \right)$ are given by

where $j=\sqrt{-1}$ and the magnitude is defined by

and the phase by

which gives

in terms of the magnitude and phase. Using the real and imaginary parts is using a rectangular coordinate system and using the magnitude and phase isusing a polar coordinate system. Often, the polar system is easier to interpret.

Mathematical problems arise from using $\left|H\right(\omega \left)\right|$ and $\Phi \left(\omega \right)$ , because $\left|H\right(\omega \left)\right|$ is not analytic and $\Phi \left(\omega \right)$ not continuous. This problem is solved by introducing an amplitude function $A\left(\omega \right)$ that is real valued and may be positive or negative. The frequency response is written as

where $A\left(\omega \right)$ is called the amplitude in order to distinguish it from the magnitude $\left|H\right(\omega \left)\right|$ , and $\Theta \left(\omega \right)$ is the continuous version of $\Phi \left(\omega \right)$ . $A\left(\omega \right)$ is a real, analytic function that is related to the magnitude by

or

With this definition, $A\left(\omega \right)$ can be made analytic and $\Theta \left(\omega \right)$ continuous. These are much easier to work with than $\left|H\right(\omega \left)\right|$ and $\Phi \left(\omega \right)$ . The relationship of $A\left(\omega \right)$ and $\left|H\right(\omega \left)\right|$ , and of $\Theta \left(\omega \right)$ and $\Phi \left(\omega \right)$ are shown in [link] .

To develop the characteristics and properties of linear-phase filters, assume a general linear plus constant form for the phasefunction as

This gives the frequency response function of a length-N FIR filter as

and

[link] can be put in the form of

if $M$ (not necessarily an integer) is defined by

or equivalently,

[link] then becomes

There are two possibilities for putting this in the form of [link] where $A\left(\omega \right)$ is real: $K_1=0$ or $K_1=\pi /2$ . The first case requires a special even symmetry in $h\left(n\right)$ of the form

which gives

where $A\left(\omega \right)$ is the amplitude, a real-valued function of $\omega$ and $e^{-jM\omega }$ gives the linear phase with $M$ being the group delay. For the case where $N$ is odd, using [link] , [link] , and [link] , we have

or with a change of variables,

which becomes

where $\widehat{h}\left(n\right)=h(M-n)$ is a shifted $h\left(n\right)$ . These formulas can be made simpler by defining new coefficients so that [link] becomes

where

and [link] becomes

with

Notice from [link] for $N$ odd, $A\left(\omega \right)$ is an even function around $\omega =0$ and $\omega =\pi$ , and is periodic with period $2\pi$ .

For the case where $N$ is even,

or with a change of variables,

These formulas can also be made simpler by defining new coefficients so that [link] becomes

where

and [link] becomes

with