0.6 Linear phase l_p filter design

Linear phase FIR filters are important tools in signal processing. As will be shown below, they do not require the user to specify a phase response in their design (since the assumption is that the desired phase response is indeed linear). Besides, they satisfy a number of symmetry properties that allow for the reduction of dimensions in the optimization process, making them easier to design computationally. Finally, there are applications where a linear phase is desired as such behavior is more physically meaningful.

Four types of linear phase filters

The frequency response of an FIR filter $h\left(n\right)$ is given by

In general, $H\left(\omega \right)=R\left(\omega \right)+jI\left(\omega \right)$ is a periodic complex function of $\omega$ (with period $2\pi$ ). Therefore it can be written as follows,

where the magnitude response is given by

and the phase response is

However $A\left(\omega \right)$ is not analytic and $\phi \left(\omega \right)$ is not continuous. From a computational point of view [link] would have better properties if both $A\left(\omega \right)$ and $\phi \left(\omega \right)$ were continuous analytic functions of $\omega$ ; an important class of filters for which this is true is the class of linear phase filters [link] .

Linear phase filters have a frequency response of the form

where $A\left(\omega \right)$ is the real, continuous amplitude response of $H\left(\omega \right)$ and

is a linear phase function in $\omega$ (hence the name); $K_1$ and $K_2$ are constants. [link] shows the frequency response for a linear phase FIR filter. The jumps in the phase response correspond to sign reversals in the magnitude resulting as defined in [link] .

Consider a length- $N$ FIR filter (assume for the time being that $N$ is odd). Its frequency response is given by

where $M=\frac{N-1}{2}$ . Equation [link] can be written as follows,

It is clear that for an odd-length FIR filter to have the linear phase form described in [link] , the term inside braces in [link] must be a real function (thus becoming $A\left(\omega \right)$ ). By imposing even symmetry on the filter coefficients about the midpoint ( $n=M$ ), that is

[link] becomes

Similarly, with odd symmetry (i.e. $h\left(k\right)=h(2M-k)$ ) equation [link] becomes

Note that the term $h\left(M\right)$ disappears as the symmetry condition requires that

Similar expressions can be obtained for an even-length FIR filter,

It is clear that depending on the combinations of $N$ and the symmetry of $h\left(n\right)$ , it is possible to obtain four types of filters [link] , [link] , [link] . [link] shows the four possible linear phase FIR filters described by [link] .

Irls-based methods

[link] introduced linear phase filters in detail. In this section we cover the use of IRLS methods to design linear phase FIR filters according to the $l_p$ optimality criterion. Recall from [link] that for any of the four types of linear phase filters their frequency response can be expressed as