Fir digital filters  (Page 2/7)

In the special case of Type 1 filters with $L$ equally spaced sample points, the samples of the frequency response are of theform

For Type 2 filters,

For Type 3 filters,

For Type 4 filters,

Although this section has primarily concentrated on linear-phase filters by taking their symmetries into account, themethod of taking the DFT of $h\left(n\right)$ to obtain samples of the frequency response of an FIR filter also holds for generalarbitrary linear phase filters.

Zero locations for linear-phase fir filters

A qualitative understanding of the filter characteristicscan be obtained from an examination of the location of the $N-1$ zeros of an FIR filter's transfer function. This transfer function is given by the z-transform of the length-N impulseresponse

which can be rewritten as

or as

where $D\left(z\right)$ is an $N-1$ order polynomial that is multiplied by an $N-1$ order pole located at the origin of the complex z-plane. $D\left(z\right)$ is defined in order to have a simple polynomial in positive powers of $z$ .

The fact that h(n) is real valued requires the zeros to all be real or occur in complex conjugate pairs. If the FIR filter islinear phase, there are further restrictions on the possible zero locations. From [link] , it is seen that linear phase implies a symmetry in the impulse response and, therefore, in thecoefficients of the polynomial $D\left(z\right)$ in [link] . Let the complex zero $z_1$ be expressed in polar form by

where $r_1$ is the radial distance of $z_1$ from the origin in the complex z-plane, and $x$ is the angle from the real axis as shown in [link] .

Using the definition of $H\left(z\right)$ and $D\left(z\right)$ in [link] and [link] and the linear-phase even symmetry requirement of

gives

which implies that if $z_1$ is a zero of $H\left(z\right)$ , then $1/z_1$ is also a zero of $H\left(z\right)$ . In other words, if

This means that if a zero exists at a radius of $r_1$ , then one also exists at a radius of $1/r_1$ , thus giving a special type of symmetry of the zeros about the unit circle. Another possibilityis that the zero lies on the unit circle with $r_1=1/r_1=1$ .

There are four essentially different cases [link] of even symmetric filters that have the lowest possible order. All higherorder symmetric filters have transfer functions that can be factored into products of these lowest order transfer functions.These are illustrated by four basic filters of lowest order that satisfy these conditions: one length-2, two length-3, and onelength-5.

The only length-2 even-symmetric linear-phase FIR filter has the form

which, for any constant $K$ , has a single zero at $z_1=-1$ .

The even symmetric length-3 filter has a form

There are two possible cases. For $\left|a\right|>2$ , two real zeros can satisfy [link] with $z_1=r$ and $1/r$ . This gives

The other length-3 case for $\left|a\right|<2$ has two complex conjugate zeros on the unit circle and is of the form