Finite Impulse Response Filters - Introduction to Digital Signal Processing and Filter Design - page 258

Phase response of type I FIR filter Phase response unwrapped

4 2

3 0

2 − 2

1 − 4

0 6

− 1 − 8

− 2

− 10

− 3 − 12

4 − 14

−

0 0.5

Normalized frequency

1 1.5 2 0 0.5

Normalized frequency

1 1.5 2

( a ) ( b )

Figure 5.3 Linear phase responses of type I FIR filter.

continuous function of ω when it is unwrapped. The result of unwrapping the
phase (Fig. 5.3a) is to remove the jump discontinuities in the phase response
such that the phase response lies within π (Fig. 5.3b). If the order N of the

FIR filter is even, its group delay is an integer multiple of samples equal to N/ 2
samples. If the order N is odd, then the group delay is equal to (an integer plus
half) a sample. We will use all of these properties before we start the design of
FIR filters with linear phase.
The linear phase FIR filters have some interesting properties in the z plane
also. As seen in the examples, their transfer functions always contain pairs of
termssuchas[ z n

±

z − n ] . Denoting the transfer function of the FIR filters with
symmetric coefficients by H(z) , we write

±

N N

h(n)z − n n)z − n H(z) = = h(N − (5.23)

n = 0 n = 0

n) , we reduce the series n = 0 h(N

By making a change of variable m = (N − −

n)z − n to

N N

h(m)z − N + m z − N h(m)z m z − N H(z − 1 ) = =

(5.24)

m = m =

0 0

so we have the following result:

z − N H(z − 1 ) H(z) = (5.25)