Finite Impulse Response Filters - Digital Signal Processing and Applications with the C6713 and C6416 DSK - page 131

From (4.50),

ka
1

== 04 .

11

0.4 can be verified using
(4.30). In the next chapter, we will continue our discussions on lattice structures in
conjunction with IIR filters.

Note that the values for the k -parameters k 2 =- 0.5 and k 1 =

4.5 FIR IMPLEMENTATION USING FOURIER SERIES

The design of an FIR filter using a Fourier series method is such that the magni-
tude response of its transfer function H ( z ) approximates a desired magnitude
response. The transfer function desired is

•
Â

() =

H w C e

jn w T

n < • (4.51)

d n

=- • n

where C n are the Fourier series coefficients. Using a normalized frequency variable

such that = f / F N , where F N is the Nyquist frequency, or F N = F s /2, the desired

transfer function in (4.51) can be written as

•
Â

() =

H C e

jn p

(4.52)

d n

=- • n

where w T = 2 p f / F s =p and | | < 1. The coefficients C n are defined as

1

()

Ú
Ú

C =

1
2

He

- jn pn

d

n d

- 1

1

() ( )

(4.53)

=

1
2

H cos n p - j sin n p d

d

- 1

Assume that H d ( ) is an even function (frequency selective filter); then (4.53)

reduces to

1

Ú 0

()

0

CH = cos n p d n (4.54)

n d

since H d ( ) sin n p is an odd function and

1

Ú

()

Hn sin p d = 0

d

- 1