PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Analog and Digital Filters 591

Observe that at W = 0.

H 3 (W = 0 ) = 0. Therefore, this function cannot be used to implement an LPF (H 3

(W = π) = 0 ). As a result, this function can be used to implement either a BS or an

HP fi lter.

The following sequence illustrates a type-3 FIR fi lter:

h(n) (n) (n ) (n )

(n )

 







0.019 0.3593 1 0.023 2

0.3593 3 0.01

 994 
(n )

R.6.123 The type-4 FIR fi lter is defi ned by

h(n) = −h (N − n) for 1 ≤ n ≤ N, with N = even

Then

H(W)
a (n)sin[W(n 1/2)]



eejNW j

n

N

22

1

2



4

∑

where

a(n) 4 
 2 h forn 1, 2,





N 

n

1 N

2

1

2









...

Observe that at W = 0, H(W = 0 ) = 0; therefore, this transfer function cannot be

used to implement an LPF.

The following sequence illustrates a type-4 FIR fi lter:

h(n) = −0.019
(n) + 0.3593
(n − 1 ) − 0.3593
(n − 3 ) + 0.019
(n − 3 )

Analytical examples of each FIR fi lter type are presented in the following points.

R.6.124 For example, analyze by hand a type-1 FIR fi lter with N = 5.

ANALYTICAL Solution

Then

H(z) = h( 0 ) + h( 1 )z−^1 + h( 2 )z−^2 + h( 3 )z−^3 + h( 4 )z−^4

Note that h( 0 ) = h( 4 ) and h( 1 ) = h( 3 ).

Then

H(z) = [h( 0 ) + h( 4 )z−^4 ] + [h( 1 )z−^1 + h( 3 )z−^3 ] + h(2)z−^2

H(z) = h( 0 )z−^2 (z^2 + z−^2 ) + h( 1 )z−^2 [z^1 + z−^1 ] + h( 2 )z−^2

substituting z by ejW(z → ejW)

H(ejW)^ = h( 0 )e−^2 jW(ejW^2 + e−jW^2 ) + h( 1 )e−jW^2 (e−jW + ejW) + h( 2 )e−^2 jW