PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Analog and Digital Filters 593

ANALYTICAL Solution

Then using the symmetry conditions and the fact that h( 4 ) must be zero the following

frequency spectrum is obtained:

H(ejW) = j[ 2 h( 0 )sin( 4 W) + 2 h( 1 )sin( 3 W) + 2 h( 2 )sin( 2 W) + 2 h( 3 )sin(W)]e−j^4 W

Clearly, ∠H(ejW) = − 4 W and the quantity inside the brackets represents its mag-

nitude spectrum.

R.6.128 Let us consider the frequency response of an FIR type-4 fi lter with N = 8, given as

follows:

H(ejW) = j[ 2 h( 0 )sin( 7 W/ 2 ) + 2 h( 1 )sin( 5 W/ 2 ) + 2 h( 2 )sin( 3 W/ 2 )

+ 2 h( 3 )sin(W/ 2 )]e−jW^7 /^2

Then ∠H(ejW) = −(7/ 2)W and the quantity inside the brackets represents the

magnitude spectrum.

R.6.129 The zeros on the z-plane of an FIR fi lter having real impulse response coeffi cients

occur in (complex conjugate) pairs. Applying symmetry and some mathematical

considerations, that are left as an exercise, the following general observations can

be stated about the locations of the fi lter’s zeros:

a. Ty p e -1 fi lter, zeros occur anywhere (inside, on, and outside the unit circle)

b. Ty p e -2 fi lter, one zero occurs at z = −1 and the remaining zeros anywhere

c. Ty p e - 3 fi lter, one zero occurs at z = 1 and one at z = −1 and the remaining zeros

occur anywhere

d. Ty p e - 4 fi lter, one zero occurs at z = 1, and the remaining zeros anywhere

R.6.130 FIR fi lters can be designed by choosing an ideal frequency select fi lter with an IIR,

and then limiting or truncating its impulse response.

R.6.131 Recall that the process of truncating or limiting a sequence is referred to as win-

dowing. A number of window models are accepted standards currently used by

engineers. Recall also that the standard windows were introduced and discussed

in Chapter 1.

R.6.132 FIR fi lters are characterized by having a fi nite number of coeffi cients in its impulse

response that affect the output of the fi lter in the following way:

a. By increasing the number of coeffi cients, the transition region decreases and

the fi lter approaches ideal brickwall conditions, introducing discontinuities in

the transition band (region between the pass and stop bands).

b. Any discontinuity when approximated by a Fourier series expansion shows

the Gibb’s effect in the form of oscillations. This effect can be reduced by

using windows, in which case the transition region becomes smoother (see

Chapter 1).

c. The use of windows generally causes a reduction of the ripple magnitude, but

also causes an increase of the transition band.

d. There are other ways to reduce the Gibb’s effects, but the simplest, easiest,

cost-effective, and the one generally preferred by engineers is the windowing

method.