Finite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 220

Notice that we obtain Equations (7.5) and (7.6) simply by treating the Fourier series expansion in the
time domain with the time variable t replaced by the normalized digital frequency variable U . The
fundamental frequency is easily found to be

u 0 ¼ 2 p=ðperiod of waveformÞ¼ 2 p= 2 p ¼ 1 (7.7)

Substituting u 0 ¼ 1 into Equation (7.6) and introducing hðnÞ¼c n , called the desired impulse
response of the ideal filter, we obtain the Fourier transform design as

Z p

1
2 p

Hðe jU Þe jUn dU hðnÞ¼ for N < n < N (7.8)

- p

Now, let us look at the possible z-transfer function. If we substitute e jU ¼ z and u 0 ¼ 1 back to
Equation (7.5) , we yield a z-transfer function in the following format:
HðzÞ¼ N

n¼ N hðnÞz n
/ þ hð 2 Þz

(7.9)

2 1

þ hð 0 Þþhð 1 Þz 1 þ hð 2 Þz 2

þ /

þ hð 1 Þz

This is a noncausal FIR filter. We will deal with this later in this section. Using the Fourier transform
design shown in Equation (7.8) , the desired impulse response approximation of the ideal lowpass filter
is solved as

Z p

1
2 p

Hðe jU Þe jU 0

For n ¼ 0 hðnÞ¼ dU

- p

Z U c

1
2 p

1 dU ¼ U c
p

¼

U c

Z p Z U c

1
2 p

1
2 p

Hðe jU Þe jUn dU ¼ e jUn dU For n s 0 hðnÞ¼

- p

U c

U c
U c ¼

¼ e jnU
2 pjn

e jnU c e jnU c
2 j

1
pn

sin ðU c nÞ
pn

¼ (7.10)

The desired impulse response hðnÞ is plotted versus the sample number n in Figure 7.3 .
Theoretically, hðnÞ in Equation (7.10) exists for N < n < N and is symmetrical about n ¼ 0; that
is, hðnÞ¼hðnÞ . The amplitude of the impulse response sequence hðnÞ becomes smaller when n
increases in both directions. The FIR filter design must first be completed by truncating the infinite-length
sequence hðnÞ to achieve the 2 M þ 1 dominant coefficients using the coefficient symmetry, that is,
HðzÞ¼hðMÞz M þ / þ hð 1 Þz þ hð 0 Þþhð 1 Þz 1 þ / þ hðMÞz M

1