Finite Impulse Response Filters - Digital Signal Processing and Applications with the C6713 and C6416 DSK - page 135

acteristics, such as selectivity. For an FIR filter to have linear phase, the coefficients
must be symmetric, as in (4.65).
The program Amplit. cpp (on the CD), described in Appendix D, plots the
magnitude and phase of a transfer function. It can be used to show that the coeffi-
cients in (4.65) yield a lowpass filter (use 1 as the coefficient for the denominator).

4.6 WINDOW FUNCTIONS

We truncated the infinite series in the transfer function equation (4.52) to arrive at
(4.55). We essentially put a rectangular window function with an amplitude of 1
between

Q and ignored the coefficients outside that window. The wider
this rectangular window, the larger Q is and the more terms we use in (4.55) to get
a better approximation of (4.52). The rectangular window function can therefore be
defined as

- Q and +

1
0

for
otherwise

nQ £

Ó

() =

(4.66)

wn

R

The transform of the rectangular window function w

R ( n ) yields a sinc function in

the frequency domain. It can be shown that

21
2

Q +

È
Í

Ê
Ë

ˆ
¯

˘
˙

sin p

Q 2 Q

Ê
Á

ˆ
˜ =

 Â

() =

W

jn
nQ

e

p

= e

- jQ p

e

jn p

(4.67) R

( )

sin p 2

=- n = 0

which is a sinc function that exhibits high sidelobes or oscillations caused by the
abrupt truncation, specifically, near discontinuities.
A number of window functions are currently available to reduce these high-
amplitude oscillations; they provide a more gradual truncation to the infinite series
expansion. However, while these alternative window functions reduce the ampli-
tude of the sidelobes, they also have a wider mainlobe, which results in a filter with
lower selectivity. A measure of a filter's performance is a ripple factor that com-
pares the peak of the first sidelobe to the peak of the mainlobe (their ratio). A com-
promise or trade-off is to select a window function that can reduce the sidelobes
while approaching the selectivity that can be achieved with the rectangular window
function. The width of the mainlobe can be reduced by increasing the width of the
window (order of the filter). Later, we will plot the magnitude response of an FIR
filter that shows the undesirable sidelobes.
In general, the Fourier series coefficients can be written as

()

CCwn
n

¢=

(4.68)

n