Signal Processing - May 2016 - 109

symmetric. Thus, H ccb (z) has a linear
phase response [5]. Because the CCF is
sufficiently simple, the computation
complexity is limited despite numerous
CCFs being used. There are K zeros for
a CCF, and its zeros are used to reshape
H pro (z) . The zeros of H ccb (z) are
obtained by solving H ccb (z) = 0 .
Applying De Moivre's formula [6] to
z K = - 1 yields [4]

K

Hsha(z ) X (z )

Hsha(z )

Hpro(z )

Shaping Filter

Y (z ) = Hsha(z ) Hpro(z ) X (z )

Prototype Filter

FIGure 2. The structure of a composite FIR filter.

0

(4)

-10
-20
-30
-40
K=1
K=3

-50

, n = 0, 1, f, K - 1.
(5)

Figure 3 displays the magnitude frequency responses of CCFs for
K = 1, 2, 3, 4 . In the following discussion, all of the filter frequency
responses at ~ = 0 rad/sample are normalized to 0 dB to facilitate comparison. Although the CCFs are simple to
implement, their frequency responses
are not satisfactory for reshaping an
LPF. We must elaborate on the combination of CCFs of various lengths to
reshape the prototype filter.
Figure 4 shows an example of cascading CCFs of orders K = 1, 2, 3, 4, 5.
Using a cascade of CCFs to reshape an
LPF is markedly more effective than
using a single CCF. Note that the shaping filter has no multiplier; it uses nine
adders and 24 memory units. A by-product of the shaping filter is that it preserves the linear phase response for the
composite filter. One trick for improving a prototype filter is to use cascaded CCFs to reshape it. Another
method, called the modified window
design method, is to deliberately
design a prototype filter that has an
overshoot at the passband edge so that
the composite filter has a flat passband. We then detail the design of a
lowpass prototype filter.

-60

0

0.5

1

1.5
2
ω (rad./Sample)

K=2
K=4
2.5

3

FIGure 3. The magnitude frequency responses of CCFs for K = 1, 2, 3, 4.

0
-50
Magnitude (dB)

r + 2rn

FIR

10

Comparing (4) with z = e j~ =
cos ^~h + j sin ^~h , we can locate zeros
of H ccb (~) as
~n =

X (z )

Magnitude (dB)

z = K 1 ;cos c r + 2rn m
K
r + 2rn
+ j sin c
mE,
K
n = 0, 1, f, K - 1.

FIR

-100
-150
-200
-250
-300

0

0.5

1

1.5
2
ω (rad./Sample)

2.5

3

FIGure 4. The magnitude frequency response of a cascade of CCFs. The transfer function is
2
2
3
H sha ^ z h = ^1 + z -1h ^1 + z -2h ^1 + z -3h ^1 + z -4 h^1 + z -5 h.

Modified window design method
The window method for FIR filter
design begins with determining the cutoff frequency of a continuous LPF,
which is plotted in a solid black line in
Figure 5(a). The continuous LPF is
assumed to be ideal with a unity gain at
low frequencies and zero (infinite attenIEEE SIgnal ProcESSIng MagazInE

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May 2016

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uation) beyond the cutoff frequency [1].
Let the sample rate of the discrete filter
be fs . The frequency response H ( f ) on
6- fs /2 fs /2@ is then simulated using a
discrete frequency-domain representation H (m) with N samples, which are
plotted in black dots in Figure 5(b). The
passband of H ( f ) is equivalent to that
109
Table of Contents for the Digital Edition of Signal Processing - May 2016
