Signal Processing - May 2016 - 110

beyond the cutoff frequency. For the
modified window design method, the
discrete frequency response is simulated using H ^mh = e c m , - ^ M - 1h /2
# m # ^ M - 1h /2 . Notice that the
parameter c controls the shape of H (m) .
When c is set at zero, H (m) is reduced
to that for the traditional window design
method. Applying H (m) as a symmetric
exponential function induces a ripple at
the passband edge so that the composite
filter has a flat passband. The timedomain filter coefficients h ∞ (n) is
obtained by applying the IDFT equation
to H (m) as

H (f )

Cutoff
Frequency

1

-fs

0
Frequency (f )

-fs /2

fs /2

fs

(a)
For Window
Design Method

m = -N
(-fs )

H (m )

m = -(M - 1)/2

For Modified Window
Design Method

m = (M - 1)/2

(0)

h ∞ (n) = 1
N

m=N
(fs )

M
Frequency (m )
(b)

FIGure 5. LPF frequency responses for the modified window design method and the window design
method: (a) continuous frequency response H(f); (b) periodic, discrete frequency response H(m).

of the M frequency-domain samples of
H (m) . Notice that H (m) is periodic
with a period identical to the sample
rate fs . We then apply the discrete frequency response H (m) to the inverse
discrete Fourier transform (IDFT)
equation to get the time domain h ∞ (n) .
The filter coefficients are obtained by
calculating h (n) = h ∞ (n) $ w (n) , where
w (n) represents a window function to
truncate h ∞ (n) .

In this article, to design the prototype filter with a deliberately generated
ripple at the passband edge, we remodel
a continuous LPF frequency response
H ( f ) by using an exponential function.
This method is called the modified window design method.
In contrast to the window design
method, the passband frequency
response of the modified window design
method is exponential and is zero

∆γ
Magnitude (dB)

0
γ Increases

∆0

γ =0
ωc
ω (rad./Sample)

FIGure 6. The comparison of the magnitude frequency responses of two lowpass FIR filters designed using the modified window design method. One filter is with c = 0 and the other is with c > 0.

110

IEEE SIgnal ProcESSIng MagazInE

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May 2016

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(M -1) /2

/

e c m e j2rmn/N ,

m =-(M -1)/2

(6)

where n is an integer. If we set c at
zero, then the modified window design
method is reduced to the traditional
window design method. Reorganizing
(6), we obtain the simplified representation of h ∞ (n) as
M -1

2
h (n) = 1 >1 + 2 / e cm cos ` 2rmn jH .
N
N
m =1
∞

(7)
The filter coefficients are obtained by
calculating
h (n) = h ∞ (n) $ w (n),

(8)

where w (n) represents a window function to truncate h ∞ (n) . More coefficients can be used for w (n) to narrow
the filter transition region. As c increases, the ripple induced at the passband
becomes more substantial.
Both the window design method and
modified method enhance the stopband
attenuation through a smooth window
function other than a rectangular window function. Some commonly used
window functions include the Blackman, Chebyshev, and Kaiser window
functions [1]. We adopt the Blackman
window for illustration.
Figure 6 presents a comparison of
the magnitude frequency responses
for difference choices of c designed
using the modified window design
method. In contrast to the traditional
Table of Contents for the Digital Edition of Signal Processing - May 2016
