FIR Filter Approximations - i3s

107 Pages·2011·0.68 MB·English

Diniz, da Silva and Netto
FIR Filter Approximations
Paulo S. R. Diniz
Eduardo A. B. da Silva
Sergio L. Netto
diniz,eduardo,[email protected]
September 2010
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Diniz, da Silva and Netto
Contents
Ideal characteristics of standard filters
•
– Lowpass, highpass, bandpass, and bandstop filters
– Differentiators
– Hilbert transformers
FIR filter approximation by frequency sampling
•
FIR filter approximation by window functions
•
– Rectangular window
– Triangular windows
– Hamming and Hann windows
– Blackman window
– Kaiser window
– Dolph-Chebyshev window
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Diniz, da Silva and Netto
Contents (cont.)
Maximally-flat FIR filter approximation
•
FIR filter approximation by optimization
•
– Weighted-least-squares method
– Chebyshev method
– WLS-Chebyshev method
Do-it-yourself: FIR filter approximations
•
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Diniz, da Silva and Netto
Ideal characteristics of standard filters
H( jω) h(n)
As seen in Section 2.8, the relationship between e and is given by the
•
following pair of equations:
H( jω) = h(n) −jωn
e e (2)
∞
"
n=−
π
1 ∞
h(n) = H( jω) jωn ω
e e d (3)
2π #
−π
H( jω) h(n)
In what follows, we determine e and related to ideal standard filters.
•
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Diniz, da Silva and Netto
Lowpass, highpass, bandpass, and bandstop filters
The ideal magnitude responses of some standard digital filters are depicted below.
•
j! j!
|H(e )| |H(e )|
1 1
(a) !c " 2" ! (b) 0 !c " 2" !
j! j!
|H(e )| |H(e )|
1 1
(c) 0 !c1 !c2 " 2" ! (d) 0 !c1 !c2 " 2" !
Figure 1: Ideal magnitude responses: (a) lowpass; (b) highpass; (c) bandpass; (d) band-
stop filters.
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Diniz, da Silva and Netto
Lowpass, highpass, bandpass, and bandstop filters
For instance, the lowpass filter, as seen in Figure 1a, is described by
•
1, |ω| ω
for
c
|H(ejω)| =  ≤ (4)
 0, for ω < |ω| π
c
≤

Using (3), the impulse response for the ideal lowpass filter is
•
ω
c
, n = 0
for
ω
1 c  π
h(n) = jωn ω =
e d (5)

2π #−ω  sin(ωcn)
c , n = 0
for
πn "



One should note that in the above inverse transform calculations we have supposed
•
that the phase of the filter is zero.
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Diniz, da Silva and Netto
Lowpass, highpass, bandpass, and bandstop filters
From Section 4.2.3, we have that the phase of an FIR filter must be of the form
•
−jωM M
e 2 , where is an integer.
M
M
Therefore, for even, it suffices to shift the above impulse response by
2
•
M
M
samples. However, for odd, is not an integer, and the impulse response must
2
be computed as
ω
1 c
h(n) = −jωM jωn ω
e 2 e d
2π #
−ω
c
ω
1 c
= jω(n−M) ω
e 2 d
2π #
−ω
c
ω n − M
sin
c
= 2
(6)
π n − M
! " #$
2
" #
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Diniz, da Silva and Netto
Lowpass, highpass, bandpass, and bandstop filters
Likewise, for bandstop filters, the ideal magnitude response, depicted in Figure 1d, is
•
given by
1, 0 |ω| ω
for
c
≤ ≤ 1

|H(ejω)| =  0, for ωc < |ω| < ωc (7)
 1 2

1, ω |ω| π
for
c
 2 ≤ ≤



Then, using (3), the impulse response for such an ideal filter is
•
ω π −ω
1 c c
1 2
h(n) = jωn ω + jωn ω + jωn ω
e d e d e d
2π # # #
% −ω ω −π &
c c
1 2
ω − ω
c c
1 + 1 2 , n = 0
for
 π
=
(8)
1

 [ (ω n) − (ω n)] , n = 0
sin sin for
c c
πn 1 2 "


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Diniz, da Silva and Netto
Differentiators
An ideal discrete-time differentiator is a linear system that, when samples of a
•
band-limited continuous signal are used as input, the output samples represent the
derivative of the continuous signal.
x (t) −π, π
More precisely, given a continuous-time signal band-limited to ,
a
T T
•
x(n) = x (nT)
when its corresponding sampled version is input to an ideal
a ! #
y(n)
differentiator, it produces the output signal, , such that
x (t)
d
a
y(n) =
(9)
t
d
’t=nT
’
’
X ( Ω)
If the Fourier transform of the continuous-time si’gnal is denoted by j , we
a
•
ΩX ( Ω)
have that the Fourier transform of its derivative is j j .
a
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Diniz, da Silva and Netto
Differentiators
Therefore, an ideal discrete-time differentiator is characterized by a frequency
•
response, up to a multiplicative constant, of the form
H( jω) = ω, −π ω < π
e j for (10)
≤
The magnitude and phase responses of a differentiator are depicted in Figure 2.
•
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