Chebyshev IIR filter sharpening implemented on FPGA - Telfor 2008

is 1:
R ( x min ) = 1, R(1)
= 1
(4)
The position of maximum can be found from
2
T r ( x max ) − k = 0
(5)
2 2
ε − µε
k = N D , µ = 2,3,4,... (6)
2 2
ε N ε D ( µ − 1)
The ripple factor (the maximal variation of the
magnitude function)
2
2
ε = R(
xmax
) − 1
(7)
becomes
2
2
µ −1
2 ε ⎛ ( 1) ⎞
⎜ µ − ε
ε = N N ⎟ − 1 (8)
µ 2 ⎜ 2 2
µ ε
⎟
D ⎝ ε N − ε D ⎠
The constant k can be expressed as a function of ε N or
ε D
i = µ
⎛ µ ⎞ 2i
∑ ⎜ ⎟ε
D
i i
k = 2 ⎝ ⎠
(9)
=
i = µ
2i
⎛ µ ⎞ 2i
( µ − 1) ε D ∑ ⎜ ⎟ε
D
i = 1 ⎝ i ⎠
All roots of equation (5) are real for 0 < ε D < 1 and
µ > 2 , which provide the equiripple character of the
magnitude function.
The positions of the minima x min are not function of
ε N or ε D . The positions of the maxima x max change
slightly with µ and ε N or ε D . Therefore, the shape of
the magnitude is practically the same for any µ and ε N
or ε D for 0 ≤ x ≤ 1 . Conversely, the magnitude function
decreases 6 µ r dB/octave.
The equiriple rational function can be used as
approximating function for <strong>filter</strong> <strong>sharpening</strong> because it is
based on processing the date µ times with the same startup
<strong>filter</strong> (<strong>Chebyshev</strong> <strong>filter</strong>).
III. DESIGN METHOD
The simplest solution of <strong>filter</strong> <strong>sharpening</strong> is <strong>filter</strong>ing the
data k times with the existing <strong>filter</strong>. If the original <strong>filter</strong>’s
transfer function is H(z), then the new transfer function is
(H(z)) k . The new <strong>filter</strong> has the same band edges and the
minimum attenuation, in decibels, is k times larger in the
stopband. The drawback is that the passband ripple is also
k times larger in decibels. This solution improves the
stopband but degrades the passband.
The <strong>filter</strong> <strong>sharpening</strong> of analog <strong>filter</strong>s [3] provides very
low passband ripple and the Q factors of the dominant
transfer-function pole pairs are reduced. An active RC
<strong>filter</strong> cannot be efficiently <strong>implemented</strong> as a selective <strong>filter</strong>
with very high pole-Q factors, while multiple low-Q factor
SOS <strong>filter</strong>s can be massively produced. Using the bilinear
transformation, analog <strong>filter</strong>s with low-Q factors are
transformed into digital <strong>filter</strong>s that are more robust for
fixed point implementation [4]. Therefore, the same
method as used in [3] can be used for digital <strong>filter</strong>
<strong>sharpening</strong>.
The standard <strong>filter</strong> H s (z) should be chosen to be simple
for implementation. For example, the standard <strong>filter</strong> can be
the fourth-order <strong>Chebyshev</strong> <strong>filter</strong> that has small passband
ripple (say A s =0.1 dB) and it consists of two SOS <strong>filter</strong>s
−1
−2
−1
−2
1 + 2z
+ z 1 + 2z
+ z
H ( z)
=
(10)
s
−1
− 2 −1
− 2
1 + as1z
+ as2
z 1 + as3z
+ as4
z
The coefficients are shown in Table 1.
The <strong>filter</strong> has a stopband attenuation of 8 dB at the
stopband edge f s =0.135; the passband edge frequency is
f p =0.1. The target <strong>filter</strong>, the cascaded connection of µ basic
<strong>filter</strong>s, should have the minimum stopband attenuation 40
dB; it also has large passband ripple 0.1•µ dB. To achieve
a stopband attenuation of 40 dB, the number of cascades µ
is equal to 5. Let us design the fourth-order <strong>Chebyshev</strong>
<strong>filter</strong> H i (z) that has the passband ripple 0.5 dB for µ=5.
The coefficients are shown in Table 1.
TABLE 1: COEFFICIENTS OF THE FILTER.
Coefficients
a s1
–1.26232
a s2
0.44004
a s3
–1.310268
a s4
0.73834
a c1
–1.46622
a c2
0.58087
a c3
–1.44789
a 0.814144
c4
The transfer function of the new <strong>filter</strong>
H(z)=(H s (z)) µ /H i (z) can be simplified H(z)=(H s (z)) µ-1 H c (z)
in such a way that a new forth order <strong>filter</strong> becomes H c (z)=
H s (z)/H i (z)
−1
−2
−1
−2
1 + a 1z
+ a 2z
1 + a
( )
3z
+ a 4z
H z = c c
c c (11)
c
−1
−2
−1
−2
1 + as1z
+ as2z
1 + as3z
+ as4
z
The transfer function of cascaded connection of m
standard and n compensation sections becomes
m
H ( z)
= ( H s ( z)
) ( H c ( z)
) n (12)
In Figures 1, 2 and 3, the attenuations of <strong>filter</strong>s are
shown for a number of combinations of m and n. The
minimum stopband attenuation increases with the number
of H s (z) sections while the passband ripple is the same as
for the standard section, in this case 0.1 dB. The passband
ripple is smaller than the ripple of the standard <strong>filter</strong>
⎛
µ −1
µ µ ⎞
⎜ 1 ( µ − 1) ( α − 1)
A = 10 n log
⎟ (13)
n 10
µ −1
µ µ −1
µ −1
⎝ ( α − 1) α µ ( α − 1) ⎠
for
A /10
α = 10 s , m = n(
µ − 1) (14)
For example, for µ=5, n=1, A s =0.1 dB, the input data is
<strong>filter</strong>ed 4 times using the standard <strong>filter</strong> H s (z) and ones
using the compensation <strong>filter</strong> H c (z); the overall passband
ripple is A n =0.0058 dB and the minimum stopband
attenuation is A s min =26 dB. For n=2, the data is <strong>filter</strong>ed 8
times using H s (z) and twice using H c (z), the overall
passband ripple is A n =0.0116 dB and A s min =52 dB.
Attenuation of <strong>filter</strong>s depends on the number of
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