Bounded Real Transfer Function
Bounded Real Transfer Function
• If the input and the output to the digital
filter are x[n] and y[n], respectively with
X (e jω ) and Y (e jω ) denoting their DTFTs,
then the equivalent condition for the BR
property is given by
• Bounded Real Transfer Function • A causal stable real coefficient transfer
function H(z) is defined to be a bounded
real (BR) function if it satisfies the
following condition:
2
Y ( e jω ) ≤ X ( e jω )
• Integrating the above from − π to π,
dividing by 2π, and applying Parseval’s
theorem we get ∞ y 2[n] ≤ ∞ x 2[n]
H (e jω ) ≤ 1, for all values of ω
∑
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• Thus, a digital filter characterized by a BR
transfer function is passive as, for all finiteenergy inputs,
Output energy ≤ Input energy
• If
H (e jω ) = 1 for all values of ω
then it is a lossless system
• A causal stable transfer function H(z) with
| H (e jω )| = 1 is called a lossless bounded
real (LBR) transfer function
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H(e jω )
1
5
ωk
ω
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Copyright © 2010, S. K. Mitra
Copyright © 2010, S. K. Mitra
Requirements for
Low Coefficient Sensitivity
• Since | H (e jω )| is bounded above by unity, the
frequencies ωk must be in the passband
0
∑
n = −∞
• A causal stable digital filter with a transfer
function H(z) has low coefficient sensitivity
in the passband if it satisfies the following
conditions:
(1) H(z) is a bounded-real transfer function
(2) There exists a set of frequencies ωk at
which | H (e jωk )| = 1
(3) The transfer function of the filter with
quantized coefficients remains bounded real
Requirements for
Low Coefficient Sensitivity
0
n = −∞
Requirements for
Low Coefficient Sensitivity
Bounded Real Transfer Function
3
2
6
• Any causal stable transfer function can be
scaled to satisfy the first two conditions
• Let the digital filter structure N realizing
the BR transfer function H(z) be
characterized by R multipliers with
coefficients mi
• Let the nominal values of these multiplier
coefficients assuming infinite precision
realization be mi 0
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Requirements for
Low Coefficient Sensitivity
Requirements for
Low Coefficient Sensitivity
• Because of the third condition, regardless of
the actual values of mi in the immediate
neighborhood of their design values mi 0 ,
the actual transfer function remains BR
• Consider | H (e jωk )| which for multiplier
values mi 0 is equal to 1
• The third condition implies that if the
coefficient mi is quantized, then | H (e jωk )|
can only become less than 1
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• A plot of | H (e jωk )| will thus appear as
indicated below
| H(e jω k) |
1
mi 0
8
Requirements for
Low Coefficient Sensitivity
jω
• Since all frequencies ωk , where the
magnitude function is exactly equal to
unity, are in the passband of the filter and if
these frequencies are closely spaced, it is
expected that the sensitivity of the
magnitude function to be very small at other
frequencies in the passband
mi =mi 0
9
11
The first-order sensitivity of | H (e jω) |
with respect to each multiplier coefficient
is zero at all frequencies ωk where | H (e jω) |
assumes its maximum value of unity
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Copyright © 2010, S. K. Mitra
Requirements for
Low Coefficient Sensitivity
• Thus the plot of | H (e k )| will have a zerovalued slope at mi = mi 0
• i.e.
∂ | H (e jωk ) |
=0
∂mi
•
mi
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Requirements for
Low Coefficient Sensitivity
Requirements for
Low Coefficient Sensitivity
• A digital filter structure satisfying the
conditions for low coefficient sensitivity is
called a structurally bounded system
• Since the output energy of such a structure
is also less than or equal to the input energy
for all finite energy input signals, it is also
called a structurally passive system
• If | H (e jω )| = 1, the transfer function H(z) is
called a lossless bounded real (LBR)
function, i.e., a stable allpass function
• An allpass realization satisfying the LBR
condition is called a structurally lossless or
LBR system implying that the structure
remains allpass under coefficient
quantization
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2
Low Passband Sensitivity
IIR Digital Filter
Low Passband Sensitivity
IIR Digital Filter
• Power-complementary property implies that
• Let G(z) be an N-th order causal BR IIR
transfer function given by
P( z ) p0 + p1z −1 + L + pN z − N
G( z) =
=
D( z ) 1 + d1z −1 + L + d N z − N
| G (e jω ) |2 + | H (e jω ) |2 = 1
• Thus, H(z) is also a BR transfer function
• We determine the conditions under which
G(z) and H(z) can be expressed in the form
G ( z ) = 12 { A0 ( z ) + A1( z )}
with a power-complementary transfer
function H(z) given by
Q ( z ) q0 + q1z −1 + L + qN z − N
H ( z) =
=
D ( z ) 1 + d1z −1 + L + d N z − N
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Low Passband Sensitivity
IIR Digital Filter
• Symmetric property of the numerator of
G(z) implies
P( z −1 ) = z N P ( z )
• Likewise, antisymmetric property of the
numerator of H(z) implies
• H ( z ) = 12 { A0 ( z ) − A1 ( z )}
• H(z) must have an anti-symmetric
numerator, i.e.,
qn = −q N −n
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Q( z −1 ) = − z N Q( z )
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Low Passband Sensitivity
IIR Digital Filter
H ( z) =
• Using the relations P( z −1) = z N P ( z ) and
Q( z −1) = − z N Q( z ) in the previous equation
we get
z − N [ P( z ) P( z ) − Q ( z )Q( z )] = D ( z −1) D ( z )
• or, equivalently
Q( z )
D( z )
P 2 ( z ) − Q 2 ( z ) = [ P ( z ) + Q( z )][ P ( z ) − Q( z )]
= z N D ( z ) D( z −1)
P( z ) P( z −1) + Q ( z )Q( z −1) = D( z −1) D( z )
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Copyright © 2010, S. K. Mitra
Low Passband Sensitivity
IIR Digital Filter
• By analytic continuation, the powercomplementary condition can be rewritten
as
G ( z −1 )G ( z ) + H ( z −1 ) H ( z ) = 1
• Substituting
P( z )
G( z) =
,
D( z )
in the above we get
Copyright © 2010, S. K. Mitra
Low Passband Sensitivity
IIR Digital Filter
1
• G ( z ) = 2 { A0 ( z ) + A1( z )}
G(z) must have a symmetric numerator, i.e.,
pn = p N − n
15
H ( z ) = 12 { A0 ( z ) − A1 ( z )}
where A0 ( z ) and A1 ( z ) are stable allpass
functions
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Low Passband Sensitivity
IIR Digital Filter
Low Passband Sensitivity
IIR Digital Filter
• Using the relations P( z ) = z − N P ( z −1) and
Q( z ) = − z − N Q( z −1) we can write
P ( z ) − Q( z ) = z − N [ P ( z −1 ) + Q( z −1 )]
• Let z = ξ k , 1 ≤ k ≤ N , denote the zeros of
[P(z) + Q(z)]
• Then z = 1 , 1 ≤ k ≤ N , are the zeros of
ξk
[ P ( z ) − Q ( z )]
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[ P ( z ) − Q ( z )] is the mirror image of
the polynomial [P(z) + Q(z)]
• From P ( z ) − Q( z ) = z − N [ P ( z −1 ) + Q ( z −1 )]
it follows that the zeros of [P(z) + Q(z)]
inside the unit circle are also zeros of D(z),
and the zeros of [P(z) + Q(z)] outside the
unit circle are zeros of D( z −1 ) , since G(z)
and H(z) are assumed to be stable functions
•
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Low Passband Sensitivity
IIR Digital Filter
Low Passband Sensitivity
IIR Digital Filter
• To identify the above zeros of D(z) with the
appropriate allpass transfer functions A0 ( z )
and A1 ( z ) we observe that these allpass
functions can be expressed as
P( z ) + Q( z )
A0 ( z ) = G ( z ) + H ( z ) =
D( z )
P( z ) − Q( z )
A1 ( z ) = G ( z ) − H ( z ) =
D( z )
• Let z = ξ k , 1 ≤ k ≤ r , be the r zeros of
[P(z) + Q(z)] inside the unit circle, and the
remaining N − r zeros, z = ξ k , r + 1 ≤ k ≤ N ,
be outside the unit circle
• Then the N zeros of D(z) are given by
1≤ k ≤ r
⎧ ξk ,
⎪
z=⎨ 1
, r +1 ≤ k ≤ N
⎪⎩ξk
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Copyright © 2010, S. K. Mitra
Low Passband Sensitivity
IIR Digital Filter
Low Passband Sensitivity
IIR Digital Filter
• Therefore, the two allpass transfer functions
can be expressed as
• In order to arrive at the above expressions,
it is necessary to determine the transfer
function H(z) that is power-complementary
to G(z)
• Let U(z) denote the 2N-th degree
polynomial
⎛ z −1 − (ξ*k ) −1 ⎞
⎟
⎜
k = r +1⎜⎝ 1 − ξ k−1z −1 ⎟⎠
r ⎛ z −1 − ξ*k ⎞
⎟
A1( z ) = ∏ ⎜
k =1⎜⎝ 1 − ξ k z −1 ⎟⎠
A0 ( z ) =
Copyright © 2010, S. K. Mitra
N
∏
2N
U ( z ) = P 2 ( z ) − z − N D( z −1 ) D ( z ) = ∑ un z − n
n =0
23
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Low Passband Sensitivity
IIR Digital Filter
Low Passband Sensitivity
IIR Digital Filter
• Then
• After Q(z) has been determined, we form
the polynomial [P(z) + Q(z)], find its zeros
z = ξk , and then determine the two allpass
functions A0 ( z ) and A1 ( z )
• It can be shown that IIR digital transfer
functions derived from analog Butterworth,
Chebyshev and elliptic filters via the
bilinear transformation can be decomposed
into the sum of allpass functions
2N
Q 2 ( z ) = ∑ un z − n
n =0
25
27
• Solving the above equation for the
coefficients qk of Q(z) we get
q0 = u0
u
q1 = 1
2q0
u − ∑lk=−11ql qn −l
qk = q N − k = k
, k≥2
2q0
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Copyright © 2010, S. K. Mitra
Low Passband Sensitivity
IIR Digital Filter
Low Passband Sensitivity
IIR Digital Filter
• For lowpass-highpass filter pairs, the order
N of the transfer function must be odd with
the orders of A0 ( z ) and A1 ( z ) differing by 1
• For bandpass-bandstop filter pairs, the order
N of the transfer function must be even with
the orders of A0 ( z ) and A1 ( z ) differing by 2
• A simple approach to identify the poles of
the two allpass functions for odd-order
digital Butterworth, Chebyshev, and elliptic
lowpass or highpass transfer functions is as
follows
• Let z = λ k , 0 ≤ k ≤ N − 1 denote the poles of
G(z) or H(z)
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Copyright © 2010, S. K. Mitra
Low Passband Sensitivity
IIR Digital Filter
Low Passband Sensitivity
IIR Digital Filter
• The pole interlacing property is illustrated
below
Im z
• Let θk denote the angle of the pole λ k
• Assume that the poles are numbered such
that θk < θk +1
• Then the poles of A0 ( z ) are given by λ 2k
and the poles of A1( z ) are given by λ 2k +1
1
θλ6
λ
0.5
6
7 θ6
0
Re z
θ0
-0.5
λ
0
-1
-1
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Copyright © 2010, S. K. Mitra
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-0.5
0
0.5
1
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Low Passband Sensitivity
IIR Digital Filter
• Example - Consider the parallel allpass
realization of a 5-th order elliptic lowpass
filter with the specifications: ω p = 0.4π ,
α p = 0.5 dB, and α s = 40 dB
• The transfer function obtained using
MATLAB is given by
G( z ) =
Low Passband Sensitivity
IIR Digital Filter
• Its parallel allpass decomposition is given
by
1 (0.491 − z −1 )(0.8828 − 0.557 z −1 + z −2 )
G( z) = [
2 (1 − 0.491z −1 )(1 − 0.557 z −1 + 0.8828 z −2 )
+
0.528 + 0.0797 z − 1 + 0.1295 z − 2
+ 0.1295 z − 3 + 0.0797 z − 4 + 0.528 z − 5
(1− 0.491z −1 )(1−0.7624 z −1 + 0.539 z − 2 )
×(1− 0.557 z −1 + 0.8828 z − 2 )
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32
• The gain response of the filter with infinite
precision multiplier coefficients and that
with quantized coefficients are shown
below
ideal - solid line, quantized - dashed line
• A 5-multiplier realization using a signed 7bit fractional sign-magnitude representation
of each multiplier coefficient is shown
below
_
_1
_
_
z1
z1
z1
_1
0
1Δ100011
Gain, dB
0Δ1
Copyright © 2010, S. K. Mitra
Low Passband Sensitivity
IIR Digital Filter
Low Passband Sensitivity
IIR Digital Filter
x[n]
0.539 − 0.762 z −1 + z − 2
]
1 − 0.762 z −1 + 0.539 z − 2
1Δ011111
0Δ111000
y[n]
_
_
z1
z1
_1
1Δ110000
-20
-40
-60
0Δ100010
33
Copyright © 2010, S. K. Mitra
34
0
0.2
0.4
0.6
0.8
1
ω/π
Copyright © 2010, S. K. Mitra
Low Passband Sensitivity
IIR Digital Filter
• Passband details of the parallel allpass
realization and a direct form realization
Direct Form
Realization
Parallel Allpass
Realization
ideal - solid line, quantized - dashed line
ideal - solid line, quantized - dashed line
0
Gain, dB
Gain, dB
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-1
-1.5
-0.5
0
35
0.1
0.2
ω/π
0.3
0.4
-2
0
0.1
0.2
0.3
0.4
0.5
ω/π
Copyright © 2010, S. K. Mitra
6