Finite Wordlength Effects
Binary number representation
Fixed Point
= 0 . 1 2 1
negative numbers can be expressed by
sign bit
twos complement
most significant bit
ones complement
a0 a1 a2
aL-1
sign magnitude
binary point
1
(1) least significant bit
(10) = 0 + 1 2 + + 1 2
Floating Point
sign bit
sign of exponent mantissa
m0 r0 r1
rE m1
mM
binary point
exponent
Quantization
Reduction of Wordlength
roundoff
add 1 to LSB if the next bit is 1
do nothing if 0
*
add 1 if 1
truncation
truncate the following bits
Quantization Error
When a number is rounded to B bit
Fixed Point
=
2 < 2, = 2
Floating Point
= ( )/
2 < 2
3
Finite Wordlength Effects
deviation of frequency response
due to finite-wordlength coefficients
sensitivity
round off operation
noise
uncorrelated error : random
limit cycles
correlated
overflow
all are related to network structure (direct,
cascade ......)
4
Round off Operation
n-bit
n-bit
2n-bit
full precision cannot be kept
n-bit
quantization
roundoff
truncation
p
error
: stepsize
for uniform distribution, zero mean and
/2
3
1
1
2 =
2 =
2 =
3
/2
-Q/2
/2
/2
0
2
=
12
error
Q/2
Linear Model of Quantization
()
Q
filter
input
()
=
2 () =
()
()
()
filter
output
()
=
2 2
=
=
Noise Gain
using Parsevals equation
2
2 ()
1
=
2
2
For multiple noise sources
if noises are uncorrelated
=
=1
2
=1
1
2
noise gain
Scaling
adjust internal signal level so that
the output SN ratio is maximized
digital
1/S
filter
scaling
multiplier
Scaling strategy
no overflow for any input sequence
---too pessimistic
lp norm scaling
0
1
1/
: transfer function from
the input to the internal node
Example of Limit Cycles
zero input case
= 0.8 1
1
2
3
4
5
6
7
8
=
=
=
=
=
=
=
=
0.8(0)
0.8(1)
0.8(2)
0.8(3)
0.8(4)
0.8(5)
0.8(6)
0.8(7)
=
=
=
=
=
=
=
=
0 = 10
Quantizer
absolute value
rounded to integer
0.8 10 = 8 = 8
0.8 (8) = 6.4 = 6
0.8 6 = 4.8 = 5
0.8 (5) = 4 = 4
0.8 4 = 3.2 = 3
0.8 (3) = 2.4 = 2
0.8 2 = 1.6 = 2
0.8 (2) = 1.6 = 2
9
Filter Structures
Direct Form I
Direct Form II
z -1
z -1
z -1
z -1
z -1
z -1
-1
-1
z -1
exchange of the order
common use of the delays
Cascade Form
factorization of the transfer function
+
+
z -1
z -1
cn
-an
z -1
z -1
-bn
dn
10
Example of Coefficient Rounding
4th-order LPF
0.0018 1 + 1 4
=
(1 1.55 1 + 0.65 2 )(1 1.50 1 + 0.85 2 )
is realized using 8bit coefficients
ideal
cascade
direct
= 1
ideal response overlaps with the cascade
11
cf. Numerical Error
For the solution of linear algebraic equations
Gauss-Jordan Elimination
Gaussian Elimination with Back substitution
LU Decomposition
etc.
Depending on the algorithm, computational
complexity and numerical accuracy are different
Choice of filter structures ~ choice of computational algorithms
12
Simulation of reactance circuit
R
V1 ~
reactance
(lossless)
available power
1 2
4
V2
consumed power
2 2
13
Matching
matching points
2
1
1
2
maximum power transfer
|| is bounded
frequency
14
at a matching point
2
1
1
2
0
sensitivity
reactance
||
is zero at matching frequencies
and low in the passband
15
voltage-current simulation
V1
R1
I1
L2
V2
R2
V3
I3
I2
C3
C1
V1
-1 V2
1
R1
I1
1
s C1
V3
-1
1
s L2
-1
I2
1
R2
1
s C3
-1
leap frog
I3
widely used in RC active and switched capacitor filters
16
Delay-free Loops
bilinear transformation
integrator
1+ 1
1 1
+
+
+
z -1
z -1
+
+
+
17
Simulation in terms of
Wave Quantities
more precisely : voltage waves
incident wave
reflected wave
A=V+RI
B=V-RI
R : port resistance
one-part elements
R
s
sources
R
sR
R
= 1 = 1 = 0 =
A
A
z
z -1
-1
=
A
= 2
A
-1
-1
B=0
2E
B
18
Interconnections and Adaptors
Parallel connection of n ports
Series connection of n ports
Kirchhoff s voltage law
Kirchhoff s current law
are interpreted in terms of waves using
= +
=
= 1, 2, ,
19
Parallel Connection
2
2
1
1
2 2
1
1
1 + 2 + + = 0,
1 = 2 = =
= 1 1 + 2 2 + + ,
where =
1 +2 ++
, =
1 + 2 + + = 2
= 1, 2, ,
1
20
Parallel
Adapters
A
B
A3
R3
A1
B1
R1
A1
1 ||2
R2
B2
B1
B1
R1
1 ||
R3
1
1
1 =
=
1 + 2 3
A3
B2
port 3 : dependent port
A1
B3
A2
A2
+
R2
A1
B3
if 3 = 1, i.e. 3 = 1 + 2
B2
A2
2
=
, = 1,2
1 + 2 + 3
A2
B3
A3
1
B1
B2
constrained three-port parallel adaptor
with port3 reflection-free
and port2 dependent
21
Series Connection
2
2
2
1
1
1
1
1 + 2 + + = 0,
1 = 2 = =
= 1 + 2 + + , = 1, 2, ,
2
where =
1 +2 ++
1 + 2 + + = 2
22
A3
Series AdaptersA
B3
A1
R3
A1
R1
B1
A2
2 R2
2 Rk
R1 R2 R3
B2
B1
R1
B2
k 1,2
B3
-1
port 3 : dependent port
B2
A2
A1
B3
R3
1
1
1 =
=
1 + 2 3
A3
B1
-1
+
+
R2
A1
B1
if 3 = 1, i.e. 3 = 1 + 2
A2
A2
1
+
-1
B3
A3
B2
constrained three-port parallel adaptor
with port3 reflection-free
and port2 dependent
23
Interconnection Rules
1. grouping of terminals into ports must remain
respected
2. waves must flow in the same direction
3. two port resistances must be the same
4. connection must not make delay-free loops
1
2
24
Wave Digital Filters
low sensitivity
limit cycles can easily be suppressed
forced response stability
stability under looped condition
25
Third-order Filter C
L2
1/C2
L2
C1
R1
R2
L2
C2
C1
R1
C3
1/C3
1/C1
C3
R2
R1
z -1
z -1
3 ||
z -1
z -1
X1
Y1
1 ||
1
1
||5
2
+
1
=1
Y2
0
1
2
1
five multipliers in general cases
four multipliers if symmetric
R2
1
1
26
Symmetric Case
Bartletts bisection theorem
I1
I2
z1
V1
Z1
R V2
R
z1
1 + 2 2 1
1
1
2
2
=
2
2 1 1 + 2 2
2
2
= +
,
= 1,2
=
Z2
1 + 2 1 2
1
2
2
=
2
2 1 1 + 2
2
2
where = +, = 1,2
1
2
reflection coefficient
if 1 and 2 are reactances
1 and 2 are all-paass fucntions.
27
Lattice Wave Digital Filters
A2
A1
+
S1
S2
B2
B1
for A2=0
S1
A1
S2
B2
parallel of two all-pass filters
for the realization of 1 and 2 themselves,
any methods can be used.
for example, Cauer
cascade of all-pass sections
28
example
L2
C2
C1
C1
C1+2C2
2
2
Z1:
Z2:
C1
1
z-1
1 ||2
-
||
z-1
1
2
three multipliers
z-1
symmetry
odd-order
Butterworth, Chebyshev and elliptic
29
WDF structures may become very complicated
when general transfer functions are realized.
Simple recursive structures + weighted taps
cascade unit elements (vocal tract)
||
-1
||
||
-1
z -1
Digital Lattice Filters
internal circuit of two-port adaptor
+
km
km
km
one-multiplier lattice
two-multiplier lattice
30
Simulation of
Physical Systems
Wave digital filters physically model passive RLC circuits
Can model any systems described by (ordinary or partial)
differential equations (both linear and non-linear) with high
degrees of parallelism and locality
A traveling wave view (at the speed of light) is much closer to
underlying physical reality than any instantaneous models
The bilinear transform is equivalent in the time domain to the
trapezoidal rule for numerical integration
In discretizing space-time continuum, we can get stability and
robustness, unlike Courant-Friedrichs-Lewy Condition which is
necessary but not strictly sufficient
Courant, R., Friedrichs, K., and Lewy, H., "ber die partiellen
Differenzengleichungen der mathematischen Physik",
Mathematische Annalen 100 (1): 3274 1928 (English versions in
31
1956 and 1967)
Exercise 3
1. For the zero-input first-order digital filter in Slide 9,
what is a multiplier coefficient range that does not
cause any limit cycle oscillation?
2. Show that the wave digital filter for a capacitor with its
impedance 1/ is a simple delay, where the port
resistance is = 1/.
3. Derive a wave digital filter structure for a voltage
source E with a series inductive impedance , where
the port resistance is = .
4. Read the following paper;
A.Nishihara & M.Murakami, Signal-Processor-Based
Digital Filters Having Low Sensitivity, Electronics
Letters, 20, 8, pp.325-326, Apr. 1984
32
