1 finite wordlength

1 Professor A G Constantinides
Finite Wordlength EffectsFinite Wordlength Effects
• Finite register lengths and A/D converters
cause errors in:-
(i) Input quantisation.
(ii) Coefficient (or multiplier)
quantisation
(iii) Products of multiplication truncated
or rounded due to machine length

• Quantisation
Q
Output
Input
2
)(
2
,
Q
ke
Q
oi ≤≤−
)(kei
)(keo

• The pdf for e using rounding
• Noise power
or
2
Q
−
2
Q
Q
1
∫ ==
−
2
2
222
}{).(
Q
Q
eEdeepeσ
12
2
2 Q
=σ

• Let input signal be sinusoidal of unity
amplitude. Then total signal power
• If b bits used for binary then
so that
• Hence
or dB
2
1
=P
b
Q 22=
32 22 b−
=σ
b
P 22
2.
2
3 +
=σ
b68.1SNR +=

• Consider a simple example of finite
precision on the coefficients a,b of second
order system with poles
• where
21
1
1
)( −−
+−
=
bzaz
zH
θ
ρ j
e±
221
..cos21
1
)( −−
+−
=
zz
zH
ρθρ
2
ρ=bθρ cos2=a

•
bit pattern
000 0 0
001 0.125 0.354
010 0.25 0.5
011 0.375 0.611
100 0.5 0.707
101 0.625 0.791
110 0.75 0.866
111 0.875 0.935
1.0 1.0 1.0
2
,cos2 ρθρ ρ

• Finite wordlength computations
+
INPUT OUTPUT
+
+

Limit-cycles; "Effective Pole"Limit-cycles; "Effective Pole"
Model; DeadbandModel; Deadband
• Observe that for
• instability occurs when
• i.e. poles are
• (i) either on unit circle when complex
• (ii) or one real pole is outside unit
circle.
• Instability under the "effective pole" model
is considered as follows
)1(
1)( 2
2
1
1
−−
++
=
zbzb
zH
12 →b

• In the time domain with
•
• With for instability we have
indistinguishable from
• Where is quantisation
)(
)()(
zX
zYzH =
)2()1()()( 21 −−−−= nybnybnxny
12 →b
[ ])2(2 −nybQ )2( −ny
[ ]⋅Q

• With rounding, therefore we have
are indistinguishable (for integers)
or
• Hence
• With both positive and negative numbers
5.0)2(2 ±−nyb )2( −ny
)2(5.0)2(2 −=±− nynyb
21
5.0
)2(
b
ny
−
±
=−
21
5.0
)2(
b
ny
−
±
=−

• The range of integers
constitutes a set of integers that cannot be
individually distinguished as separate or from the
asymptotic system behaviour.
• The band of integers
is known as the "deadband".
• In the second order system, under rounding, the
output assumes a cyclic set of values of the
deadband. This is a limit-cycle.
21
5.0
b−
±






−
+
−
−
22 1
5.0
,
1
5.0
bb

• Consider the transfer function
• if poles are complex then impulse response
is given by
)1(
1)( 2
2
1
1
−−
++
=
zbzb
zG
2211 −− −−= kkkk ybybxy
kh
[ ]θ
θ
ρ
)1(sin.
sin
+= kh
k
k

• Where
• If then the response is sinusiodal
with frequency
• Thus product quantisation causes instability
implying an "effective “ .
2b=ρ 




−
= −
2
11
2
cos
b
b
θ
12 =b




−
= −
2
cos
1 11 b
T
ω
12 =b

• Consider infinite precision computations for
21 9.0 −− −+= kkkk yyxy
0;0
100
≠=
=
kx
x
k
-10 -5 0 5 10
-10
-8
-6
-4
-2
0
2
4
6
8
10

• Now the same operation with integer
precision
-10 -5 0 5 10
-10
-8
-6
-4
-2
0
2
4
6
8
10

• Notice that with infinite precision the
response converges to the origin
• With finite precision the reponse does not
converge to the origin but assumes
cyclically a set of values –the Limit Cycle

• Assume , ….. are not
correlated, random processes etc.
Hence total output noise power
• Where and
{ })(1 ke { })(2 ke
∑=
∞
=0
222
0 )(
k
iei khσσ
12
2
2 Q
e =σ
[ ]
∑
+
=+=
∞
=
−
0
2
2
2
2
2
02
2
01
2
0
sin
)1(sin
.
12
2
.2
k
k
b
k
θ
θ
ρσσσ
b
Q −
= 2
[ ] 0;
sin
)1(sin
.)()( 21 ≥
+
== k
k
khkh k
θ
θ
ρ

• ie






−+−
+
=
−
θρρρ
ρ
σ
2cos21
1
.
1
1
6
2
242
22
2
0
b

• For FFT
A(n)
B(n)
B(n+1)
B(n+1)-
A(n)
B(n+1)
B(n)W(n)B(n)
A(n+1)
)().()()1(
)().()()1(
nBnWnAnB
nBnWnAnA
−=+
+=+
W(n)

• FFT
• AVERAGE GROWTH: 1/2 BIT/PASS
2)1()1(
22
=+++ nBnA
)(2)(
)(2)1(
22
nAnA
nAnA
=
=+

• FFT
• PEAK GROWTH: 1.21.. BITS/PASS
IMAG
REAL
1.0
1.0
-1.0
-1.0
....414.2)()(0.1
)(
)1(
)()()()()()1(
)()()()()()1(
=−+<
+
−+<+
−+=+
nSnC
nA
nA
nSnBnCnBnAnA
nSnBnCnBnAnA
x
x
yxxx
yxxx

• Linear modelling of product quantisation
• Modelled as
x(n) )(~ nx
x(n)
q(n)
+ )()()(~ nqnxnx +=
[ ]⋅Q

• For rounding operations q(n) is uniform
distributed between , and where Q is
the quantisation step (i.e. in a wordlength of
bits with sign magnitude representation or
mod 2, ).
• A discrete-time system with quantisation at
the output of each multiplier may be
considered as a multi-input linear system
2
Q
−
2
Q
b
Q −
= 2

• Then
• where is the impulse response of the
system from the output of the multiplier
to y(n).
h(n)
)()...()...( 21 nqnqnq p
{ })(nx { })(ny
∑ 



∑ −+∑ −=
=
∞
=
∞
=
p
rr
rnhrqrnhrxny
1 00
)().()().()(
λ
λλ
)(nhλ
λ

• For zero input i.e. we can write
• where is the maximum of
which is not more than
• ie
nnx ∀= ,0)(
∑ ∑ −≤
=
∞
=
p
r
rnhqny
1 0
)(.ˆ)(
λ
λλ
λqˆ rrq ,,)( λλ ∀
2
Q
∑ 



∑ −≤
=
∞
=
p
n
rnh
Q
ny
1 0
)(.
2
)(
λ
λ

• However
• And hence
• ie we can estimate the maximum swing at
the output from the system parameters and
quantisation level
∑ ∑≤
∞
=
∞
=0 0
)()(
n n
nhnhλ
∑≤
∞
=0
)(.
2
)(
n
nh
pQ
ny

1 finite wordlength

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to 1 finite wordlength

Similar to 1 finite wordlength (20)

More from Naatchammai Ramanathan

More from Naatchammai Ramanathan (20)

Recently uploaded

Recently uploaded (20)

1 finite wordlength