Stability for chaotic sigma delta quantization
Lauren Bandklayder
October 15, 2010
Analog-to-digital (A/D) conversion
Goal: Represent audio signals by a sequence of bits, or by a
sequence of ±1.
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Audio signals can be modeled as a sequence of their sample
values fn = f (tn ) ∈ [−1, 1] at time tn .
0.1
0.04
0.05
0.02
0
0
−0.05
−0.1
−0.02
0
.5
1
Seconds
1.5
−0.04
.4
.4125
.425
Seconds
Figure: the phrase ”hi, how are you”, at two time scales
.4375
Quantization schemes
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Pulse Code Modulation (PCM): Replace each sample fn by
(qj )M
j=1 the first M bits in its binary expansion,
P
−j
fn ≈ M
j=1 qj 2
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Sigma delta (Σ∆) quantization: Sample audio at higher
rate, then replace each sample fn by a single value
qn ∈ {−1, 1} or qn ∈ {−1, 0, 1} such that fn is approximated
by local averages of the qn ,
fn ≈
j=n+M
X
j=n−M
cj−n qj
Σ∆ quantization
The standard second-order Σ∆ scheme can be reformulated as a
dynamical system; set u0 = v0 = 0 and iterate for n ≥ 1:
qn = Q un−1 + γvn−1 ,
un = un−1 + fn − qn
vn = vn−1 + un
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One-bit quantization qn ∈ {−1, 1}, Q(x) =sign(x)
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Tri-level quantization: qn ∈ {−1, 0, 1},

 −1, x < −.5,
0,
−.5 ≤ x ≤ .5,
Q(x) =

1,
x > .5
Idle tones in Σ∆ quantization
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Problem: Periodicities in the output qn often occur in Σ∆
schemes, producing audible idle tones
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One proposed solution: Modify the standard Σ∆ scheme by
adding amplification to break up periodicities: fix λ > 1, and
consider
qn = Q un−1 + γvn−1 ,
un = λun−1 + fn − qn
vn = vn−1 + un ,
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This modification is called chaotic Σ∆ quantization in
practice. So far a proof of stability was missing. By stability,
we mean that the iterates (un , vn ) do not blow up.
Output of quantization scheme
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Here is a sample of a sequence of points (un , vn ) when the
input sequence fn is constant.
2.5
2.5
2
2
1.5
1.5
1
0.5
1
0
0.5
−0.5
−1
0
−1.5
−0.5
−1.5
−1
−0.5
0
0.5
1
1.5
−2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Figure: Output of standard scheme (left) versus chaotic scheme (right).
Proof of stability
I
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Ozgur Yilmaz proved the standard second-order Σ∆ scheme
was to be stable within a specific convex region, as shown
below.
If (un , vn ) ∈ Sα and |fn | < α, then (un+1 , vn+1 ) ∈ Sα .
15
C
u+ ! v =0
10
"B
1
5
0
−5
"B
2
−10
−15
−5
0
5
Figure: ΓB1 and ΓB2 are the graphs of two quadratic functions,
symmetric about the origin.
Proof of stability
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Restricting the input sequence (fn ) to |fn | ≤ α < 1, then
δn = |fn − qn | can take values from L = 1 − α to H = 1 + α,
where |fn | ≤ α < 1, we can rewrite the system:
(un , vn ) =
(λun−1 − δn , λun−1 + vn−1 − δn ); if qn = 1,
(λun−1 + δn , λun−1 + vn−1 + δn ); if qn = −1
(1)
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To extend this, we suppose |fn | ≤ α0 < α < 1, where
α0 = α − (α), for a small nonnegative value dependent on
α.
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If (un , vn ) ∈ Sα , and |fn | ≤ α0 < α, then (un+1 , vn+1 ) ∈ Sα .
Bounds on expansion parameter λ
Previous constraints on C imply that our stability results will
L
. However, in practice, values of
only hold for λ ≤ 1 + 2H
lambda could hypothetically be much larger than this.
1.5
1.45
1.4
1.35
1.3
"
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1.25
1.2
1.15
1.1
1.05
1
0
0.2
0.4
0.6
0.8
1
!
Figure: Lambda as a function of α for fixed .
Extensions of the chaotic Σ∆ scheme
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Trilevel Quantizer: A slight adjustment to the conditions on
C , λ, and γ allowed us to extend the proof of stability of the
system to the case where qn can take values of 0, 1, or -1.
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Finite Memory Quantizer: We similarly extended the proof
to the ”leaky” scheme, described by
(un , vn ) = (βλun−1 + fn − qn , βvn−1 + βλun−1 + fn − qn )
where β ≤ 1.
Open problem
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There is still no rigorous proof that the second order Σ∆
scheme with expansion parameter λ > 1 is chaotic.