Waveform design and Sigma-Delta quantizationjjb/JBWaveSDprosper2006.pdfOutline and collaborators 1....

Waveform design and Sigma-Deltaquantization

John J. BenedettoNorbert Wiener Center, Department of Mathematics

University of Maryland, College Park

http://www.norbertwiener.umd.edu

. – p.1/49

Outline and collaborators1. CAZAC waveforms2. Finite frames3. Sigma-Delta quantization – theory and implementation4. Sigma-Delta quantization – number theoretic estimates

Collaborators: Jeff Donatelli (waveform design); Matt Fickus (frameforce); Alex Powell and Özgür Yilmaz (Σ −∆ quantization); Alex Powell,Aram Tangboondouangjit, and Özgür Yilmaz (Σ − ∆ quantization andnumber theory).

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Processing

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CAZAC waveformsConstant Amplitude Zero Autocorrelation (CAZAC)

WaveformsA K-periodic waveform u : ZK = 0, 1, . . . , K − 1 → C is CAZAC if|u(m)| = 1, m = 0, 1, . . . , K − 1, and the autocorrelation

Au(m) =1

K

K−1∑

k=0

u(m + k)u(k) is 0 for m = 1, . . . , K − 1.

The crosscorrelation of u, v : ZK → C is

Cu,v(m) =1

K

K−1∑

k=0

u(m + k)v(k) for m = 0, 1, . . . , K − 1.

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Properties of CAZAC waveformsu CAZAC ⇒ u is broadband (full bandwidth).

There are different constructions of different CAZAC waveformsresulting in different behavior vis à vis Doppler, additive noises,and approximation by bandlimited waveforms.u CA ⇐⇒ DFT of u is ZAC off dc. (DFT of u can have zeros)u CAZAC ⇐⇒ DFT of u is CAZAC.User friendly software: http://www.math.umd.edu/∼jjb/cazac

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Properties of CAZAC waveformsu CAZAC ⇒ u is broadband (full bandwidth).There are different constructions of different CAZAC waveformsresulting in different behavior vis à vis Doppler, additive noises,and approximation by bandlimited waveforms.

u CA ⇐⇒ DFT of u is ZAC off dc. (DFT of u can have zeros)u CAZAC ⇐⇒ DFT of u is CAZAC.User friendly software: http://www.math.umd.edu/∼jjb/cazac

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Properties of CAZAC waveformsu CAZAC ⇒ u is broadband (full bandwidth).There are different constructions of different CAZAC waveformsresulting in different behavior vis à vis Doppler, additive noises,and approximation by bandlimited waveforms.u CA ⇐⇒ DFT of u is ZAC off dc. (DFT of u can have zeros)

u CAZAC ⇐⇒ DFT of u is CAZAC.User friendly software: http://www.math.umd.edu/∼jjb/cazac

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Properties of CAZAC waveformsu CAZAC ⇒ u is broadband (full bandwidth).There are different constructions of different CAZAC waveformsresulting in different behavior vis à vis Doppler, additive noises,and approximation by bandlimited waveforms.u CA ⇐⇒ DFT of u is ZAC off dc. (DFT of u can have zeros)u CAZAC ⇐⇒ DFT of u is CAZAC.

User friendly software: http://www.math.umd.edu/∼jjb/cazac

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Properties of CAZAC waveformsu CAZAC ⇒ u is broadband (full bandwidth).There are different constructions of different CAZAC waveformsresulting in different behavior vis à vis Doppler, additive noises,and approximation by bandlimited waveforms.u CA ⇐⇒ DFT of u is ZAC off dc. (DFT of u can have zeros)u CAZAC ⇐⇒ DFT of u is CAZAC.User friendly software: http://www.math.umd.edu/∼jjb/cazac

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Rationale for CAZAC waveformsCA allows transmission at peak power. (The system does not haveto deal with the suprise of greater than expected amplitude.)

Distortion amplitude variations can be detected using CA. (WithCA amplitude variations during transmission due to additive noisecan be theoretically eliminated at the receiver without distortingmessage.)A sharp unique peak in Au is important because of distortion andinterference in received waveforms, e.g., in radar andcommunications–more later.

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Rationale for CAZAC waveformsCA allows transmission at peak power. (The system does not haveto deal with the suprise of greater than expected amplitude.)Distortion amplitude variations can be detected using CA. (WithCA amplitude variations during transmission due to additive noisecan be theoretically eliminated at the receiver without distortingmessage.)

A sharp unique peak in Au is important because of distortion andinterference in received waveforms, e.g., in radar andcommunications–more later.

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Rationale for CAZAC waveformsCA allows transmission at peak power. (The system does not haveto deal with the suprise of greater than expected amplitude.)Distortion amplitude variations can be detected using CA. (WithCA amplitude variations during transmission due to additive noisecan be theoretically eliminated at the receiver without distortingmessage.)A sharp unique peak in Au is important because of distortion andinterference in received waveforms, e.g., in radar andcommunications–more later.

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Examples of CAZAC WaveformsK = 75 : u(x) =

(1, 1, 1, 1, 1, 1, e2πi 1

15 , e2πi 2

15 , e2πi 1

5 , e2πi 4

15 , e2πi 1

3 , e2πi 7

15 , e2πi 3

5 ,

e2πi 11

15 , e2πi 13

15 , 1, e2πi 1

5 , e2πi 2

5 , e2πi 3

5 , e2πi 4

5 , 1, e2πi 4

15 , e2πi 8

15 , e2πi 4

5 ,

e2πi 16

15 , e2πi 1

3 , e2πi 2

3 , e2πi, e2πi 4

3 , e2πi 5

3 , 1, e2πi 2

5 , e2πi 4

5 , e2πi 6

5 ,

e2πi 8

5 , 1, e2πi 7

15 , e2πi 14

15 , e2πi 7

5 , e2πi 28

15 , e2πi 1

3 , e2πi 13

15 , e2πi 7

5 , e2πi 29

15 ,

e2πi 37

15 , 1, e2πi 3

5 , e2πi 6

5 , e2πi 9

5 , e2πi 12

5 , 1, e2πi 2

3 , e2πi 4

3 , e2πi·2, e2πi 8

3 ,

e2πi 1

3 , e2πi 16

15 , e2πi 9

5 , e2πi 38

15 , e2πi 49

15 , 1, e2πi 4

5 , e2πi 8

5 , e2πi 12

5 , e2πi 16

5 ,

1, e2πi 13

15 , e2πi 26

15 , e2πi 13

5 , e2πi 52

15 , e2πi 1

3 , e2πi 19

15 , e2πi 11

5 , e2πi 47

15 , e2πi 61

15 )

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Autocorrelation of CAZAC K = 75

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Finite ambiguity function

Given K-periodic waveform, u : ZK → C let em(n) = e−2πimn

K .The ambiguity function of u, A : ZK × ZK → K is defined as

Au(j, k) = Cu,uek(j) =

1

K

K−1∑

m=0

u(m + j)u(m)e2πimk

K .

Analogue ambiguity function for u ↔ U , ‖u‖2 = 1, on R:

Au(t, γ) =

∫

bR

U(ω − γ

2)U(ω +

γ

2)e2πit(ω+ γ

2)dω

=

∫u(s + t)u(s)e2πisγds.

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Finite ambiguity function and DopplerStandard Doppler tolerance problems:

How well does Au(·, k) behave as k varies from 0?

Standard Doppler frequency shift problem: Construct a statistic todetermine unknown Doppler frequency shift. Do this for multiplefrequencies.Provide rigorous justification for CAZAC simulations associatedwith the Doppler tolerance question and frequency shift problem.

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How well does Au(·, k) behave as k varies from 0?Standard Doppler frequency shift problem: Construct a statistic todetermine unknown Doppler frequency shift. Do this for multiplefrequencies.

Provide rigorous justification for CAZAC simulations associatedwith the Doppler tolerance question and frequency shift problem.

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How well does Au(·, k) behave as k varies from 0?Standard Doppler frequency shift problem: Construct a statistic todetermine unknown Doppler frequency shift. Do this for multiplefrequencies.Provide rigorous justification for CAZAC simulations associatedwith the Doppler tolerance question and frequency shift problem.

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RemarksUnder certain conditions, the Doppler statistic

|Cu,uek(j)|

is excellent and provable for detecting deodorized Dopplerfrequency shift.

If one graphs only

Re A(j, k) = Re Cu,uek(j)

then the statistic sometimes fails.There are unresolved “arithmetic” complexities which are affectedby waveform structure and length.Noise analysis is ongoing.

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|Cu,uek(j)|

is excellent and provable for detecting deodorized Dopplerfrequency shift.If one graphs only


then the statistic sometimes fails.

There are unresolved “arithmetic” complexities which are affectedby waveform structure and length.Noise analysis is ongoing.

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|Cu,uek(j)|



then the statistic sometimes fails.There are unresolved “arithmetic” complexities which are affectedby waveform structure and length.

Noise analysis is ongoing.

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|Cu,uek(j)|



then the statistic sometimes fails.There are unresolved “arithmetic” complexities which are affectedby waveform structure and length.Noise analysis is ongoing.

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Doppler Statistic

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Doppler Statistic

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Finite frames

FramesFrames F = enN

n=1for d-dimensional Hilbert space H, e.g., H = Kd, where K = C or

K = R.

Any spanning set of vectors in Kd is a frame for Kd.

F ⊆ Kd is A-tight if

∀x ∈ Kd, A‖x‖2 =

NX

n=1

|〈x, en〉|2

If enNn=1

is a finite unit norm tight frame (FUN-TF) for Kd, with frame constant A, thenA = N/d.

Let en be an A-unit norm TF for any separable Hilbert space H. A ≥ 1, andA = 1 ⇔ en is an ONB for H (Vitali).

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Properties and examples of FUN-TFsFrames give redundant signal representation to compensate forhardware errors, to ensure numerical stability, and to minimize theeffects of noise.

Thus, if certain types of noises are known to exist, then theFUN-TFs are constructed using this information.Orthonormal bases, vertices of Platonic solids, kissing numbers(sphere packing and error correcting codes) are FUN-TFs.

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Properties and examples of FUN-TFsFrames give redundant signal representation to compensate forhardware errors, to ensure numerical stability, and to minimize theeffects of noise.Thus, if certain types of noises are known to exist, then theFUN-TFs are constructed using this information.

Orthonormal bases, vertices of Platonic solids, kissing numbers(sphere packing and error correcting codes) are FUN-TFs.

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Properties and examples of FUN-TFsFrames give redundant signal representation to compensate forhardware errors, to ensure numerical stability, and to minimize theeffects of noise.Thus, if certain types of noises are known to exist, then theFUN-TFs are constructed using this information.Orthonormal bases, vertices of Platonic solids, kissing numbers(sphere packing and error correcting codes) are FUN-TFs.

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Recent applications of FUN-TFs (1)Robust transmission of data over erasure channels such as theinternet [Casazza, Goyal, Kelner, Kovacevic]

Multiple antenna code design for wireless communications[Hochwald, Marzetta,T. Richardson, Sweldens, Urbanke]Multiple description coding [Goyal, Heath, Kovacevic,Strohmer,Vetterli]

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Recent applications of FUN-TFs (1)Robust transmission of data over erasure channels such as theinternet [Casazza, Goyal, Kelner, Kovacevic]Multiple antenna code design for wireless communications[Hochwald, Marzetta,T. Richardson, Sweldens, Urbanke]

Multiple description coding [Goyal, Heath, Kovacevic,Strohmer,Vetterli]

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Recent applications of FUN-TFs (1)Robust transmission of data over erasure channels such as theinternet [Casazza, Goyal, Kelner, Kovacevic]Multiple antenna code design for wireless communications[Hochwald, Marzetta,T. Richardson, Sweldens, Urbanke]Multiple description coding [Goyal, Heath, Kovacevic,Strohmer,Vetterli]

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Recent applications of FUN-TFs (2)Quantum detection [Bölkskei, Eldar, Forney, Oppenheim, Kebo, B]

Σ∆-quantization (better linear reconstruction than MSE of PCM)[Daubechies, Devore, Gunturk, Powell, N. Thao, Yilmaz, B]Grassmannian “min-max” waveforms [Calderbank, Conway,Sloane, et al., Kolesar, B]Grassmannian analysis gives another measure of thecrosscorrelation. A FUN frame unN

n=1 ⊆ H is Grassmannian ifmaxk 6=l |〈uk, ul〉| = inf maxk 6=l |〈xk, xl〉|, where the infimum is overall FUN frames.

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Recent applications of FUN-TFs (2)Quantum detection [Bölkskei, Eldar, Forney, Oppenheim, Kebo, B]Σ∆-quantization (better linear reconstruction than MSE of PCM)[Daubechies, Devore, Gunturk, Powell, N. Thao, Yilmaz, B]

Grassmannian “min-max” waveforms [Calderbank, Conway,Sloane, et al., Kolesar, B]Grassmannian analysis gives another measure of thecrosscorrelation. A FUN frame unN


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Recent applications of FUN-TFs (2)Quantum detection [Bölkskei, Eldar, Forney, Oppenheim, Kebo, B]Σ∆-quantization (better linear reconstruction than MSE of PCM)[Daubechies, Devore, Gunturk, Powell, N. Thao, Yilmaz, B]Grassmannian “min-max” waveforms [Calderbank, Conway,Sloane, et al., Kolesar, B]

Grassmannian analysis gives another measure of thecrosscorrelation. A FUN frame unN


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Recent applications of FUN-TFs (2)Quantum detection [Bölkskei, Eldar, Forney, Oppenheim, Kebo, B]Σ∆-quantization (better linear reconstruction than MSE of PCM)[Daubechies, Devore, Gunturk, Powell, N. Thao, Yilmaz, B]Grassmannian “min-max” waveforms [Calderbank, Conway,Sloane, et al., Kolesar, B]Grassmannian analysis gives another measure of thecrosscorrelation. A FUN frame unN


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Positive-operator-valued measures

Let B be a σ-algebra of sets of X. A positive operator-valued measure (POM) is a functionΠ : B → L(H) such that

1. ∀U ∈ B, Π(U) is a positive self-adjoint operator,

2. Π(∅) = 0 (zero operator),

3. ∀ disjoint Ui∞i=1⊂ B and x, y ∈ H,

*

Π

∞[

i=1

Ui

!

x, y

+

=∞X

i=1

〈Π(Ui)x, y〉,

4. Π(X) = I (identity operator).

A POM Π on B has the property that given any fixed x ∈ H, px(·) = 〈x, Π(·)x〉 is ameasure on B. (Probability if ‖x‖ = 1).

A dynamical quantity Q gives rise to a measurable space (X,B) and POM. Whenmeasuring Q, px(U) is the probability that the outcome of the measurement is in U ∈ B.

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Parseval frames correspond to POMsLet F = enN

n=1be a Parseval frame for a d-dimensional Hilbert space H and let

X = ZN .

For all x ∈ H and U ⊆ X define

Π(U)x =X

i∈U

〈x, ei〉ei.

Clear that Π satisfies conditions (1)-(3) for a POM. Since F is Parseval, we havecondition (4) (Π(X)x =

P

i∈X 〈x, ei〉ei = x). Thus Π defines a POM.

Conversely, let (X,B) be a measurable space with corresponding POM Π for ad-dimensional Hilbert space H. If X is countable then there exists a subset K ⊆ Z, aParseval frame eii∈K , and a disjoint partition Bjj∈X of K such that for all j ∈ X

and y ∈ H,Π(j)y =

X

i∈Bj

〈y, ei〉ei.

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DFT FUN-TFsN × d submatrices of the N × N DFT matrix are FUN-TFs for Cd.These play a major role in finite frame Σ∆-quantization.

“Sigma-Delta” Super Audio CDs - but not all authorities are fans.

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Frame force and potential energy

F : Sd−1 × Sd−1 \ D −→ Rd

P : Sd−1 × Sd−1 \ D −→ R,

where P (a, b) = p(‖a − b‖), p′(x) = −xf(x)

Coulomb force

CF (a, b) = (a − b)/‖a − b‖3, f(x) = 1/x3

Frame force

FF (a, b) =< a, b > (a − b), f(x) = 1 − x2/2

Total potential energy for the frame force

TFP (xn) = ΣNm=1Σ

Nn=1| < xm, xn > |2

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Characterization of FUN-TFs

For the Hilbert space H = Rd and N , consider xnN1 ∈ Sd−1 × ... × Sd−1

and

TFP (xn) = ΣNm=1Σ

Nn=1| < xm, xn > |2.

Theorem Let N ≤ d. The minimum value of TFP , for the frame forceand N variables, is N ; and the minimizers are precisely theorthonormal sets of N elements for Rd.

Theorem Let N ≥ d. The minimum value of TFP , for the frame forceand N variables, is N2/d; and the minimizers are precisely theFUN-TFs of N elements for R

d.

Problem Find FUN-TFs analytically, effectively, computationally.

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Sigma-Delta quantization − theory and implementation

+ + +D Qxn qn

-

un= un-1 + xn-qn

First Order Σ∆

Given u0 and xnn=1

un= un-1 + xn-qnqn= Q(un-1 + xn)

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A quantization problemQualitative Problem Obtain digital representations for class X, suitable forstorage, transmission, recovery.Quantitative Problem Find dictionary en ⊆ X:

1. Sampling [continuous range K is not digital]

∀x ∈ X, x =∑

xnen, xn ∈ K (R or C).

2. Quantization. Construct finite alphabet A and

Q : X → ∑

qnen : qn ∈ A ⊆ K

such that |xn − qn| and/or ‖x − Qx‖ small.Methods Fine quantization, e.g., PCM. Take qn ∈ A close to given xn.Reasonable in 16-bit (65,536 levels) digital audio.

Coarse quantization, e.g., Σ∆. Use fewer bits to exploit redundancy.. – p.24/49

Quantization

AδK = (−K + 1/2)δ, (−K + 3/2)δ, . . . , (−1/2)δ, (1/2)δ, . . . , (K − 1/2)δ

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

δ

δ

δ

3δ/2δ/2

u−axis

f(u)=u

(K−1/2)δ

(−K+1/2)δ

u

qu

Q(u) = arg min|u − q| : q ∈ AδK = qu

. – p.25/49

SettingLet x ∈ Rd, ‖x‖ ≤ 1. Suppose F = enN

n=1 is a FUN-TF for Rd. Thus,we have

x =d

N

N∑

n=1

xnen

with xn = 〈x, en〉. Note: A = N/d, and |xn| ≤ 1.Goal Find a “good” quantizer, given

AδK = (−K +

1

2)δ, (−K +

3

2)δ, . . . , (K − 1

2)δ.

Example Consider the alphabet A21 = −1, 1, and E7 = en7

n=1, with

en = (cos( 2nπ7 ), sin( 2nπ

7 )).

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A2

1= −1, 1 and E7

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

ΓA2

1(E7) = 2

7

∑7n=1 qnen : qn ∈ A2

1

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PCM

Replace xn ↔ qn = argmin |xn − q| : q ∈ AδK. Then x =

d

N

N∑

n=1

qnen

satisfies

‖x − x‖ ≤ d

N‖

N∑

n=1

(xn − qn)en‖ ≤ d

N

δ

2

N∑

n=1

‖en‖ =d

2δ.

Not good!

Bennett’s “white noise assumption”

Assume that (ηn) = (xn − qn) is a sequence of independent, identicallydistributed random variables with mean 0 and variance δ2

12 . Then the meansquare error (MSE) satisfies

MSE = E‖x − x‖2 ≤ d

12Aδ2 =

(dδ)2

12N. – p.28/49

Remarks1. Bennett’s “white noise assumption” is not rigorous, and not true in

certain cases.2. The MSE behaves like C/A. In the case of Σ∆ quantization of

bandlimited functions, the MSE is O(A−3) (Gray, Güntürk andThao, Bin Han and Chen). PCM does not utilize redundancyefficiently.

3. The MSE only tells us about the average performance of aquantizer.

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A2

1= −1, 1 and E7

Let x = ( 13 , 1

2 ), E7 = (cos( 2nπ7 ), sin( 2nπ

7 ))7n=1. Consider quantizers with

A = −1, 1.

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A2

1= −1, 1 and E7

Let x = ( 13 , 1

2 ), E7 = (cos( 2nπ7 ), sin( 2nπ


A = −1, 1.

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

. – p.31/49

A2

1= −1, 1 and E7

Let x = ( 13 , 1

2 ), E7 = (cos( 2nπ7 ), sin( 2nπ


A = −1, 1.

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

xPCM

. – p.32/49

A2

1= −1, 1 and E7

Let x = ( 13 , 1

2 ), E7 = (cos( 2nπ7 ), sin( 2nπ


A = −1, 1.

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

xPCM

xΣ∆

. – p.33/49

Σ∆ quantizers for finite framesLet F = enN

n=1 be a frame for Rd, x ∈ Rd.Define xn = 〈x, en〉.Fix the ordering p, a permutation of 1, 2, . . . , N.Quantizer alphabet Aδ

K

Quantizer function Q(u) = argmin |u − q| : q ∈ AδK

Define the first-order Σ∆ quantizer with ordering p and with thequantizer alphabet Aδ

K by means of the following recursion.

un − un−1 = xp(n) − qn

qn = Q(un−1 + xp(n))

where u0 = 0 and n = 1, 2, . . . , N .

. – p.34/49

Stability

The following stability result is used to prove error estimates.

Proposition If the frame coefficients xnNn=1 satisfy

|xn| ≤ (K − 1/2)δ, n = 1, · · · , N,

then the state sequence unNn=0 generated by the first-order Σ∆

quantizer with alphabet AδK satisfies |un| ≤ δ/2, n = 1, · · · , N.

The first-order Σ∆ scheme is equivalent to

un =n∑

j=1

xp(j) −n∑

j=1

qj , n = 1, · · · , N.

Stability results lead to tiling problems for higher order schemes.

. – p.35/49

Error estimateDefinition Let F = enN

n=1 be a frame for Rd, and let p be apermutation of 1, 2, . . . , N. The variation σ(F, p) is

σ(F, p) =

N−1∑

n=1

‖ep(n) − ep(n+1)‖.

Theorem Let F = enNn=1 be an A-FUN-TF for Rd. The approximation

x =d

N

N∑

n=1

qnep(n)

generated by the first-order Σ∆ quantizer with ordering p and with thequantizer alphabet Aδ

K satisfies

‖x − x‖ ≤ (σ(F, p) + 1)d

N

δ

2.

. – p.36/49

Order is important

0 0.05 0.1 0.15 0.2 0.250

0.1

0.2

Approximation Error

Rel

ativ

e Fr

eque

ncy

0 0.05 0.1 0.15 0.2 0.250

0.1

0.2

Approximation ErrorR

elat

ive

Freq

uenc

yLet E7 be the FUN-TF for R2 given by the 7th roots of unity. Randomlyselect 10,000 points in the unit ball of R

2. Quantize each point usingthe Σ∆ scheme with alphabet A1/4

4 . The figures show histograms for||x − x|| when the frame coefficients are quantized in their natural orderx1, x2, x3, x4, x5, x6, x7 (left) and order x1, x4, x7, x3, x6, x2, x5 (right).

. – p.37/49

Even – odd

100 101 102 103 10410−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

Frame size N

Approx. Error5/N5/N1.25

EN = eNn N

n=1, eNn = (cos(2πn/N), sin(2πn/N)). Let x = ( 1

π ,√

317 ).

x =d

N

N∑

n=1

xNn eN

n , xNn = 〈x, eN

n 〉.

Let xN be the approximation given by the 1st order Σ∆ quantizer withalphabet −1, 1 and natural ordering. log-log plot of ||x − xN ||.

. – p.38/49

Improved estimatesEN = eN

n Nn=1, N th roots of unity FUN-TFs for R2, x ∈ R2,

||x|| ≤ (K − 1/2)δ.

Quantize x =d

N

N∑

n=1

xNn eN

n , xNn = 〈x, eN

n 〉

using 1st order Σ∆ scheme with alphabet AδK .

Theorem If N is even and large then ||x − x|| . δ log NN5/4

.

If N is odd and large then δN . ||x − x|| ≤ (2π+1)d

Nδ2 .

Remark The proof uses the analytic number theory approach developed

by Sinan Güntürk, and the theorem is true more generally.

. – p.39/49

Harmonic framesZimmermann and Goyal, Kelner, Kovacevic, Thao, Vetterli.

H = Cd. An harmonic frame en

Nn=1 for H is defined by the rows of the

Bessel map L which is the complex N -DFT N × d matrix with N − d columnsremoved.

H = Rd, d even. The harmonic frame en

Nn=1 is defined by the Bessel map L

which is the N × d matrix whose nth row is

eNn =

r

2

d

„

cos(2πn

N), sin(

2πn

N), . . . , cos(

2π(d/2)n

N), sin(

2π(d/2)n

N)

«

.

Harmonic frames are FUN-TFs.

Let EN be the harmonic frame for Rd and let pN be the identity permutation.

Then∀N, σ(EN , pN ) ≤ πd(d + 1).

. – p.40/49

Error estimate for harmonic frames

Theorem Let EN be the harmonic frame for Rd with frame bound N/d.Consider x ∈ Rd, ‖x‖ ≤ 1, and suppose the approximation x of x isgenerated by a first-order Σ∆ quantizer as before. Then

‖x − x‖ ≤ d2(d + 1) + d

N

δ

2.

Hence, for harmonic frames (and all those with bounded variation),

MSEΣ∆ ≤ Cd

N2δ2.

This bound is clearly superior asymptotically to

MSEPCM =(dδ)2

12N.

. – p.41/49

Σ∆ and “optimal” PCM

The digital encoding

MSEPCM =(dδ)2

12N

in PCM format leaves open the possibility that decoding (reconstruction)could lead to

“MSEoptPCM” O(

1

N).

Goyal, Vetterli, Thao (1998) proved

“MSEoptPCM” ∼ Cd

N2δ2.

Theorem The first order Σ∆ scheme achieves the asymptotically optimal

MSEPCM for harmonic frames.

. – p.42/49

Sigma-Delta quantization–number theoretic estimates

Proof of Improved Estimates theorem

If N is even and large then ||x − x|| . δ log NN5/4 .

If N is odd and large then δN . ||x − x|| ≤ (2π+1)d

Nδ2 .

∀N, eNn N

n=1 is a FUN-TF.

x − xN =d

N

( N−2∑

n=1

vNn (fN

n − fNn+1) + vN

N−1fNN−1 + uN

NeNN

)

fNn = eN

n − eNn+1, vN

n =n∑

j=1

uNj , uN

n =uN

n

δ

To bound vNn .

. – p.43/49

Koksma InequalityDiscrepancyThe discrepancy DN of a finite sequence x1, . . . , xN of realnumbers isDN = DN (x1, . . . , xN ) = sup0≤α<β≤1

∣∣∣∣1N

∑Nn=1

[α,β)(xn)−(β−α)

∣∣∣∣,

where x = x − bxc.

Koksma Inequalityg : [−1/2, 1/2) → R of bounded variation andωjn

j=1 ⊂ [−1/2, 1/2) =⇒

∣∣∣∣1

n

n∑

j=1

g(ωj) −∫ 1

2

− 1

2

g(t)dt

∣∣∣∣ ≤ Var(g)Disc(ωjn

j=1

).

With g(t) = t and ωj = uNj , |vN

n | ≤ nδDisc(uN

j nj=1

).

. – p.44/49

Koksma InequalityDiscrepancyThe discrepancy DN of a finite sequence x1, . . . , xN of realnumbers isDN = DN (x1, . . . , xN ) = sup0≤α<β≤1

∣∣∣∣1N

∑Nn=1

[α,β)(xn)−(β−α)

∣∣∣∣,

where x = x − bxc.Koksma Inequalityg : [−1/2, 1/2) → R of bounded variation andωjn

j=1 ⊂ [−1/2, 1/2) =⇒

∣∣∣∣1

n

n∑

j=1

g(ωj) −∫ 1

2

− 1

2

g(t)dt

∣∣∣∣ ≤ Var(g)Disc(ωjn

j=1

).

With g(t) = t and ωj = uNj , |vN

n | ≤ nδDisc(uN

j nj=1

).

. – p.44/49

Erdös-Turán Inequality

∃C > 0, ∀K, Disc(uN

n jn=1

)≤ C

(1

K+

1

j

K∑

k=1

1

k

∣∣∣j∑

n=1

e2πikeuNn

∣∣∣)

.

To approximate the exponential sum.

. – p.45/49

Approximation of Exponential Sum

(1) Güntürk’sProposition∀N, ∃XN ∈ BΩ/N

such that∀n = 0, . . . , N ,XN (n) = uN

n + cnδ

2, cn ∈ Z

and ∀t,∣∣∣X ′

N (t) − h( t

N

)∣∣∣ .1

N

(2) Bernstein’s InequalityIf x ∈ BΩ, then ‖x(r)‖∞ ≤Ωr‖x‖∞

(1)+(2)

∀t,∣∣∣X ′′

N (t) − 1

Nh′

( t

N

)∣∣∣ .1

N2

bBΩ = T ∈ A′(bR) : suppT ⊆ [−Ω, Ω ]

MΩ = h ∈ BΩ : h′ ∈ L∞(R) and all zeros of h′ on [0, 1] are simple

We assume ∃h ∈ MΩ such that ∀N and ∀ 1 ≤ n ≤ N, h(n/N) = xNn .

. – p.46/49

Van der Corput LemmaLet a, b be integers with a < b, and let f ∈ C2([a, b]) withf ′′(x) ≥ ρ > 0 for all x ∈ [a, b] or f ′′(x) ≤ −ρ < 0 for all x ∈ [a, b] then

∣∣∣b∑

n=a

e2πif(n)∣∣∣ ≤

(∣∣f ′(b) − f ′(a)∣∣ + 2

)( 4√ρ

+ 3).

∀0 < α < 1, ∃Nα such that ∀N ≥ Nα,

∣∣∣j∑

n=1

e2πikeuNn

∣∣∣ . Nα +

√kN1−α

2

√δ

+k

δ.

. – p.47/49

Choosing appropriate α and K

Putting α = 3/4, K = N1/4 yields

∃N such that∀N ≥ N , Disc(uN

n jn=1

).

1

N1

4

+N

3

4 log(N)

j

Conclusion∀n = 1, . . . , N, |vN

n | . δN3

4 log N

. – p.48/49

. – p.49/49

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