SPECTRUM LAB

! !SPECTRUM LAB

! !SPECTRUM LAB

! ! Spectrum Lab

1. Overview

3. Continuous-Domain Analysis

7. Simulation Results

References

4. Discrete-Domain Analysis

Definition 1 (Vanishing moments)

A wavelet has p+1 vanishing moments

if

Rtk (t) dt = 0, for k 2 J0, pK.

Lemma 1 Let x(t) be a polynomial of de-

gree p, (t) be a wavelet with p + 1 van-

ishing moments, and y(t) = M�{x(t)}.

Then, y (t) := (y ⇤ )(t) =P

k(↵k �

↵k�1)⇠(t�tk), where ⇠(t) = �Z t

�1 (⌧) d⌧ .

Acknowledgements

2. Problem Formulation

5. Main Result: WAVE-BUS

6. Reconstruction Algorithm

WAVE-BUS with � = 0.15: (a) Signal x(nT ), its modulo sam-

ples y(nT ), and the reconstruction x̄(nT ); (b) wavelet filter

output yg(nT ), and the estimate z̄(nT ). The reconstruction

error was computed to be �330 dB.

15 20 25 300

20

40

60

SNRin (dB)

SRN

R (d

B)

WAVE−BUSRFD

Modulo operation: (a) Input signal x(t) and its modulo version

y(t); and (b) piecewise-constant signal z(t).

An illustration of the transfer characteristics of a clipping-ADC

and a self-reset-ADC with the thresholds at ±�.

where z(t) =P

k ↵k1[tk,tk+1](t) is a

piecewise-constant signal.

1. Given y(t) = M�{x(t)}, reconstruct x(t).

2. Given y(nT ) = M�{x(nT )}, reconstruct

x(nT ).

y(nT ) = M�{x(nT )} = x(nT )� z(nT )

where z(nT ) =P

k ↵k1Jnk,nk+1K(nT ).

Definition 1 ( Proof: )

A wavelet has p+1 vanishing moments

if

Rtk (t) dt = 0, for k 2 J0, pK.

WAVE-BUS: WAVElet-Based UnlimitedSampling

Selection of sampling interval (T ):

• Sufficient condition for exact reconstruction:

Sampling interval T Tf

2p 2�L(2p) .

Output:y(t) = M�{x(t)} := mod (x(t) + �, 2�)� �,

= x(t)� z(t),

Discrete version:

y(nT ) = M�{x(nT )} = x(nT )� z(nT ),

Algorithm 1 : WAVE-BUS.

• Input: y(nT ) = M�{x(nT )}, L, �, p, T

• Output: x̄(nT )

• Method:

1. Wavelet filtering: (yg)[n] = (y ⇤ g)[n]

2. LASSO (compute nks and ↵ks):arg min

hkAh� ygk2 + �kygk1

3. Compute z̄[n] :P

k ↵k1Jnk,nk+1K

4. Reconstruct x[n] : x̄[n] = y[n] + z̄[n]

Algorithm 1 : WAVE-BUS.

• Input: y(nT ) = M�{x(nT )}, L, �, p, T

• Output: x̄(nT )

• Method:

1. Wavelet filtering: (yg)[n] = (y ⇤ g)[n]

2. LASSO (compute nks and ↵ks):arg min

hkAh� ygk2 + �kygk1

3. Compute z̄[n] :P

k ↵k1Jnk,nk+1K

4. Reconstruct x[n] : x̄[n] = y[n] + z̄[n]

Definition 1 ( Key idea: )

A wavelet has p+1 vanishing moments

if

Rtk (t) dt = 0, for k 2 J0, pK.

Use wavelets to annihilate the

smooth parts of y(t).

• Estimation of ↵ks: by matched filtering.

• Sufficient condition for exact reconstruction:

(tk � tk�1) > (2p� 1) =) 2�L < (2p� 1).

• Let Tf := min

k(nk � nk�1) be the minimum

sampling interval.

• For no overlap: Tf > supp{m[n]}.

• Support of Daubechies wavelet filter of order

p is 2p.

• A sufficient condition on T : T < Tf

2p .

From Lipschitz continuity,|x(t+ T̃f )� x(t)| LT̃f < 2�, 8t.Hence, at any tk, no folding happens inthe interval (tk, tk + T̃f ). Thus T̃f Tf .

From Lipschitz continuity,|x(t+ T̃f )� x(t)| LT̃f < 2�, 8t.Hence, at any tk, no folding happens inthe interval (tk, tk + T̃f ). Thus, T̃f Tf .

�

�����

OUTPUT

INPUT

x(t)

ySR-ADC(t)

yC-ADC(t)

x(t)

z(t)

y(t)(a)

(b)

t1 t2 t3 t4 t5 t6 t7

�2�

2�

�4�

0

���

Sunil Rudresh, Aniruddha Adiga, Basty Ajay Shenoy, and Chandra Sekhar Seelamantula Department of Electrical Engineering, Indian Institute of Science, Bangalore - 560012, India

Email: {sunilr, aaniruddha, bastys}@iisc.ac.in, chandra.sekhar@ieee.org

WAVELET-BASED RECONSTRUCTION FOR UNLIMITED SAMPLING

• Signal-to-reconstruction-noise ratio (SRNR)

vs input SNR for WAVE-BUS and repeated

finite-difference (RFD)

1method.

−0.5 0 0.5−0.5

0

0.5

1

x(nT ) y(nT ) x̄(nT )

−0.5 0 0.5−0.5

0

0.5

TIME (s)

yg(nT ) z̄(nT )

(a)

(b)

−0.5 0 0.5−0.5

0

0.5

1

x(nT ) y(nT ) x̄(nT )

−0.5 0 0.5−0.5

0

0.5

TIME (s)

yg(nT ) z̄(nT )

(a)

(b)

−0.5 0 0.5−0.5

0

0.5

1

x(nT ) y(nT ) x̄(nT )

−0.5 0 0.5−0.5

0

0.5

TIME (s)

yg(nT ) z̄(nT )

(a)

(b)

LASSO: arg minh

= kyg �Ahk22 + �khk1

Algorithm 1 : WAVE-BUS.

• Input: y(nT ) = M�{x(nT )}, L, �, p, T

• Output: x̄(nT )

• Method:

1. Wavelet filtering: (yg)[n] = (y ⇤ g)[n]

2. LASSO (compute nks and ↵ks):arg min

hkyg �Ahk22 + �khk1

3. Compute z̄[n] :P

k ↵k1Jnk,nk+1K

4. Reconstruct x[n] : x̄[n] = y[n] + z̄[n]

Compactly supported g[n]: Daubechies

filter of order p with support J0, 2p� 1K.

Computing {nk}:

A � convolutional dictionary of time-shifted

versions of m[n]

h � sparse vector with h[nk] = �k and

supp{h}=cardinality of {nk}

Lemma 3 Let y(t) = M�{x(t)}, where

x(t) is a Lipschitz-continuous signal that

satisfies |x(a)� x(b)| L|b� a|. Then, we

have T̃f := 2�L Tf .

• Sum of sinusoids of frequency 4 Hz and7 Hz with T = 2 ms.

• The work has been funded by the Science and Engineering ResearchBoard (SERB), Government of India through the project “Sub-NyquistSampling.”

• Conference travel funded by Indian Institute of Science, Bangaloreand SPCOM 2018 travel grant.

Lemma 2 Let x[n] be the samples of a

polynomial of degree p, and let g[n] be a

discrete wavelet filter with p + 1 vanish-

ing moments. Then, yg[n] := (y ⇤ g)[n] =X

k

�k m[n � nk] with �k = (↵k � ↵k�1)

and m[n] := �nX

k=�1g[k].

Annihilation of a sampled polynomial of

degree p:

X

n2Znkg[n] = 0, for k 2 J0, pK.

Self-reset analog-to-digital converter

(SR-ADC):

[1] A. Bhandari, F. Krahmer, and R. Raskar, “On unlimited sampling,”in Proceedings of International Conference on Sampling Theory and

Applications (SampTA), July 2017, pp. 31– 35.

[2] K. Sasagawa, T. Yamaguchi, M. Haruta, Y. Sunaga, H. Takehara, T.Noda, T. Tokuda, and J. Ohta, “An implantable CMOS image sensorwith self-reset pixels for functional brain imaging,” IEEE Transactions

on Electron Devices, vol. 63, no. 1, pp. 215–222, Jan 2016.

[3] I. Daubechies, “Ten Lectures on Wavelets,” SIAM, 1992.

[4] S. Mallat, A Wavelet Tour of Signal Processing, Academic Press,2008.

[5] R. Tibshirani, “Regression shrinkage and selection via the lasso,”Journal of the Royal Statistical Society, Series B, vol. 58, pp. 267–288,1994.