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, [email protected]
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.