1

Feedback Acquisition and Reconstruction of
Spectrum-Sparse Signals by Predictive Level
Comparisons
arXiv:1711.09658v1 [] 27 Nov 2017

Mahdi Boloursaz Mashhadi, Student Member, IEEE, Saeed Gazor, Nazanin
Rahnavard, and Farokh Marvasti, Senior Members, IEEE

Abstract—In this letter, we propose a sparsity promoting feedback acquisition and reconstruction scheme for sensing, encoding
and subsequent reconstruction of spectrally sparse signals. In the
proposed scheme, the spectral components are estimated utilizing
a sparsity-promoting, sliding-window algorithm in a feedback
loop. Utilizing the estimated spectral components, a level signal
is predicted and sign measurements of the prediction error are
acquired. The sparsity promoting algorithm can then estimate
the spectral components iteratively from the sign measurements.
Unlike many batch-based Compressive Sensing (CS) algorithms,
our proposed algorithm gradually estimates and follows slow
changes in the sparse components utilizing a sliding-window
technique. We also consider the scenario in which possible
flipping errors in the sign bits propagate along iterations (due to
the feedback loop) during reconstruction. We propose an iterative
error correction algorithm to cope with this error propagation
phenomenon considering a binary-sparse occurrence model on
the error sequence. Simulation results show effective performance
of the proposed scheme in comparison with the literature.
Index Terms—Sparse Signal Acquisition, 1-Bit Compressive
Sensing (CS), Level Comparison (LC) Sign Measurements,
Binary-Sparse Error Correction.

I. I NTRODUCTION
PECTRUM sparse signals arise in many applications such
as cognitive radio networks, frequency hopping communications, radar/sonar imaging systems, musical audio signals
and many more. In such cases, the signal components maybe
sparsely spread over a wide spectrum and need to be acquired
without prior knowledge of their frequencies. This is a major
challenge in spectrum sensing that is an essential block in any
spectrum-aware communication system. In this research, we
propose a scheme and the corresponding signal processing
algorithms for acquisition of spectrally sparse signals. The
proposed scheme utilizes tools from the general theory of
Compressive Sensing (CS) [1], [2] to address spectral sparsity.
Several schemes have already been proposed for sparse signal acquisition. These include the Random Demodulator (RD)
[3], the Multi-coset Sampler [4] and the Modulated Wideband
Converter (MWC) [5]. However, the acquired measurements

S

Mahdi Boloursaz Mashhadi is both with the ECE Department, University of
Central Florida, Orlando, USA and the Advanced Communications Research
Institute (ACRI), Sharif University of Technology, Tehran, Iran, e-mail:
boloursaz@eecs.ucf.edu.
Saeed Gazor is with the ECE Department, Queens’ University, Kingston,
Canada, Nazanin Rahnavard is with the ECE Department, University of
Central Florida, Orlando, USA, and Farokh Marvasti is with the Advanced
Communications Research Institute (ACRI), Sharif University of Technology,
Tehran, Iran.

need to be quantized and encoded to bits for subsequent
transmission or processing. This is addressed in the Quantized
Compressive Sensing [6], [7], [8] literature.
The extreme case of 1-bit compressive sensing has been
extensively studied [9], [10], [11], [12], [13], [14] and proved
to be robust against high levels of additive noise on the
measurements [8]. However, the 1-bit measurements acquired
in these works provide no information on the norm of the
sparse signal. Hence in these works, reconstruction is possible
only up to a scale factor.
In the proposed scheme, the input signal is compared with
a level signal [15], [16], [17] and sign measurements of the
error are acquired. The level signal is estimated adaptively in a
feedback loop utilizing a sparse reconstruction algorithm. The
reconstruction algorithm utilizes the previously acquired sign
values to estimate the sparse signal components and predict the
level signal, subsequently. This overcomes the scale ambiguity
of 1-bit CS reconstruction.
The idea of acquiring sign measurements of level comparisons was also applied in [18], [19], [20]. Previous studies on
one-bit sigma-delta quantization [21], [22], [23] investigate
how adaptivity in the level values can improve the reconstruction error bound in terms of the number of measurements.
The approach in [24] achieves exponential decay in the reconstruction error as a function of the number of measurements but requires the levels themselves to be transmitted
for reconstruction. This is in contrast to our proposed scheme
where the adaptive levels are estimated from the sequence of
previously acquired sign measurements themselves. Moreover,
unlike many previously proposed batch-based reconstruction
algorithms, our proposed algorithm applies one iteration on
each sliding window on the input signal using the previous
estimate of the sparse vector as an initial estimate. This not
only can decrease the computational complexity for large
values of batch sizes and iteration counts, but also enables
the proposed algorithm to better follow possible slow changes
in the sparse components along iterations. In Section IV, we
provide performance comparisons with state-of-the-art techniques in [23], [24] and show effective performance of the
proposed scheme by simulations.
In case the acquired sign bits are subsequently transmitted
over a channel, the sign bits available to the receiver may
contain flipping errors. Due to the feedback, these errors will
propagate and make reconstruction unstable. To cope with
this, we propose an iterative algorithm for correcting possible

2

sign flip errors assuming a binary-sparse occurrence model
on the error sequence. The iterations for error correction are
performed along iterations of the main sparse component
estimation algorithm at the receiver to gradually estimate
the error sequence and avoid error propagation. Unlike the
previously proposed error-robust 1-bit CS reconstruction techniques [25], [26], [27], our proposed error correction algorithm
alleviates the need for prior knowledge of the number of errors
by applying a binary-sparse occurrence model on the error
sequence.
This paper is organized as follows. In section II we describe
our proposed feedback acquisition and the corresponding
reconstruction scheme. Section III presents the algorithms performed in the main building blocks of our proposed scheme.
Section IV provides the simulation results and finally section
V concludes the paper.
II. T HE P ROPOSED ACQUISITION AND R ECONSTRUCTION
S CHEME
In this research, we adopt the sparse exponential model in
order to accommodate the general class of spectrally sparse
signals that arise in many real world applications. Assuming
that power
P spectrum of x(t) is sparse, we may approximate
x(t) =
z∈Z sz (t) as the sum of exponential components
for Z = {z1 , z2 , · · · , zN }, zi ∈ C where each component can
be predicted by szi (t + ǫ) = ezi ǫ szi (t). Also assume that
x(t) is sparse in the sense that only a few of its components
have significant amplitudes |sz (t)| at any time. Note that the
adopted model allows non-equidistant frequencies and hybrid
real/imaginary exponentials.
Fig. 1a shows the block diagram of the proposed feedback
acquisition scheme. In this figure, the complex input signal
x(t) is compared with the level signal ℓ(t) utilizing a simple
comparator. The error signal e(t) goes through the complex
sign 1 block and is then sampled uniformly at t = mτ resulting
the output sequence of sign values bm ∈ {±1 ± 1j}. To
encode the signal more efficiently, ℓ(t) is calculated from bm
in a feedback loop utilizing a sparse component estimation
algorithm followed by prediction.
In many cases, the acquired signal needs to be subsequently
transmitted over a channel. In these cases, the sign bits
available for reconstruction at the receiver experience flipping
errors. These errors cause the receiver to estimate inaccurate
level values. If the levels estimated at the receiver are inaccurate, the subsequent sign bits received will be wrongly
interpreted which introduces further errors to reconstruction.
In other words, due to the feedback, the error propagates and
may unstabilize the whole reconstruction. To prevent error
propagation, we propose secondary iterations that are applied
along iterations of the main sparse component estimation
algorithm at the receiver to correct the sign-flip errors as
depicted in Fig. 1b.
In the next section, we elaborate the algorithms performed
in the main building blocks of the proposed scheme.

x(t) +

+

csgn

e(t)

bm

−

t = mτ

ℓ(t)
Sparse
Component
Estimation

Predict
&
Hold

(a) Block diagram for the proposed acquisition scheme.

b̂m

Sparse
Component
Estimation

Predict
&
Hold

ℓ̂(t)

Sparse
Error
Correction
(b) Block diagram for reconstruction at the receiver.

III. T HE P ROPOSED A LGORITHMS
In this section, we first elaborate our proposed algorithm
to be performed in the sparse component estimation block to
reconstruct the spectral components from the sign bits. Then,
we introduce our proposed sparsity-promoting algorithm to
correct sign-flip errors at the receiver.
A. Sparse Component Estimation
Consider a sliding window on the input samples as Xm =
[x(mτ ), x((m−1)τ ), · · · , x((m−M +1)τ )]T in which τ is the
sampling period. Moreover, denote the corresponding level and
sign values by Lm = [ℓ(mτ ), ℓ((m − 1)τ ), · · · ℓ((m − M +
1)τ )]T and Bm = [bm , bm−1 , · · · , bm−M+1 ]T , respectively.
Utilizing this vector notation, we get Bm = csgn(Xm − Lm ).
Now define Sm = [sz1 (mτ ), sz2 (mτ ), · · · , szN (mτ )]T as the
state vector for the observed signal x(t), we can write Xm =
ΦSm , where Φ is a Vandermond matrix defined by


1
1
···
1
 e−z1 τ
e−z2 τ
···
e−zN τ 


Φ=
(1)
.
..
..
..
..


.
.
.
.
e−z1 (M−1)τ e−z2 (M−1)τ · · · e−zN (M−1)τ
The exponential modeling szi (t + ǫ) = ezi ǫ szi (t) simplifies
to a one step predictor as Sm = P ⊙ Sm−1 where P =
[ez1 τ , ez2 τ , · · · , ezN τ ] and ⊙ is element wise multiplication
of two vectors. To estimate and update the sparse state vector
Sm , we propose to iteratively minimize
Ŝm = arg min kB̂m − csgn(ΦS − Lm )k22
S

+ λ1 kS − P Ŝm−1 k22 + λ2

N
X

(2)

gσ ([S]i ),

i=1

1 The

complex sign function(is defined as csgn(.) = sgn(Re(.)) +
√
1,
x≥0
, j = −1. csgn(.) operates
jsgn(Im(.)) where sgn(x) =
−1, x < 0
element-wisely on vectors.

where Ŝm and Ŝm−1 represent estimates of the vector of
sparse components for the sliding windows corresponding to
t = mτ and t = (m − 1)τ , respectively, and [Sm ]i denotes

3

the ith element of the vector Sm . Note that B̂m is the vector
of observed sign bits and is different from the true Bm in
the sense that it may contain bit-flip errors. The first term of
the cost function in (2) enforces consistency with the encoded
sequence of sign values, the second term guarantees smooth
update of the solution and the last term promotes sparsity.
For the sparsity promoting term, we set gσ (s) = arctan(σ|s|)
P arctan(σ)
[28], [29], [30]. It P
is easy to show that limσ→∞ i gσ ([S]i ) =
kSk0 and limσ→0 i gσ ([S]i ) = kSk1 . Thus, by starting from
a small σ value and increasing it along the iterations, we
migrate from the convex ℓ1 to the non-convex l0 norm gradually. Similarly, for ease of calculating the gradient, we replace
the sign function with an S-shaped, infinitely differentiable
function [31], [32], [33]. We set fδ (s) = π2 arctan(δs), for
some δ > 0 which is differentiable with the derivative f ′ (s) =
2
δ
d
ds f (s) = π 1+δ 2 s2 . It is obvious that limδ→∞ fδ (s) = sgn(s)
and hence we increase δ value exponentially along the iterations. Making these substitutions we get (3) 2
Ŝm = arg min C(S)

(3)

S

= arg min kB̂m − cf (ΦS − Lm )k22
S
PN
arctan(σ|[S]i |)
2
.
+ λ1 kS − P Ŝm−1 k2 + λ2 i=1
arctan(σ)
∂
C(S) = 0. In
To solve (3), we shall find the roots of ∂S
order to decrease the computational cost, we apply only one
iteration on each sliding window but gradually increase the σ
and δ parameters along temporal iterations. Utilizing a slidingwindow approach also enables following possible changes in
the spectral components along iterations. We get,

2ΦH cf ′ (ΦS − Lm ) ⊙ (cf (ΦS − Lm ) − B̂m )
λ2
+ 2λ1 (S − P Ŝm−1 ) +
G ⊙ S = 0,
arctan(σ)

(4)

where
[G]i =

1
,
|[S]i |(1 + σ 2 |[S]i |2 )

for i = 1, · · · , N.

(5)

To solve this non-linear equation, we approximate the first
term in (4) by its value at the prior state estimate and denote
Ym−1 = 2λ1 P Ŝm−1 − 2ΦH f ′ (ΦŜm−1 − Lm ) ⊙ (f (ΦŜm−1 −
Lm ) − B̂m ), we get (6)
(2λ1 1N +

λ2
G) ⊙ S = Ym−1 ,
arctan(σ)

(6)

where 1N = [1, · · · , 1] ∈ RN . The elements of 2λ1 1 +
λ2
λ2
arctan(σ) G are 2λ1 + arctan(σ)|[S]i |(1+σ2 |[S]i |2 ) , which are all
real positive values, therefore, from (6) we obtain
∠[S]i = ∠[Ym−1 ]i ,
2λ1 |[S]i | +
2 For

λ2
= |[Ym−1 ]i |.
arctan(σ)(1 + σ 2 |[S]i |2 )

(7)
(8)

a function f : R 7→ R, we denote cf (.) = f (Re(.)) + jf (Im(.)).

λ2
and αi = |[Ym−1 ]i |, we can
By denoting β = arctan(σ)
rewrite (8) as a cubic polynomial equation in terms of ri =
|[S]i | given by

2λ1 σ 2 ri3 − αi σ 2 ri2 + 2λ1 ri + (β − αi ) = 0.

(9)

The coefficients of the cubic polynomial (9) are real. Hence (9)
has either three real roots or a single real root and a complex
conjugate pair. To enforce sparsity, coefficients with smaller
amplitudes are encouraged and hence (3) is minimized by
choosing the smallest non-negative real root of (9). We propose
to solve (9) as follows
Case 1 All
roots of (9) are real: The sum of the three roots
αi
αi σ2
=
2λ1 σ2
2λ1 > 0 is always positive and hence there
exists at least a positive root. The smallest positive root
is feasible for the algorithm.
Case 2 One of the roots is real and the other two are a
complex conjugate pair: If the product of the roots is posαi −β
itive, i.e., 2λ
2 > 0, the real root is positive and hence
1σ
λ2
< αi
feasible. Hence we must enforce β = arctan(σ)
or equivalently increase σ such that σ > arctan( λα2i ).
Note that σ is already increased along the iterations,
hence if this situation happens, σ is further increased till
σ > arctan( λα2i ) holds.
As described above, the magnitude and phase of [Ŝm ]i are
given by the solution of (9) and (7), respectively.
Using the state estimate Ŝm , the predict &P
hold block calN
culates the next level value as ℓ((m + 1)τ ) = i=1 [P ⊙ Ŝm ]i .
Finally, to get ℓ(t) from its samples, each ℓ(mτ ) is holded by
this block at the output for as long as the sampling period τ .
B. Sparse Error Correction
r
Let us define the real and imaginary error vectors Em
i
r
i
r
and Em with elements em , em ∈ {0, 1}. Define em = 1 if
Re(bm ) is flipped and erm = 0, otherwise. Hence, we get
Re(bm ) = Re(b̂m )(1−2erm ) and Im(bm ) = Im(b̂m )(1−2eim ).
Note that for ease of calculations, we consider the real and
imaginary error vectors separately and provide our algorithm
for the real part. The imaginary part is similar. It is obvious that
r
Em
itself, is a sparse vector with elements in {0, 1}. Hence, we
r
propose secondary iterations to update and estimate Em
along
the primary iterations of the sparse component estimation
ˆr
r
= S(Êm−1
), in which the
algorithm. Let us denote Êm−1
S(.) operator denotes sliding the estimated error vector for
one sample and inserting a zero as the initial estimate of its
r
new element. Now to estimate Em−1
, we solve the following
r
Êm
= arg min h(E) + θ
E

s.t.

M
X

[E]i

(10)

i=1

ˆr
k2 ≤ ǫ,
kE − Êm−1

[E]i ∈ [0, 1],

r
]i is relaxed to be the convex interval
where the range for [Em
[0, 1] and the second term of the cost function is the l1 norm
r
r
which promotes sparsity in Em
since the elements of Em
are
non-negative. h(E) = kRe(B̂m ) ⊙ (1 − 2E) − sgn(Re(ΦŜm −
Lm ))k22 is a quadratic convex term with regard to E.

4

p=0
p = 0.0125
p = 0.025
p = 0.05

w/o EC
w/ EC
w/o EC
w/ EC
w/o EC
w/ EC

k = 2.5%
-19.9
-16.3
-19.4
-10.2
-18.2
-4.1
-17.8

k = 5%
-19.6
-12.1
-18.3
-8.7
-17.4
-2.3
-16.5

k = 10%
-17.9
-10.3
-14.5
-4.6
-12.5
F
-10.2

k = 20%
-10.4
-6.3
-7.8
F
-7.1
F
-5.8

To solve (10), we use the gradient descent algorithm followed by projection onto [0, 1] and stochastic rounding [34],
[35] to {0, 1}. Note that both the projected gradient and
stochastic rounding techniques have convergence guarantees
for the convex case as in (10). The gradient descent step is
given by
Tm

D
ˆr
,
=Êm−1
−ǫ
kDk2

(12)

− sgn(ΦŜm − Lm )) + θ1M ,
The projection and stochastic rounding are performed by
(
0, [Tm ]i ≤ u
[Em ]i =
,
(13)
1, [Tm ]i > u,
where u is generated as a uniformly distributed random
variable over the interval [0, 1].
IV. S IMULATION

M = 50
-13.34
-15.76
-16.12

[23]
[24]
This Work

M = 100
-18.54
-29.61
-29.73

M = 200
-25.88
-57.21
-56.96

5

p=0.0125
p=0.025
p=0.05

0

−5

−10

−15

−20

0

20

40

60

80

100

Iteration

Fig. 2: MSE vs. Iteration for the Off-Grid Scenario (k = 0.05).

(11)

where ǫ is an small step-size and
ˆr
D = − 4Re(B̂m ) ⊙ (Re(B̂m) ⊙ (1 − 2Êm−1
)

TABLE II: MSE Comparisons (dB) between our Proposed
Scheme and the Literature

Reconstruction MSE (dB)

TABLE I: The MSE Values (dB) Achieved by the Proposed
Scheme

RESULTS

To numerically evaluate the performance of our proposed
scheme, we generate random spectrally sparse signals according to the model presented in Section II with N = 500,
M = 50, τ = 5 × 10−4 sec, and Z = {1j, 2j, · · · , 500j} × ω0,
ω0 = 10 rad/sec. The non-zero spectral components are
selected uniformly at random and the corresponding amplitudes come from a N (0, 1) distribution. For comparisons, the
final normalized reconstruction Mean Square Error (M SE =
kS−Ŝk2
10 log10 ( kSk2 2 )) values averaged over 100 runs are reported
2
in Table I for different sparsity factors. The sparsity factor
k is defined as the ratio of the number of nonzero spectral
components over the total number of components N . The
algorithm parameters are experimentally optimized for the best
performance as δm = 1.01 × δm−1 , σm = 1.1 × σm−1 .In
this table, p denotes the rate at which sign-flip errors occur,
”w/ EC” and ”w/o EC” represent the results with and without
the proposed error correction (EC) iterations and the letter
”F” shows divergence of the proposed algorithm (MSE>-5dB)
due to the error propagation phenomenon. As shown, EC is
necessary to avoid error propagation.
Next, we investigate the general scenario in which x(t) both
contains frequencies that do not lie on any of the quantized
frequencies Z = {1j, 2j, · · · , 500j} × 10 rad/sec (the offgrid problem) and may also have stable real exponential
parts. Note that exp(γt + j(Kω0 + ∆ω)t) = exp((γ +

j∆ω)t) × exp(jKω0 t), γ ∈ R− which is the grid frequency
exp(jKω0 t) with an amplitude that varies with time according
to exp((γ + j∆ω)t). Hence, if γ and ∆ω are small, the
algorithm will still be able to converge and follow the smooth
changes in the component amplitudes. To investigate this, we
generate x(t) with a sparsity factor of k = 0.05 that contains
components on ω = j214.8 × 10, −1.5 + j442.1 × 10 rad/sec
and provide the MSE curves versus iteration in Fig. 2. These
curves confirm effective performance of the proposed algorithm to follow smooth changes in the component amplitudes.
Finally in Table II, we compare the performance of our
proposed algorithm with state-of-the-art techniques in [23],
[24] for different values of the window length M where
k = 5%, p = 0 and the other simulation parameters are
fixed as previously. This table provides the final normalized
reconstruction MSEs (dB) achieved by the three acquisition/reconstruction schemes averaged over 20 runs when there
exists an additive zero-mean Gaussian pre-quantization noise
with standard deviation 0.1 and [24] is applied in a hard
thresholding scheme. As observed in this table, both our
proposed algorithm and [24] outperform [23] especially for
larger values of M . This is due to an exponential error decay
bound for our proposed algorithm and [24] in comparison with
a root exponential decay bound for the Σ∆ scheme in [23].
Our proposed scheme shows a slightly improved performance
in comparison with [24] for smaller values of M which may
be due to improved robustness to noise by the proposed error
correction algorithm.
V. C ONCLUSION
In this letter, we proposed a feedback acquisition scheme for
encoding of spectrally sparse signals to a stream of 1-bit measurements. We proposed a sparsity promoting reconstruction
algorithm to predict comparison levels in a feedback loop to
facilitate more efficient 1-bit measurements of the input signal.
We also proposed a sparse error correction technique to cope
with possible sign flip errors during transmission. Finally, we
reported simulation results to confirm effective performance
of the proposed scheme and algorithms.

5

R EFERENCES
[1] D.L. Donoho, “Compressed sensing,” IEEE Transactions on Information
Theory, vol. 52, no. 4, pp. 1289–1306, 4 2006.
[2] E.J. Candes and M.B. Wakin, “An Introduction To Compressive
Sampling,” IEEE Signal Processing Magazine, vol. 25, no. 2, pp. 21–30,
3 2008.
[3] Joel A. Tropp, Jason N. Laska, Marco F. Duarte, Justin K. Romberg, and
Richard G. Baraniuk, “Beyond Nyquist: Efficient Sampling of Sparse
Bandlimited Signals,” IEEE Transactions on Information Theory, vol.
56, no. 1, pp. 520–544, 1 2010.
[4] M. Mishali and Y.C. Eldar, “Blind Multiband Signal Reconstruction:
Compressed Sensing for Analog Signals,” IEEE Transactions on Signal
Processing, vol. 57, no. 3, pp. 993–1009, 3 2009.
[5] M. Mishali and Y.C. Eldar, “From Theory to Practice: Sub-Nyquist
Sampling of Sparse Wideband Analog Signals,” IEEE Journal of
Selected Topics in Signal Processing, vol. 4, no. 2, pp. 375–391, 4 2010.
[6] A. Zymnis, S. Boyd, and E. Candes, “Compressed Sensing With
Quantized Measurements,” IEEE Signal Processing Letters, vol. 17,
no. 2, pp. 149–152, 2 2010.
[7] Wei Dai and Olgica Milenkovic, “Information Theoretical and Algorithmic Approaches to Quantized Compressive Sensing,” IEEE Transactions
on Communications, vol. 59, no. 7, pp. 1857–1866, 7 2011.
[8] Jason N. Laska and Richard G. Baraniuk, “Regime Change: Bit-Depth
Versus Measurement-Rate in Compressive Sensing,” IEEE Transactions
on Signal Processing, vol. 60, no. 7, pp. 3496–3505, 7 2012.
[9] Petros T. Boufounos, “Greedy sparse signal reconstruction from sign
measurements,” in 2009 Conference Record of the Forty-Third Asilomar
Conference on Signals, Systems and Computers. 2009, pp. 1305–1309,
IEEE.
[10] J. N. Laska, Zaiwen Wen, Wotao Yin, and R. G. Baraniuk, “Trust, But
Verify: Fast and Accurate Signal Recovery From 1-Bit Compressive
Measurements,” IEEE Transactions on Signal Processing, vol. 59, no.
11, pp. 5289–5301, 11 2011.
[11] Yaniv Plan and Roman Vershynin, “Robust 1-bit Compressed Sensing
and Sparse Logistic Regression: A Convex Programming Approach,”
IEEE Transactions on Information Theory, vol. 59, no. 1, pp. 482–494,
1 2013.
[12] Laurent Jacques, Jason N. Laska, Petros T. Boufounos, and Richard G.
Baraniuk, “Robust 1-Bit Compressive Sensing via Binary Stable
Embeddings of Sparse Vectors,” IEEE Transactions on Information
Theory, vol. 59, no. 4, pp. 2082–2102, 4 2013.
[13] Fuwei Li, Jun Fang, Hongbin Li, and Lei Huang, “Robust One-Bit
Bayesian Compressed Sensing with Sign-Flip Errors,” IEEE Signal
Processing Letters, vol. 22, no. 7, pp. 857–861, 7 2015.
[14] Sohail Bahmani, Petros T Boufounos, and Bhiksha Raj, “Robust 1-bit
Compressive Sensing via Gradient Support Pursuit,” .
[15] Mahdi Boloursaz Mashhadi and Farokh Marvasti, “Iterative Methods
for Sparse Signal Reconstruction from Level Crossings,” 11 2016.
[16] F. Marvasti and M.B. Mashadi, “Wideband analog to digital conversion
by random or level crossing sampling,” Aug. 8 2017, US Patent
9,729,160.
[17] M. B. Mashhadi, N. Salarieh, E. S. Farahani, and F. Marvasti, “Level
crossing speech sampling and its sparsity promoting reconstruction using
an iterative method with adaptive thresholding,” IET Signal Processing,
vol. 11, no. 6, pp. 721–726, 2017.
[18] U. S. Kamilov, A. Bourquard, A. Amini, and M. Unser, “Onebit measurements with adaptive thresholds,” IEEE Signal Processing
Letters, vol. 19, no. 10, pp. 607–610, Oct 2012.
[19] K. Knudson, R. Saab, and R. Ward, “One-bit compressive sensing with
norm estimation,” IEEE Transactions on Information Theory, vol. 62,
no. 5, pp. 2748–2758, May 2016.
[20] C. Qian and J. Li, “Admm for harmonic retrieval from one-bit sampling
with time-varying thresholds,” in 2017 IEEE International Conference
on Acoustics, Speech and Signal Processing (ICASSP), March 2017, pp.
3699–3703.
[21] C. S. Gntrk, M. Lammers, A. Powell, R. Saab, and . Yilmaz, “Sigma
delta quantization for compressed sensing,” in 2010 44th Annual
Conference on Information Sciences and Systems (CISS), March 2010,
pp. 1–6.
[22] Felix Krahmer, Rayan Saab, and zgr Ylmaz, “Sigma-Delta quantization
of sub-Gaussian frame expansions and its application to compressed
sensing,” 6 2013.
[23] Rayan Saab, Rongrong Wang, and Ozgr Ylmaz, “Quantization of
compressive samples with stable and robust recovery,” 2015.

[24] R. G. Baraniuk, S. Foucart, D. Needell, Y. Plan, and M. Wootters,
“Exponential decay of reconstruction error from binary measurements
of sparse signals,” IEEE Transactions on Information Theory, vol. 63,
no. 6, pp. 3368–3385, June 2017.
[25] M. Yan, Y. Yang, and S. Osher, “Robust 1-bit compressive sensing using
adaptive outlier pursuit,” IEEE Transactions on Signal Processing, vol.
60, no. 7, pp. 3868–3875, July 2012.
[26] A. Movahed, A. Panahi, and G. Durisi, “A robust rfpi-based 1-bit compressive sensing reconstruction algorithm,” in 2012 IEEE Information
Theory Workshop, Sept 2012, pp. 567–571.
[27] A. Movahed, A. Panahi, and M. C. Reed, “Recovering signals with
variable sparsity levels from the noisy 1-bit compressive measurements,”
in 2014 IEEE International Conference on Acoustics, Speech and Signal
Processing (ICASSP), May 2014, pp. 6454–6458.
[28] Y. E. Salehani, S. Gazor, I. M. Kim, and S. Yousefi, “Sparse hyperspectral unmixing via arctan approximation of l0 norm,” in 2014 IEEE
Geoscience and Remote Sensing Symposium, July 2014, pp. 2930–2933.
[29] I. W. Selesnick and . Bayram, “Sparse signal estimation by maximally
sparse convex optimization,” IEEE Transactions on Signal Processing,
vol. 62, no. 5, pp. 1078–1092, March 2014.
[30] Yaser Esmaeili Salehani, Saeed Gazor, Il-Min Kim, and Shahram
Yousefi, “0-norm sparse hyperspectral unmixing using arctan smoothing,” Remote Sensing, vol. 8, no. 3, 2016.
[31] Stanley Osher Zhaohui Guo, Todd Wittman, “L1 unmixing and its
application to hyperspectral image enhancement,” 2009.
[32] H. Zayyani, M. Korki, and F. Marvasti, “Dictionary learning for blind
one bit compressed sensing,” IEEE Signal Processing Letters, vol. 23,
no. 2, pp. 187–191, Feb 2016.
[33] H. Zamani, H. Zayyani, and F. Marvasti, “An iterative dictionary
learning-based algorithm for doa estimation,” IEEE Communications
Letters, vol. 20, no. 9, pp. 1784–1787, Sept 2016.
[34] Prabhakar Raghavan and Clark D. Tompson, “Randomized rounding:
A technique for provably good algorithms and algorithmic proofs,”
Combinatorica, vol. 7, no. 4, pp. 365–374, 12 1987.
[35] “Probabilistic construction of deterministic algorithms: Approximating
packing integer programs,” Journal of Computer and System Sciences,
vol. 37, no. 2, pp. 130–143, 10 1988.

