1

Asymptotic Signal Detection Rates
with 1-bit Array Measurements

arXiv:1711.00739v2 [] 20 Feb 2018

Manuel S. Stein

Abstract—This work considers detecting the presence of a
band-limited random radio source using an antenna array
featuring a low-complexity digitization process with single-bit
output resolution. In contrast to high-resolution analog-to-digital
conversion, such a direct transformation of the analog radio
measurements to a binary representation can be implemented
hardware and energy-efficient. However, the probabilistic model
of the binary receive data becomes challenging. Therefore, we
first consider the Neyman-Pearson test within generic exponential
families and derive the associated analytic detection rate expressions. Then we use a specific replacement model for the binary
likelihood and study the achievable detection performance with 1bit radio array measurements. As an application, we explore the
capability of a low-complexity GPS spectrum monitoring system
with different numbers of antennas and different observation
intervals. Results show that with a moderate amount of binary
sensors it is possible to reliably perform the monitoring task.
Index Terms—1-bit ADC, analog-to-digital conversion, array
processing, exponential family, GPS, Neyman-Pearson test, quantization, detection, spectrum monitoring

I. I NTRODUCTION
Since, in 1965, Gordon E. Moore predicted a doubling
in computational capability every two years, chip companies
have kept pace with this prognosis and set the foundation
for digital systems which today allow processing high-rate
radio measurements by sophisticated algorithms. In conjunction with wireless sensor arrays, this results in advanced
signal processing capabilities, see, e.g. [1]. Unfortunately, in
the last decades, the advances regarding the analog circuits
forming the radio front-end were much slower. Therefore, in
the advent of the Internet of things (IoT), where small and
cheap objects are supposed to feature radio interfaces, cost
and power consumption of wireless sensors are becoming an
issue. In particular, analog-to-digital conversion [2], [3] shows
to set constraints on the digitization rate and the number of
antenna elements under strict hardware and power budgets.
In this context, we consider detection of a band-limited
source with unknown random structure by a sensor array
providing single-bit radio measurements. We formulate the
processing task as a binary hypothesis test regarding exponential family models and derive expressions for the asymptotic
detection rates. The results are used to determine the design
of a low-complexity GPS spectrum monitoring system.
This work was supported by the German Academic Exchange Service
(DAAD) with funds from the German Federal Ministry of Education and
Research (BMBF) and the People Program (Marie Skłodowska-Curie Actions)
of the European Union’s Seventh Framework Program (FP7) under REA grant
agreement no. 605728 (P.R.I.M.E. - Postdoctoral Researchers International
Mobility Experience).
M. S. Stein is with the Chair for Stochastics, Universität Bayreuth, Germany
(e-mail: manuel.stein@uni-bayreuth.de).

Note that detection with quantized signals has found attention in distributed decision making [4] where data is collected
through sensors at different locations and quantization is used
to diminish the communication overhead between sensors and
terminal node [5]–[7]. Optimum quantization for detection
is the focus of [8]–[11], while [12] considers the detection
performance degradation due to hard-limiting. For discussions
on symbol detection for quantized communication see, e.g.,
[13]–[18]. Detection for cognitive radio with single-antenna
1-bit receivers is considered in [19] while array processing
with 1-bit measurements is analyzed in [20]–[22].
II. P ROBLEM F ORMULATION
We consider a receive situation where a narrow-band random wireless signal is impinging on an array of S sensors,
y = γAx + η.

(1)

Each sensor features two outputs (in-phase and quadrature),
M
such that the receive
 TvectorT yT ∈ Y = R , M =S 2S, can be
with y I , y Q ∈ R . Likewise,
decomposed y = y I y Q
T

∈ R2 consists of two
the random source x = xI xQ
zero-mean signal components with covariance matrix


(2)
Rx = Ex xxT = I,

where Eu [·] denotes the expectation concerning the probability distribution p(u) and I the identity matrix. The steering
T

matrix A = AT
∈ RM×2 , with AI , AQ ∈ RS×2
AT
Q
I
models a uniform linear sensor array response (half carrierwavelength inter-element distance) for a narrow-band signal
arriving from direction ζ ∈ R, such that




cos 0 
sin 0 


cos π sin (ζ)
sin π sin (ζ)


AI = 
 (3)
..
..


.
.


cos (S − 1)π sin (ζ) sin (S − 1)π sin (ζ)

and





AQ = 



− sin 0 
− sin π sin (ζ)
..
.
− sin (S − 1)π sin (ζ)


cos 0 
cos π sin (ζ)
..
.


cos (S − 1)π sin (ζ)





.



(4)

The parameter γ ∈ R characterizes the source strength in
relation to the additive zero-mean sensor noise η ∈ RM with


(5)
Rη = Eη ηη T = I.

2

Due to the properties of the source and noise signals, the
receive data (1) can be modeled by a Gaussian distribution

exp − 12 y T R−1
y (γ)y
y ∼ py (y; γ) = p
,
(6)
(2π)M det (Ry (γ))

with covariance matrix



Ry (γ) = Ey;γ yy T = γ 2 AAT + I.

(7)

Based on the likelihood (6), the problem of signal detection
can be stated as the decision about which of the two models
H0 : y ∼ py (y; γ0 ),

H1 : y ∼ py (y; γ1 )

has generated the data (K independent array snapshots)


Y = y 1 y2 . . . yK ∈ Y K .

(8)

H0 : z ∼ pz (z; θ0 ),

(9)

(10)

where sign (u) is the element-wise signum function, i.e.,
(
+1 if [y]i ≥ 0
[z]i =
(11)
−1 if [y]i < 0.
Note, that (10) characterizes a low-complexity digitization
process without feedback and is, therefore, distinct from
sigma-delta conversion [23], where a single fast comparator
with feedback is used to mimic a high-resolution ADC.
Modeling the output of (10) by its exact parametric probability distribution function, requires computing the integral
Z
pz (z; γ) =
py (y; γ)dy
(12)

C

while the probability of correctly deciding for H1 is given by
Z
PD =
pZ (Z; θ1 )dZ.
(16)

Approaching the decision problem (14) under the desired test
size PFA and maximum PD , the Neyman-Pearson theorem
shows that it is optimum to use the likelihood ratio test [24]
L(Z) =

pZ (Z; θ1 )
> ξ′
pZ (Z; θ0 )

IN THE

E XPONENTIAL FAMILY

Consider the multivariate parametric exponential family


pz (z; θ) = exp β T (θ)φ(z) − λ(θ) + κ(z) ,
(13)

where θ ∈ RD constitute its parameters, β(θ) : RD → RL the
natural parameters, φ(z) : RM → RL the sufficient statistics,
λ(θ) : RD → R the log-normalizer and κ(z) : RM → R the

(17)

for the assignment of the critical region
C = {Z : L(Z) > ξ ′ },

(18)

′

while the decision threshold ξ is determined through
Z
PFA =
pZ (Z; θ 0 )dZ.

(19)

{Z:L(Z)>ξ ′ }

Based on the ratio (17), a test statistic T (Z) : Z K → R can
be formulated such that the binary decision is performed by
(
H0 if T (Z) ≤ ξ
decide
.
(20)
H1 if T (Z) > ξ
To analyze the performance of (20), it is required to characterize the distribution of T (Z) and evaluate (15) and (16). As the
data Z consists of K independent samples, the test statistic
can be factorized into a sum of independent components
T (Z) =

M

for all 2 points in Z = B , where Y(z) characterizes the
subset in Y which by (10) is transformed to z. Additionally,
(12) requires the orthant probability, which for M > 4 is an
open problem. As the multivariate Bernoulli model resulting
from (10) is part of the exponential family like (6), in the
following we resort to discussing the considered processing
task for generic data models within this broad class. Without
exactly specifying the quantized likelihood (12), this will allow
us to analyze the asymptotically achievable detection rates
with low-complexity 1-bit array measurements.
III. D ECISIONS

(14)

is to be performed. To this end, we assign a critical region
C ⊂ Z K and decide in favor of H1 if the observed data
satisfies Z ∈ C. The probability of erroneously deciding for
H1 while H0 is the true data-generating model is calculated
Z
PFA =
pZ (Z; θ0 )dZ,
(15)

Y(z)

M

H1 : z ∼ pz (z; θ1 )

C

Using model (1) in practical applications implies that a highresolution analog-to-digital converter (ADC) is available for
each output channel. Significant savings are possible when
using a converter with 1-bit output resolution. Such a receiver
can be modeled by an ideal sampling device with infinite
resolution followed by a hard-limiter
z = sign (y),

carrier measure. Given a data set Z ∈ Z K of the form (9),
the simple binary hypothesis test between

K
X

t(z k )

(21)

k=1

such that, by the central limit theorem, the test statistic in the
large sample regime follows the normal distribution


 a
(T (Z) − µi )2
1
exp −
,
(22)
p T (Z)|Hi = √
2σi2
2πσi
a

where by = we denote asymptotic equality. Through the mean
and standard deviation of the test statistic
µi = EZ;θi [T (Z)] ,
r
h
σi =

EZ;θi

2 i
T (Z) − µi ,

the asymptotic performance is then given by


ξ − µ1
a
,
PD = Pr {T (Z) > ξ|H1 } = Q
σ1


ξ − µ0
a
PFA = Pr {T (Z) > ξ|H0 } = Q
,
σ0

(23)
(24)

(25)
(26)

3

where Q(u) denotes the Q-function. Consequently, for a
desired PFA , the decision threshold is
a

ξ = Q−1 (PFA ) σ0 + µ0 ,
resulting in the asymptotic probability of detection


µ1 − µ0
σ0
a
−1
.
−
PD (PFA ) = Q Q (PFA )
σ1
σ1

(27)

(28)

2
arcsin (Σy (θ)) ,
π
−1
−1
Σy (θ) = diag (Ry (θ)) 2 Ry (θ) diag (Ry (θ)) 2 .

Rz (θ) =

Further, the evaluation of (35) requires determining the matrix
h
T i

(43)
C(θ) = Ez;θ vec zz T vec zz T

(29)
IV. R ESULTS

the log-likelihood ratio for exponential family models is
ln L(Z) =

K
X

k=1

T

b φ(z k ) − K(λ(θ 1 ) − λ(θ 0 )),

(30)

such that with the empirical mean of the sufficient statistics
φ̄ =

K
1 X
φ(z k ),
K

(42)

which is possible [21] by the arcsine law and the orthant
probability of the quadrivariate Gaussian distribution [26].

Writing
b = β(θ1 ) − β(θ0 ),

is obtained by the arcsine law [25, pp. 284],

(31)

We apply the results to discuss the design of a 1-bit array
GPS spectrum monitoring system. The task is to check a spectral band with two-sided bandwidth B = 2.046 Mhz, centered
at 1.57 GHz, for a source signal from a specific direction.
The receiver samples the low-pass filtered receive signal at
Nyquist rate, i.e., fs = B, such that K = 2046 array snapshots
are available within a millisecond. The upper triangular area

k=1

a likelihood-based test statistic is

0

T (Z) = b φ̄.
The mean and standard deviation of the test are
r
1 T
T
b Rφ (θi )b,
µi = b µφ (θ i ), σi =
K

(32)
−2
(33)

χ [dB]

T

−4

where
µφ (θ) = Ez;θ [φ(z)] ,
h
i
Rφ (θ) = Ez;θ φ(z)φT (z) − µφ (θ)µT
φ (θ).

(34)
(35)

For Gaussian models (6),


1
β(θ) = − vec R−1
y (θ) ,
2

φ(y) = vec yy T ,

µφ (θ) = vec (Ry (θ)) ,

(36)
(37)
(38)

where the matrix Φ eliminates duplicate and diagonal statistics. This is potentially suboptimal as (39) does, in general,
not contain all sufficient statistics of a multivariate Bernoulli
distribution. The missing statistics are absorbed in the carrier
of (13) and, therefore, do not contribute to the decision
process. For the natural parameter difference (29), we use
(40)

as it maximizes the distance of the asymptotic outcome in (32)
under both hypotheses. The mean of the statistics (39),
µφ (θ) = Ez;θ [φ(z)] = Φ vec (Rz (θ)) ,

−8
5

and the matrix (35) can be determined through Isserlis’ theorem. If the model (13) is unspecified, a favorable choice of
φ(z) and b has to be found. For the 1-bit analysis, we use

(39)
φ(z) = Φ vec zz T ,

−1
b = R−1
φ (θ 1 )µφ (θ 1 ) − Rφ (θ 0 )µφ (θ 0 )

SNR = −15
SNR = −18
SNR = −21
SNR = −24

−6

(41)

10
15
Number of Sensors S

dB
dB
dB
dB
20

Fig. 1. Quality vs. Array Size (K = 2046, ζ = 45◦ )

under the receiver operating characteristic (ROC) curve,
Z 1
χ=2
PD (u)du − 1,
(44)
0

determines the system quality regarding the detection task.
√ For
different signal-to-noise ratios (SNR), γ0 = 0 vs. γ1 = SNR,
Fig. 1 shows the 1-bit system quality for an exemplary setting
with K = 2046, ζ = 45◦ , versus the number of array elements
S. While for very weak sources (SNR = −24 dB) more than
S = 20 sensors are required to provide high performance, at a
power level of SNR = −15 dB already S = 5 antennas suffice
to operate close to a perfect monitoring system with χ = 1. To
determine a favorable observation length, for S = 8, ζ = 30◦ ,
Fig. 2 shows χ for different numbers of samples. While
reliable detection with SNR = −24 dB requires sampling
more than 10 ms, the decision at SNR = −15 dB can be
made trustworthy within less than 1 ms. Fig. 3 depicts the
analytic and simulated performance (using 105 realizations)
with S = 8, K = 100, ζ = 15◦ . At moderate sample size, the

4

R EFERENCES

χ [dB]

0

−2
−4

SNR = −15
SNR = −18
SNR = −21
SNR = −24

−6
2

dB
dB
dB
dB

4
6
8
Observation Time [ms]

10

Fig. 2. Quality vs. Observation Time (S = 8, ζ = 30◦ )

asymptotic results already show a good correspondence with
the Monte-Carlo simulations.
1
0.8

PD

0.6
0.4
0.2
0

PFA = 10−3
PFA = 10−4
−18 −16 −14 −12 −10 −8
Signal-to-Noise Ratio [dB]

−6

Fig. 3. Analysis vs. Simulation (S = 8, K = 100, ζ = 15◦ )

V. C ONCLUSION
We have derived achievable detection rates with a large
number of radio measurements obtained with a lowcomplexity sensor array performing 1-bit analog-to-digital
conversion. Discussing the simple binary hypothesis test in the
framework of the exponential family enables circumventing
the intractability of the 1-bit array likelihood. Its difficult
characterization forms a fundamental obstacle to the application of analytic tools in statistical signal and information
processing to problems with coarsely quantized multivariate
data. Using the analytic results to determine the spectrum
monitoring capability with wireless multi-antenna receivers
shows that, under the right system design, radio frequency
measurements from binary arrays are sufficient to perform the
processing task of signal detection in a fast and reliable way.

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