Multi-Bit Cooperative Spectrum Sensing Strategyin Closed Form

Xiaoyuan Fan1, Dongliang Duan1, and Liuqing Yang2

1. Department of Electrical and Computer Engineering, University of Wyoming

Dept. 3295, 1000 E. University Ave., Laramie, WY 82071

Emails: xfan3@uwyo.edu, dduan@uwyo.edu

2. Department of Electrical and Computer Engineering, Colorado State University

1373 Campus Delivery, Fort Collins, CO 80523

Email: lqyang@engr.colostate.edu

Abstract—Spectrum sensing is one of the most importanttasks in cognitive radio system. In order to combat fading,cooperation among different sensing users is usually adopted.In our previous work [1], we quantified the performance gainof cooperative spectrum sensing by the notion of diversity. Inaddition, we have shown that even with local binary decisions, thecooperative spectrum sensing can achieve the maximum diversityby appropriately selecting local and fusion rules. However, thereis a significant signal-to-noise ratio (SNR) loss compared with thesoft information fusion scenario due to local binary quantization.Intuitively, increasing the number of bits of local quantizationwill improve the sensing performance. Most work in the literatureon multi-bit cooperative sensing are mathematically intractableand can only be solved numerically with high complexity. Inthis paper, by jointly maximizing diversity and SNR gain, weprovide a generalized multi-bit cooperative sensing strategy withthe local and fusion decision rules in explicit closed form.Simulations show that even with small number of bits, ourproposed cooperative sensing strategy can significantly improvethe sensing performance.

I. INTRODUCTION

Cognitive radios, also known as opportunistic spectrum

access, is recently proposed as a promising technique to deal

with the shortage of spectrum by allowing the unlicensed

secondary users to access the licensed band when it is re-

leased by the licensed primary user [2], [3]. Among various

tasks involved, spectrum sensing is the foremost by detecting

the spectrum occupancy. One major bottleneck of spectrum

sensing performance is the fading between the transmitters of

primary users and the receivers of the secondary sensing users.

In order to combat fading and improve the agility of spectrum

sensing, cooperation among different secondary sensing users

is usually adopted. In this case, different sensing users observe

independently faded signals transmitted from the primary user,

and the probability for the signal at all cooperating users to be

weak is very low. Therefore, the sensing performance can be

significantly improved. There are many efforts in the literature

on the cooperative spectrum sensing [1], [4]–[13].

Theoretically, in order to achieve the best cooperative sens-

ing performance, all the information received at local sensing

users need to be collected to the fusion center, which is

known as the cooperative sensing with soft information fusion

(SCoS). While the performance of SCoS is desirable, however,

this requires large communication bandwidths and is imprac-

tical. Therefore, quantization is usually made at local sensors

in practical systems. Our previous work [1], [11] optimized

and evaluated the system performance of SCoS, binary and

ternary information fusion (BCoS & TCoS) with the notion

of diversity and SNR gains. There are also other existing

works on cooperative sensing with local quantization. For

example, in [14], a softened two-bit hard combination scheme

is proposed to achieve a better detection of the primary user

and showed to achieve a good tradeoff between performance

and complexity by experiments; in [15], an algorithm with

tri-state hard information fusion and erasure-transmission is

proposed to eliminate the propagation of unreliable messages.

However, most of these work highly rely on high-complexity

numerical algorithms to solve for the sensing strategy, and

performance improvements are only shown by simulations

rather than theoretical analyses. Furthermore, they cannot be

easily extended to the case with larger number of bits. In

contrast, our previous work results in closed-form sensing

strategies and indicates the performance by diversity and SNR

gains, which lays a good foundation to optimize cooperative

sensing with local quantization of higher number of bits. In

this paper, we assume a general log2 M -bit (or equivalently,

M -array) local quantization and by jointly considering the

diversity and SNR gains, we derive simple and closed-form

expressions for both the local thresholds and fusion rule.

The remaining of this paper is organized as follows: the

problem formulation with the preliminaries of cooperative

sensing with M -array information fusion (MCoS) is intro-

duced in Section II. Then by finding the relationship between

BCoS and MCoS and based on our previous results for BCoS,

we determine the closed-form local and fusion rules for MCoS

in Section III. Numerical results are presented in Section IV,

followed by concluding remarks in Section V.

II. PROBLEM FORMULATION

A. Signal Model

During the spectrum sensing process, the sensing users

observe signals under the following two hypotheses:

H0 : absence of primary users at the band of interest,

H1 : presence of primary users at the band of interest.

We assume that the channels between the primary and the

secondary sensing users are Rayleigh fading with additive

white circular symmetric complex Gaussian (CSCG) noise.

Normalized by the noise variance, the signal at each local

sensing user follows a complex Gaussian distribution under

each hypothesis [1], [6]:

ri|H0 = ni ∼ CN(0, 1),

ri|H1 = hix+ ni ∼ CN(0, γi + 1),

where ni is noise, hi is the Rayleigh fading channel, x is the

signal from primary user, and γi is the average SNR at sensing

user i. Therefore, the sufficient statistics at the local users is

the energy ‖ri‖2.

With geographically distributed sensing users, it is reason-

able to assume that they experience independent fadings. In

addition, we study the case where N secondary users are

geographically clustered to cooperate. This is based on the

consideration that the cooperative detection of the primary user

presence is only meaningful when the cooperative secondary

users are subject to the same primary user activity. In this case,

the average SNR at all secondary sensors are the same, i.e.,

γi = γ. Hence, ris are conditionally independent identically

distributed (i.i.d.) under each hypothesis. In addition, without

loss of generality, we assume that H0 and H1 are equi-

probable.

To quantify the cooperative gain for spectrum sensing, the

diversity order is defined as [1]

d∗ = − limγ→+∞

logP∗log γ

, (1)

where ∗ can be false alarm (‘f’), missed detection (‘md’), or

average error probability (‘e’).

B. Cooperative Sensing With Multi-bit Information Fusion(MCoS)

Due to the bandwidth limit between local sensing users and

the fusion center, received signal energy ‖ri‖2s are quantized

to obtain log2 M -bit (or equivalently M -array) local decisions.

In general, the quantizations at local sensing users can be set

as

qi=

⎧⎪⎨⎪⎩0, if 0 ≤ ‖ri‖2 < θ1,

m, if θm ≤ ‖ri‖2 < θm+1, ∀m ∈ [1,M − 2],

M − 1, if ‖ri‖2 ≥ θM−1.(2)

where 0 < θ1 < θ2 < . . . < θM−1 are the M − 1 local

decision thresholds. Then, the log2 M -bit local decisions qisare sent to the fusion center for a global decision:

φ(q1, q2, . . . , qN )→ {H0, H1} . (3)

In order to determine the optimum fusion rule, we first

obtain the conditional probabilities under the H0 hypothesis

of qis as:

αm=P (qi=m|H0)=

⎧⎪⎨⎪⎩1− e−θ1 if m = 0,

e−θm − e−θm+1 if 1 ≤ m ≤M−2,

e−θM−1 if m = M−1.(4)

and the conditional probabilities under the H1 hypothesis as:

βm=P (qi=m|H1)=

⎧⎪⎪⎨⎪⎪⎩1− e−

θ1γ+1 if m = 0,

e−θmγ+1 − e−

θm+1γ+1 if 1 ≤ m ≤M−2,

e−θM−1γ+1 if m = M − 1.

(5)

Then, at the fusion center, the sufficient statistics for global

decision is the histogram n = [n0, n1, ..., nM−1]T , where nm

is the number of local decisions with qi = m collected at the

fusion center. Evidently, n follows the distribution under each

hypothesis as:

n|H0 ∼ Mun(α0, α1, . . . , αM−1),

n|H1 ∼ Mun(β0, β1, . . . , βM−1),(6)

where Mun(p0, p1, . . . , pM−1) denotes a multinomial distribu-

tion with

P (n) =N !∏M−1

m=0 (nm!)

M−1∏m=0

pnmm , (7)

and meanwhileM−1∑m=0

nm = N . (8)

As a result, the global decision can be made as:

φ(n)→ {H0, H1} , (9)

instead of Eq. (3). It should be also noted that although the

sufficient statistic is M -dimensional, due to the constraint of

Eq. (8), the actual degree of freedom is M − 1.

DenotingR1 andR0 as the set of n = [n0, n1, . . . , nM−1]T

to make global decision φ = H1 and φ = H0, respectively,

we can write the global false alarm and missed detection

probabilities as:

Pf =∑n∈R1

P (n|H0),

Pmd =∑n∈R0

P (n|H1).(10)

Based on Eqs. (4), (5), (6), (7) and (10), the optimum fusion

rule can be obtained by jointly optimizing Pe =12 (Pf +Pmd)

over the local quantization thresholds (θ1, θ2, . . . , θM−1) and

the global fusion rule R1. However, this general joint opti-

mization is mathematically intractable and can only be solved

numerically with very high complexity, especially when M is

large.

III. CLOSED-FORM SOLUTIONS TO MCOS

In order to obtain a simple but effective closed-form solution

for both the local quantization and global fusion strategy, we

try to optimize the MCoS by studying the diversity and SNR

gains. Due to the nice expression of local probabilities αms

and βms, we select the local thresholds as

θm = kmθo , (11)

where θo =(1 + 1

γ

)log(1 + γ) is the optimal threshold for

single-user detection. Accordingly, we denote this cooperative

sensing strategy as MCoS(k1, k2, . . . , kM−1) with 0 < k1 <k2 < . . . < kM−1.

In order to study the diversity gain, as γ → +∞, our

resultant local probabilities will be

αm ∼

⎧⎪⎨⎪⎩1− γ−k1 if m = 0

γ−km − γ−km+1 if 1 ≤ m ≤M−2

γ−kM−1 if m = M−1

, (12)

and

βm ∼

⎧⎪⎨⎪⎩k1γ

−1 if m = 0

(km+1 − km) γ−1 if 1 ≤ m ≤M−2

1− kM−1γ−1 if m = M−1

. (13)

According to Eq. (7), at the fusion center, as γ → +∞P (n|H0) ∼ γ−∑M−1

m=1 (nmkm) , (14)

and

P (n|H1) ∼ γ−(N−nM−1) . (15)

Then accordingly, the global false alarm probability and

missed detection probability can be expressed as:

Pf ∼∑n∈R1

γ−∑M−1m=1 (nmkm) , (16)

and

Pmd ∼∑n∈R0

γ−(N−nM−1) . (17)

In order to develop the fusion rule, we check the simplest

case with M = 2, also denoted as BCoS where ‘B’ stands

for “binary”. We have the following results from our previous

work [1], [11]:

Result 1 For MCoS(k), or equivalently BCoS(k), the fusionrule to maximize the diversity is:

R1=

{n = [n0, n1]

T : n1 ≥ θf,B ,where θf,B =N + 1

k + 1

},

(18)

with diversity order

de = df = dmd =k

k + 1(N + 1) . (19)

Evidently, the larger the k is, the higher the diversity order will

be. However, checking the expressions for the probabilities in

Eq. (13), we know that there is a higher SNR loss for larger

k for the terms in the missed detection probability. Therefore,

although the maximum diversity can be achieved by selecting

k = N , there is a huge SNR loss compared with the soft

information fusion case.

We expect that MCoS with larger M has the ability to

achieve the same diversity gain and at the same time improve

the SNR gain with appropriate selection of local thresholds and

fusion rule. In order to develop the fusion rule for this M -

dimensional sufficient statistic (n0, n1, . . . , nM−1), we first

establish the link between MCoS and BCoS in the following:

Proposition 1 An MCoS(k0, k1, . . . , kM−1) is equivalent toBCoS(kM−1) if the fusion rule for MCoS is set as

R1=

{n :nM−1 ≥ N + 1

kM−1 + 1

},

R0=

{n :nM−1=N−

M−2∑m=0

nm<N + 1

kM−1 + 1

}.

(20)

Proof: See Appendix A.

The equivalence in Proposistion 1 is made by applying a

one-dimensional fusion rule on the M -dimensional decision

statistics n = [n0, n1, . . . , nM−1]T for MCoS. By adopting

a higher-dimensional fusion rule, we expect to achieve a

better performance than BCoS(k). Since the missed detection

probability is the bottleneck of the performance as analyzed

before, we want to increase the decision region of R1, or

equivalently decrease R0 given in Eq. (20). Meanwhile, the

false alarm diversity should be preserved.

We know from our analysis of BCoS(kM−1) that the false

alarm diversity df,BCoS(kM−1) = kM−1

kM−1+1 (N + 1). Then,

from Eq. (16), if the set ΔR ⊂ R0 includes all points

n = [n0, n1, . . . , nM−1]T with

M−1∑m=1

(nmkm) ≥ df,BCoS(kM−1) =kM−1

kM−1 + 1(N + 1) , (21)

then ΔR can be moved from R0 to R1 without affecting

the false alarm diversity. Hence we obtain the resultant fusion

rule for MCoS(k1, k2, . . . , kM−1). In addition, by selecting

kM−1 = N , the maximum diversity can be achieved.

In summary, for MCoS,

• Local thresholds: θm = kmθo ∀m ∈ [1,M − 1], and

0 < k1 < k2 < . . . < kM−2 < kM−1 = N .• Global fusion rule:

R∗0= R0−ΔR =

{n :nM−1=0 and

M−1∑m=1

(nmkm) < N

},

R∗1= R1+ΔR =

{n :

M−1∑m=1

(nmkm)≥N}.

IV. NUMERICAL RESULTS

To verify the performance, we first plot the performance

comparisons of MCoS with BCoS and SCoS in Fig. 1 with

the number of cooperative sensing users N = 8. We can

see that all three strategies achieve the same diversity since

the error curves have the same slop. In addition, we see that

the average error probability performance is dominated by the

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10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

SNR (dB)

Err

or P

roba

bilit

ies

N=8

Pf,soft

Pmd

,soft

Pe,soft

Pf, MCoS(2,4,6,8)

Pmd

, MCoS(2,4,6,8)

Pe, MCoS(2,4,6,8)

Pf, BCoS(8)

Pmd

, BCoS(8)

Pe, BCoS(8)

Fig. 1. BCoS, MCoS, and soft information fusion. (N = 8)

missed detection performance, which verifies our motivation

in our fusion rule design. Furthermore, we can clearly see that

by increasing the number of bits at local sensing users from 1to log2 5 ≈ 2.32, the SNR gain is improved by almost 5 dB.

Compared with the soft information fusion case, the MCoS

with M = 5 only has a 3 dB performance loss. This means

that even with pretty small number of bits sent from local

sensing users to the fusion center, our proposed strategy can

achieve very good performance.

To illustrate the effect of the number of bits log2 M on the

performance, we plot the average probabilities for different

MCoS strategies. Since our current work does not look into

the optimization of the other quantization thresholds except the

largest threshold kM−1, we select the thresholds to be located

evenly from 0 to N to obtain the results in Fig. 2. It can be

seen that the more bits transmitted from the local sensing users

to the fusion center, the better the performance will be.

In our current MCoS design, we only have the selection

of the largest threshold. One can arbitrarily select all other

thresholds to achieve the same diversity gain. However, the

selection of the remaining thresholds will affect the SNR

gain as shown in Fig. 3. It seems that the smaller the other

thresholds are, the better the performance is, although it is

quite close to MCoS with evenly distributed thresholds. In

our future work, we will try to optimize the local thresholds

to maximize the SNR gain.

V. CONCLUSIONS AND FUTURE WORK

In this paper, we proposed a strategy for multi-bit cooper-

ative spectrum sensing (MCoS) to provide both diversity and

SNR gain. Both the local quantization thresholds to obtain the

multi-bit local decisions and the M -dimensional global fusion

rule are in simple closed form. Simulations confirmed that,

as the middle ground between BCoS and the soft information

fusion, MCoS provides a practical yet effective solution. In

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10−6

10−5

10−4

10−3

10−2

10−1

100

SNR (dB)

Glo

bal A

vera

ge E

rror

Pro

babi

lity

Pe

BCoS(8)MCoS(4,8)MCoS(8/3,16/3,8)MCoS(2,4,6,8)Soft

Fig. 2. MCoS with different M . M increases from 2 to 5 along the directionof the arrow. (N = 8)

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−7

10−6

10−5

10−4

10−3

10−2

10−1

100

SNR (dB)

Glo

bal A

vera

ge E

rror

Pro

babi

lity

Pe

MCoS(2,4,6,8)MCoS(5,6,7,8)MCoS(1,2,3,8)

Fig. 3. MCoS with the same M and kM−1, but different kms. (N = 8)

the future, we will look into the optimization of the local

thresholds k1, k2, . . . kM−2.

APPENDIX A

PROOF OF PROPOSITION 1

At the fusion center, BCoS has a one-dimensional suffi-cient statistics n = [n0, n1]

T with n0 + n1 = N whileMCoS has a (M − 1)-dimensional sufficient statistics n =[n0, n1, . . . , nM−1] with

∑M−2m=0 nm + nM−1 = N . When

the fusion center with MCoS makes a global decision on-ly based on the largest local threshold θM−1 as R1 ={n : nM−1 ≥ θf,B}, then the global false alarm probability

at the fusion center can be written as

Pf,M =∑

nM−1≥θf,B

[N !∏M−1

m=0 (nm!)

M−1∏m=0

αnmm

]

=∑

nM−1≥θf,B

⎡⎢⎣ N !

nM−1!αnM−1

M−1 ·∑

∑M−2m=0 nm=N−nM−1

1∏M−2m=0 (nm!)

αnmm

⎤⎥⎦

=∑

nM−1≥θf,B

⎡⎣ N !

nM−1!αnM−1

M−1 · 1

(N − nM−1)!

(M−2∑m=0

αm

)N−nM−1⎤⎦

=∑

nM−1≥θf,B

[N !

nM−1!(N − nM−1)!αnM−1

M−1 (1− αM−1)N−nM−1

].

(22)

Meanwhile, for BCoS,

Pf,B =∑

n1≥θf,B

[N !

n1!(N − n1)!αn11 (1− α1)

N−n1

]. (23)

Similarly, the missed detection probability at the fusioncenter under the proposed R1 can be written as

Pmd,M =∑

nM−1<θf,B

[N !∏M−1

m=0 (nm!)

M−1∏m=0

βnmm

]

=∑

nM−1<θf,B

⎡⎢⎣ N !

nM−1!βnM−1

M−1 ·∑

∑M−2m=0 nm=N−nM−1

1∏M−2m=0 (nm!)

βnmm

⎤⎥⎦

=∑

nM−1<θf,B

⎡⎣ N !

nM−1!βnM−1

M−1 · 1

(N − nM−1)!

(M−2∑m=0

βm

)N−nM−1⎤⎦

=∑

nM−1<θf,B

[N !

nM−1!(N − nM−1)!βnM−1

M−1 (1− βM−1)N−nM−1

].

(24)

Meanwhile, for BCoS,

Pmd,B =∑

n1<θf,B

N !

n1!(N − n1)!βn11 (1− β1)

N−n1 . (25)

So comparing Eq. (22) with (23) and Eq. (24) with (25),

we conclude that MCoS(θ1, . . . , θM−1) with the fusion rule

R1 = {n : nM−1 ≥ θf,B} is equivalent to BCoS with local

threshold θl,B = θM−1 and fusion threshold θf,B .

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