Impact of Reporting-Channel Coding on the

Performance of Distributed Sequential Sensing

Sachin Chaudhari and Visa Koivunen

SMARAD CoE, Department of Signal Processing and Acoustics,

Aalto University School of Electrical Engineering, Finland

Email: sachin.chaudhari@aalto.fi,visa@signal.hut.fi

Abstract—This paper compares the impact of using an errordetection code and an error correction code for cooperativesequential sensing in cognitive radios. The analyzed systemmodel is a distributed parallel detection network in which eachsecondary user (SU) sends a soft decision in the form of quantizedlocal log-likelihood ratio (LLR) to the fusion center (FC), whichmakes the final decision in a sequential manner. At each SU, thequantized LLRs are converted to bits using Gray mapping. Thesebits are then channel coded and transmitted using binary phaseshift keying (BPSK) to the FC. The reporting-channels betweenthe SUs and the FC may cause errors in the decision statistics. Incase of an error detecting code, the decision statistics detected inerror are dropped while in case of an error correcting code, thedecision statistics are used anyway. The performances of thesetwo schemes are also compared to the case where no channelcoding is used. It is shown that even a simple error correctingcode gives huge improvement in the performance of sequentialsensing while error detecting code may even perform worse thanthe case where no channel coding scheme is used.

I. INTRODUCTION

Cognitive radios hold the promise of efficient spectrum uti-

lization through dynamic spectrum access. Available spectrum

opportunities can be efficiently found by multiple secondary

users (SUs) through collaborative sensing. We consider decen-

tralized cooperative sensing scenario where each SU sends a

local decision statistic to a fusion center (FC) which makes

a final decision regarding the presence of primary user (PU)

transmission in the band of interest. Sequential detection and

in particular, sequential probability ratio test (SPRT) [1]–[3] is

an appealing candidate for such cooperative sensing scenario

as it minimizes the detection time given the constraints on

the probabilities of false alarm and missed detection. In this

paper, we consider SPRT at the FC. Sequential fusion at the

FC based on the decision statistics from the local sensors has

been considered in the cognitive radio literature [4], [5].

In [6], the effects of quantization and channel errors on

the performance of soft decision based sequential sensing

have been studied. The reporting channel are modeled using

a general bit error probability model in [6] and it is assumed

that the corrupted soft decisions will be used anyway. In many

cases, the transmitted packet incorporates an error detection

code such that the received packets detected in error are

dropped altogether. If the detection of errors is conveyed to the

SUs, they can retransmit the data if an error control mechanism

such as automatic repeat request (ARQ) is in use. Note that the

feedback and retransmissions increase the reporting overhead

and the overall sensing time. In this paper we are interested

in finding if any significant delays are experienced in decision

making (at same reliability level) by using error detecting or

error correcting codes. Moreover we want to compare their

performances in different channel conditions and coding rates.

We consider sequential detection at the FC for distributed

detection using local log-likelihood ratio (LLR) based soft

decisions. Each SU quantizes its LLR and sends it to the FC

over an erroneous reporting channel. The FC makes the final

decision sequentially. We consider maximum output entropy

(MOE) quantization [7] which is asymptotically similar to

the optimal quantizer for Gaussian signals in the low signal-

to-noise ratio (SNR) regime for independent and identically

distributed (i.i.d.) observations [8]. The reporting channels

are assumed to be additive white Gaussian noise (AWGN).

Later the case of Rayleigh faded multipath channels is also

considered. The contributions of this paper are as follows:

• Expressions for the average sample number (ASN) are

derived for the sequential detection scheme while using

error detection and correction codes for the reporting

channels.

• Comparison of the sequential detection schemes is carried

out with following different channel coding schemes: no

channel coding, error detection and error correction.

• It is shown that the use of error correcting codes as

compared to the case of no channel coding gives huge

savings in the number of samples required to achieve the

same error probabilities at the FC. On the other hand, the

use of error detecting codes and subsequent dropping of

decision statistics may result in increase in the ASN and

their performance may even become worse than the case

of not using any channel coding.

The paper is organized as follows. In Section II, system

model for the soft decision based SPRT is presented in the

considered scenario followed by their performance analysis in

Section III. Simulation results are presented in IV. Finally,

concluding remarks are given in Section V.

II. SYSTEM MODEL

In this paper, we consider that there are several spatially-

located SUs which cooperate to sense a common PU. The

problem of PU sensing is formulated as a binary hypothesis

test with H0 being the null hypothesis that the PU is inactive

while H1 being the alternative hypothesis that the PU is

978-1-4673-5577-3/13/$31.00 ©2013 IEEE

2013 IEEE 14th Workshop on Signal Processing Advances in Wireless Communications (SPAWC)

185

active. For cooperative sensing (CS), a distributed sequential

approach is employed such that each of the SUs summarize

the observations and send a decision statistic to the FC which

employs sequential detection.

Fig. 1 presents the considered framework for studying the

cooperative sensing performance in the presence of quantiza-

tion and reporting-channel errors. The nth SU evaluates an

LLR Ln from the received observations. The LLR Ln is then

quantized to one of D levels using maximum output entropy

(MOE) scheme such that each of the levels has the same

probability mass function (pmf) value of 1/D. The quantized

LLR, denoted by Lsun , is then converted to d = log2 D

bit sequence Ssun using Gray mapping. These bits are then

coded using a channel coding scheme and transmitted using

binary phase shift keying (BPSK) over the non-ideal reporting

channel to the FC. The received d-bit sequence from the nth

SU at the FC and corresponding soft decision are denoted by

Sfcn and Lfc

n , respectively.

FC

SU

n

SU

1

SU

N

D-level

MOE

Quantizer

PU Rx

PU Tx

su

NL

d-bit

Gray

Mapping

suL

)(fc

NL

su

NS

fc

NS

fcS

Reporting Channels Listening Channels

)(fcL

L

NL

Channel

Coding

D-level

MOE

Quantizer

d-bit

Gray

Mapping

Channel

Coding

D-level

MOE

Quantizer

d-bit

Gray

Mapping

Channel

Coding

fcS

0H

1Hor

Fig. 1. Considered scenario for cooperative sensing: each SU evaluates itssufficient statistic (Ln), quantizes it to form a soft decision Lsu

n using maxi-mum output entropy (MOE) quantization and then transmits a correspondingd-bit sequence Ssu

n using a channel coding scheme. Due to reporting-channel

errors, the fusion center (FC) receives a symbol Sfcn corresponding to soft

decision Lfcn . The FC then combines the received soft decisions from the

cooperating SUs to make a final decision in a sequential manner.

At the FC, the optimal fusion rule for a given quantization

scheme under the assumption of independent sensor obser-

vations in the presence of channel errors is the sum of log

likelihood ratios (LLRs) of the received soft decisions [2].

The LLR of the received decision statistic (Lfcn ), denoted by

λn, is given by

λn , logP(Lfc

n |H1)

P(Lfcn |H0)

∀ n = 1, .., k. (1)

In terms of the LLRs, the sequential test or the SPRT [1] after

receiving k statistics is

k∑

n=1

λn ≤ log B, Decide H0

k∑

n=1

λn ≥ log A, Decide H1 (2)

Otherwise, Take Next User’s Statistics.

In the above expressions A = 1−βcs

αcsand B = βcs

1−αcs. Here αcs

and βcs denote the constraints on the false alarm probability

and missed detection probability, respectively.

III. PERFORMANCE ANALYSIS

In this section, we evaluate the performance parameters of

interest for the SPRT, namely, the average sample number

(ASN). The ASN for a SPRT scheme is defined as the number

of decision statistics required on average for arriving at a

decision under either hypothesis such that the performance

constraints on βcs and αcs are satisfied. To do so, we need to

evaluate the distribution of the LLR λn for the SPRT in terms

of a given channel condition and the distributions of LLRs

Ln. The first step in this process is to evaluate the distribution

of the quantized LLR Lsun at the nth SU in terms of the given

distribution of Ln. The reporting channel errors may cause the

distribution of the received quantized LLR Lfcn to differ from

that of the transmitted quantized LLR. Therefore, the next

step is to evaluate the distribution of Lfcn . The final step is to

evaluate the distributions of λn. Consequently, the distribution

of λn under both hypotheses can be evaluated in terms of a

given channel statistics and the distributions of the LLRs Ln.

Given the distribution of Ln, the pmf of the transmitted soft

decision Lnsu can be calculated using the fact that each of the

levels has the same probability mass function (pmf) value of

1/D. See [8] for details. The pmf of the received soft decision

Lfcn at the FC under either hypothesis is given in [9] by

P(Lfcn =li,n |Hj , Pb) =

D∑

k=1

Pdik,n

b (1−Pb)d−dik,nP(Lsu

n =lk,n |Hj),

(3)

where dik,n is the Hamming distance between si,n and sk,n,

while si,n is the d-bit sequence corresponding to li,n. Here

Pb is bit error probability (BEP) of the reporting-channel. The

pmf of Lfcn can be evaluated from (3) under the assumption

that Pb is known. This is a reasonable assumption as we know

the modulation scheme and the SNR values for the reporting

channel. For example, Pb for using BPSK in AWGN channel

is given [10] by

Pb = Q(√

2γ). (4)

Similarly the BEP for BPSK in Rayleigh fading channel is

given [10] by

Pb =1

2

(

1−

√

γ

γ + 1

)

, (5)

where γ = γE[a2] with a being the amplitude of the channel

impulse response.

Let Ks denote the number of decision statistics after which

SPRT makes a final decision. Note that Ks is a random

variable and the ASNs for the SPRT conditioned on either

of the two hypotheses are given by [2]

E[Ks |H0, Pb] =αcs logA+ (1− αcs) logB

E[λn |H0, Pb]

and E[Ks |H1, Pb] =(1− βcs) logA+ βcs logB

E[λn |H1, Pb], (6)

978-1-4673-5577-3/13/$31.00 ©2013 IEEE

2013 IEEE 14th Workshop on Signal Processing Advances in Wireless Communications (SPAWC)

186

where E[·] is the expectation operator. Here E[Ks |H0, Pb]and E[Ks |H1, Pb] can be numerically computed from the

distribution of Ln, Pb, definition of λn and the pmf of λn,

which is the same as that for Lfcn . Now the ASN (or the

average number of SU statistics) for the SPRT is given by

Kb(Pb) = max {E[Ks |H0, Pb], E[Ks |H1, Pb]} . (7)

A. Error Detection Code

For the case of error detection code, the d-bit decision

statistic is dropped if any of the d decoded bits are detected

in error. Let PA denote the probability of the event A that

the error is not detected in a decoded bit. Since the errors

in the decoded bits of a d-bit decision statistic are i.i.d.,

the probability that the d-bit decision statistic is not detected

in error is P dA

. Since the decision statistic is dropped if it

is detected in error, the probability of dropping a decision

statistic, denoted by Pdrop, is given by

Pdrop = 1− P dA. (8)

An error detection code has limits on the number of bits

it can detect. For example, a repetition code (r, 1) can detect

only r−1 bit errors. Therefore certain amount of errors in the

decoded bit may not be detected. If we denote Ae to be the

event that the decoded bit is in error, then the probability of

error Ped for the decoded bit given that the decision statistic

is not dropped is given by

Ped = P(Ae|A) =P(Ae ∩ A)

P(A). (9)

The ASN for using the considered scheme in a channel with

BEP Ped can be evaluated from (7) by substituting Pb by Ped.

Moreover, taking into account the rate of dropping the decision

statistics, the ASN for a error detection code is given by

Ked =Kb(Ped)

1− Pdrop

. (10)

B. Error Correction Code

In case of an error correction code, the decision statistics are

used anyway. Therefore if we denote the bit error probability

for the coded bit as Pec, the ASN Kec for an error correcting

code can be calculated from (7) by substituting Pb by Pec,

i.e.,

Kec = Kb(Pec). (11)

IV. RESULTS

For the results, an orthogonal frequency division multiplex-

ing (OFDM) based PU signal is considered while the local

detector is an autocorrelation detector [4]. The autocorrelation

coefficient based LLR for the nth SU is given by [4]

Ln = −M log(1− ρ2n) +2Mρn(ρn − ρn)

1− ρ2n, (12)

where

• ρn is the maximum likelihood estimate of the autocorre-

lation coefficient,

• M + Td (M ≫ Td) is the number of received observa-

tions,

• Td is the useful symbol length in an OFDM symbol,

• ρn = Tc

Td+Tc· SNRl

1+SNRlis the true autocorrelation coefficient

under the alternative hypothesis, and

• Tc is the cyclic prefix length in an OFDM symbol.

• SNRl is the SNR on the listening channel at a SU.

Under the two hypotheses, the distributions of Ln can be

derived from the distribution of ρn [4] and are given by

H0 : Ln ∼ N (mn0, σ2n0)

H1 : Ln ∼ N (mn1, σ2n1), (13)

where mn0=−M log(1−ρ2n)−2Mρ2

n

1−ρ2n

, mn1=−M log(1−ρ2n),

σ2n0=

2Mρ2

n

(1−ρ2n)

2 and σ2n1=2Mρ2n.

All the listening channels are considered AWGN channels

with SNRl = −4dB while Tc = 8, Td = 32, and M = 400.

The constraints on the probabilities of false alarm and missed

detection are αcs = 0.01 and βcs = 0.01, respectively.

Without loss of any generality, a simple repetition code

(r, 1) with repetition rate r is assumed for channel coding

for convenience. The repetition code (r, 1) can detect r − 1errors while it can correct

⌊

r−12

⌋

errors. For a repetition code

in the error detection mode, no error is detected in the decoded

bit if all the r bits are same (either all or none of the bits in

error). Therefore we have

PA = (1− Pb)r + P r

b . (14)

Consequently the rate of dropping a d-bit decision statistic

while using the repetition code (r, 1) for error detection is

given by

Pdrop = 1− P dA

= 1− [(1− Pb)r + P r

b ]d. (15)

while the probability of error for a decoded bit at the FC is

given by

Ped =P(all r bits in error)

P(A)

=P rb

(1− Pb)r + P rb

. (16)

The probability of error for the decoded bit using the repetition

code (r, 1) for error correction is given by

Pec = 1−B(

⌊

r − 1

2

⌋

, r, Pb), (17)

where B(k, n, p) is the Binomial cumulative distribution func-

tion with parameters k, n, and p.

Note that we will be considering the definition of signal-to-

noise-ratio (SNR) to be γ = Eb/N0 based on energy per bit

Eb instead of energy per coded bit Ec = rEb. Here N0 = 2σ2n

is the noise spectral density with σ2n being the noise variance.

The presence of a frame header and additional information like

interference levels, channel states and occupancy information,

or probabilities for the channel occupancy [11], may result

978-1-4673-5577-3/13/$31.00 ©2013 IEEE

2013 IEEE 14th Workshop on Signal Processing Advances in Wireless Communications (SPAWC)

187

−12 −10 −8 −6 −4 −2 0 2 410

0

101

102

103

ASN vs Eb/N

0 (Rep.Chan.); AWGN Channel; d=2;α

cs=0.01, β

cs=0.01

Eb/N

0 (rep. chan.)

AS

N

No Coding; theory

No Coding; sim

ED r=3; theory

ED r=3; sim

EC r=3; theory

EC r=3; sim

Fig. 2. Theoretical and simulated average sample number (ASN) vs. SNR(dB) for SPRT schemes in AWGN reporting channels for the following threecases: no coding, error detection (r = 3), and error correction (r = 3). Hered = 2, αcs = 0.01, and βcs = 0.01. It can be seen that the simulatedvalues get closer to the theoretical values as ASN increase for all the threecases. For low SNR values, the required ASN values for same performanceparameters increase in the following order: error correction (r = 3), errordetection (r = 3) and no coding. However, the performances are similar athigh SNR values in all the three cases.

in a significant overhead. Therefore, the relative increase in

the transmitted data and energy needed while using the above

definition of the SNR for the channel coding may be small.

The major savings take place if one can refrain from sending

the decision statistics as in the case of censoring [12].

Fig. 2 shows the theoretical and simulated ASN curves

as a function of SNR (dB) in AWGN reporting channels

for three cases: no coding, error detecting code (r = 3),

and error correcting code (r = 3). The number of bits for

quantizing the LLRs is d = 2. For simulations, number of

realization used for evaluating the ASN is 2000. The ASN

for the three cases can be evaluated from (6), (7), (10), and

(11), which in turn depend on the pmf of Lfcn . The pmf of

Lfcn is implemented numerically using (3) for convenience as

a closed form expression for it is difficult to obtain. It can be

seen that simulation results are close to theoretical values. Note

that ASN given by (6) for asymptotic case, i.e., Ks → ∞.

That explains why the curves for theory and simulation get

close as average ASN increase. There are huge savings in

ASN while using the repetition code (3, 1) for correcting one

error as compared to the case when no coding is used. On

the other hand, the performance of the repetition code (3, 1)

for detecting two errors is worse than the no coding case.

However, the performances are similar at high SNR values in

all the three cases.

Fig. 3 shows the theoretical ASN curves as a function

of SNR (dB) in AWGN reporting channels for d = 1 and

d = 3. For d = 1, there are huge savings in ASN while

using repetition code for error correction (r = 3) and error

detection (r = 3) as compared to the case of no channel

coding. However, the performance for error detection scheme

degrades as the number of bits for quantization is increased

−5 0 5 1010

0

101

102

ASN vs Eb/N

0 (Rep.Chan.); , AWGN, α

cs=0.01, β

cs=0.01

Eb/N

0 (rep. chan.)

AS

N

No Coding; d=1

No Coding; d=3

ED r=3; d=1

ED r=3; d=3

EC r=3; d=1

EC r=3; d=3

Fig. 3. Theoretical average sample number (ASN) vs. SNR (dB) for SPRTschemes in AWGN reporting channels with d = 1 and d = 3 for αcs = 0.01and βcs = 0.01. The performance of error correction code (r = 3) and no-channel coding schemes improve with increase in the number of bits forquantization while that of error detection code (r = 3) degrades.

to d = 3 and its performance is worse than the case of no-

channel coding. On the other hand, the performance of error

correction code (r = 3) improves when the number of bits for

quantization are increased.

Fig. 4 shows the theoretical ASN curves as a function of

SNR (dB) in AWGN and Rayleigh fading reporting channels

for the three cases: no coding, error detection (r = 5), and

error correction (r = 5). Here E[a2] = 1 for the Rayleigh

fading channel and d = 2. The performances of all the

schemes degrade in the Rayleigh channels as compared to the

AWGN channels. However the performance of error detecting

code (r = 5), which can detect 4 errors, is worse than no

coding case in both the channel conditions. On the other hand,

there are huge performance gain for using error correction

code (r = 5), which can detect 2 errors. For Rayleigh channel,

the error correction code (r = 5) performs better than the no-

coding case even for high SNR values.

From Figs. 2, 3, and 4, it can be seen that the error

correcting codes outperform the no-coding case in the low

SNR regime as expected. However it is surprising that the

error detecting code performs worse than the no-coding case

in most of the cases except d = 1, when its performance

is better than that in the no-coding case and is very close to

performance of the error correcting case. This difference in the

performances of the error detection and error correction codes

is because of the use of soft decision based local decision

statistics. Soft decisions convey level of confidence with which

a decision is made. With the use of channel coding and Gray

mapping, the reporting-channel errors lower the confidence of

the decision without changing the decision most of the time.

On the other hand, error detection code throws away even a

decision statistic which has a small error. With increase in

number of bits for quantization d and repetition rate r, the

probability of error Ped decreases but the rate of dropping

soft decisions increases rapidly. Therefore the error correcting

978-1-4673-5577-3/13/$31.00 ©2013 IEEE

2013 IEEE 14th Workshop on Signal Processing Advances in Wireless Communications (SPAWC)

188

−5 0 5 1010

0

101

102

ASN vs Eb/N

0 (Rep.Chan.); d=2, α

cs=0.01, β

cs=0.01

Eb/N

0 (rep. chan.)

AS

N

No coding; AWGN

No Coding; Rayleigh

ED r=5; AWGN

ED r=5; Rayleigh

EC r=5; AWGN

EC r=5; Rayleigh

Fig. 4. Comparison of average sample number (ASN) vs. SNR (dB) forsequential detection in AWGN and Rayleigh fading channels for three casesof no coding, error detection (r = 5), and error correction (r = 5). Hered = 2, αcs = 0.01 and βcs = 0.01. The performances of all the schemesdegrade in the Rayleigh channels as compared to the AWGN channels. Theerror correction code (r = 5) also performs better than the no-coding case inRayleigh channel conditions even for high SNR values. On the other hand,the performance of error detection code (r = 5) is worse than the no-codingcase.

code gives improvements for soft decision based cooperative

sequential sensing with increase in r and d while the ASN for

the error detecting code becomes worse.

V. CONCLUSION

In this paper, comparison of error correcting codes and error

detecting codes has been carried out to find their suitability

to the decentralized sequential sensing in the presence of im-

perfect reporting channels. Each SU is considered to quantize

the LLR to d-bits using MOE quantization. The FC employs

sequential detection. As an example, a simple repetition code

(r, 1), which can detect r − 1 bit errors and correct⌊

r−12

⌋

errors, has been considered for encoding these bits for sending

them from the SUs to the FC.

It is shown that there are huge savings in terms of average

number of decision statistics required for achieving the same

detection performance by using simple repetition code for

error correction instead of no-coding at all. On the other hand,

error detection improves performance only when the number

of bits for quantization d and repetition rate r are both low

while the performance is worse than a no-coding case for high

d or high r.

REFERENCES

[1] A. Wald, “Sequential tests of statistical hypotheses,” Ann. Math. Stat.,vol. 16, no. 2, pp. 117–186, Jun. 1945.

[2] P. K. Varshney, Distributed Detection and Data Fusion, New York:Springer, 1997.

[3] R. S. Blum, S. A. Kassam and H. V. Poor, “Distributed Detection withMultiple Sensors: Part II - Advanced Topics,” Proc. IEEE, vol. 85, pp.64–79, Jan. 1997.

[4] S. Chaudhari, V. Koivunen and H. V. Poor, “Autocorrelation-Based De-centralized Sequential Detection of OFDM signals in Cognitive Radios,”IEEE Trans. Signal Process., vol. 57, pp. 2690–2700, Jul. 2009.

[5] Q. Zou, S. Zheng and A. Sayed, “Cooperative sensing via sequentialdetection,” IEEE Trans. Signal Process., vol. 58, pp. 6266–6283, Dec.2010.

[6] S. Chaudhari, J. Lunden, and V. Koivunen, “Effects of Quantization andChannel Errors on Sequential Detection in Cognitive Radios,” in Proc.

46th Conf. on Information Science and Systems (CISS), Princeton, USA,Mar. 21-23, 2012.

[7] D. Messerschmitt, “Quantizing for Maximum Output Entropy,” IEEE

Trans. Inform. Theory, vol. 17, p. 612, Sep. 1971.[8] S. Chaudhari, J. Lunden and V. Koivunen, “Cooperative Sensing With

Imperfect Reporting Channels: Hard Decisions or Soft Decisions?,” IEEE

Trans. Signal Process., vol. 60, no. 1, pp. 1–11, Jan. 2012.[9] S. Chaudhari and V. Koivunen, “Effect of Quantization and Channel

Errors on Collaborative Spectrum Sensing,” in Proc. 43rd Ann. Asilomar

Conf. Signals, Systems, and Computers, Pacific Grove, CA, USA, Nov.1-4, 2009.

[10] T. K. Moon, Error Correcting Coding, New Jersey: John Wiley & Sons,2005.

[11] Deliverable 5.1, “Report on Fundamental Limits,” SENDORA, May 2009[Online]. Available: http://www.sendora.eu/node/123.

[12] J. Lunden, V. Koivunen, A. Huttunen, and H. V. Poor, “Collaborativecyclostationary spectrum sensing for cognitive radio systems,” in IEEE

Trans. on Signal Process., vol. 57, no. 11, pp. 4182-4195, Nov. 2009.

978-1-4673-5577-3/13/$31.00 ©2013 IEEE

2013 IEEE 14th Workshop on Signal Processing Advances in Wireless Communications (SPAWC)

189