Sensing-Throughput Tradeoff for OFDM-Based

Cognitive Radio Under Outage Constraints

Youssif Fawzi Sharkasi1, Mounir Ghogho1,2 and Des McLernon,1

1School of Electronic and Electrical Engineering, The University of Leeds, Leeds, UK.2International University of Rabat, Morocco.

Email: ml08yfs@leeds.ac.uk, m.ghogho@ieee.org and d.c.mclernon@leeds.ac.uk.

Abstract—Sensing-throughput tradeoff under outage con-straints has been studied before with an energy detector (ED).However, the ED does not work well in the presence of noisepower uncertainty. So we study sensing-throughput tradeoff byusing an autocorrelation detector (AD) for an OFDM signal. Inthis paper, we derive a closed form expression for the false alarmprobability over both Nakagami-m and Rician fading channels.This false alarm probability relies upon a new threshold whichtakes into account an outage constraint on the probability ofdetection due to fading environments. The results show that thethroughput improves with an increase in the Nakagami-m fadingparameter and the Rician factor. In addition, the throughputimproves dramatically with an initial increase of the number ofmultipath and then levels out.

I. INTRODUCTION

Due to the increasing proliferation of wireless technologies

there is a growing shortage of frequency spectrum. One ap-

proach to alleviate this scarcity is the development of cognitive

radio [8], [11]. In a cognitive radio network there are two

operators. The first operator is a primary user who is defined as

the owner (or the licensee) of a particular part of the frequency

spectrum and has higher primacy rights to access this part of

the spectrum. The second operator is a secondary user (or

unlicensed user) who has lower rights on the usage of this

spectrum (i.e., it must not cause interference to the primary

user).

A recent study implemented by the US Federal communica-

tions commission (FCC) on spectrum utilisation showed that

a large part of the spectrum is not efficiently exploited [7]

and this observation supports the cognitive radio concept. But

in order to make cognitive radio function we need efficient

spectrum sensing. That entails identifying spectrum holes

(white spaces) or other frequency bands that are not being used

at certain times and so can be exploited by the secondary user.

In addition, the secondary user needs to vacate these frequency

bands as soon as the primary user starts its transmission.

Various techniques have been described in the literature as

regards spectrum sensing - for example, the energy detector

(ED), matched filter (MF) detection and cyclostationarity

detection [2], [9] and [13].

In IEEE 802.22 each medium access control (MAC) frame

has two subframes [10]. The first subframe is for spectrum

sensing and the second subframe is for data transmission, and

the spectrum sensing is performed periodically. In [10] the

tradeoff between the spectrum sensing and the throughput has

been studied and an optimum sensing time has been found

that maximizes the throughput of the secondary user given

that the primary user is not significantly affected. However, in

[10], the wireless channel is assumed to be known, and this

is not realistic. In a cognitive network, for example, in IEEE

802.22, there are requirements in which the secondary network

pledges to respect the contract, e.g., the detection probability

PD should not be less than a target probability of detection,

δ. But there are several factors that prevent (PD ≥ δ) from

being achieved. For example, some channel realizations in a

low signal to noise ratio have PD < δ.

Some papers have recently been examining this situation.

For example, in [4] and [5] the sensing-throughput tradeoff has

been studied under outage constraints using the ED. However,

the ED severely degrades due to noise uncertainty [12], [14].

Consequently, in this paper we seek to employ a detector

that does not depend on the noise power. In the literature,

many papers have tried detectors that do not depend on, or are

insensitive to, noise power. Most of those detectors are based

on OFDM signals. For example in [1], [3] and [6] the authors

have proposed detectors that can exploit the autocorrelation

property (due to the existence of a cyclic prefix) in order to

detect the presence of a primary user. Therefore, we will use

the autocorrelation detector (AD) in [6] (and not the AD of [1],

which does not have closed-form expressions for probabilities

of both detection and false alarm) to evaluate the sensing-

throughput tradeoff under outage constraints over Nakagami-

m and Rician fading channels. To the best of authors’ knowl-

edge, the sensing-throughput tradoff using an AD under outage

constraints has not been examined by previous papers. In this

paper, we show that the throughput improves with an increase

in the Nakagami-m fading parameter (m), the Rician factor

(K) and the number of multipaths (L).

The remainder of the paper is organized as follows. In

Section II, we introduce the system model. Section III presents

spectrum sensing using an AD. In Section IV the outage con-

straint on the probability of detection in fading environments

is presented. In Section V the sensing-throughput problem

is examined. In Section VI, we present simulation results.

Finally, we give our conclusions in Section VII.

II. SYSTEM MODEL

A. Primary User Signal

Let Sm = [Sm(0) Sm(1) Sm(2) ... Sm(Nd−1)] represents

the Nd complex PSK/QAM symbols of the m-th OFDM

978-1-4673-0762-8/12/$31.00 ©2012 IEEE 66

symbol. After the IFFT, the OFDM symbol is described by

the following Nd complex values:

sm(n) =1√Nd

Nd−1∑

k=0

Sm(k)ej2πnk

Nd , n = 0, ..., Nd − 1.

where n and k are respectively discrete time and frequency

indexes. Adding the last Nc elements of sm(n) as a cyclic pre-

fix we get the m-th cyclic-prefixed OFDM symbol [sm(Nd −Nc) ... sm(Nd − 1) sm(0) ... sm(Nd − 1)]. An OFDM frame

consists of several OFDM symbols which are transmitted

sequentially. For notation simplicity, we will denote each

element of the transmitted OFDM frame s(n). For a large IFFT

size, then by the central limit theorem, s(n) ∼ CN(0, σ2s) [6].

B. Channel Model

The frequency-selective channel between the primary user

and the secondary user is modeled as an FIR filter with impulse

response h = [h0 h1 h2 ... hL−1]T whose elements are i.i.d.

In this work, we assume the channel has an exponential power

delay profile. Also, in this model the power of the channel taps

is normalised such that∑L−1

l=0 E|hl|2 = 1.

C. Received Signal

The purpose of spectrum sensing is to inform the secondary

user about the existence of the primary user. So our goal is

to discriminate between two hypotheses, namely: H0 when

the primary user is absent and H1 when the primary user is

present. Thus

H0 : x(n) = w(n)

H1 : x(n) =

L−1∑

l=0

hls(n − l) + w(n) (1)

where n = 0, 1, 2, ..., M -1; x(n) is the signal received by

the secondary user and w(n) is i.i.d. circularly symmetric

complex Gaussian noise, CN (0, σ2w). Finally we define the

instantaneous signal to noise ratio as γ = σ2s

∑L−1l=0 |hl|2

/σ2

w.

In this work, we assume that the secondary user is time

synchronized with the primary user, and also that both Nc

and Nd are known.

III. SPECTRUM SENSING

Our proposed detector follows the approach of [6] which

exploits the property of OFDM signals (provided by the cyclic

prefix (CP)) such that the autocorrelation coefficients are non-

zero at lags ±Nd and they are also the log-likelihood ratio

test (LLRT) statistic for a low signal to noise ratio (SNR). So

our proposed test statistic is [6]

T =1M

∑M−1n=0 Rx(n)x∗(n + Nd)1

2(M+Nd)

∑M+Nd−1n=0 |x(n)|2

H1

RH0

λ (2)

where M (M >> Nd) is the number of samples used in

autocorrelation estimation, R. denotes the real part of a

complex number and λ is a threshold value used to determine

whether the primary user is present (T ≥ λ) or not (T < λ).

The distribution of the test statistic in (2) can be approximated

(for sufficiently large M ) as [6]

H0 : T ∼ N(0,

1

2M

)

H1 : T ∼ N(

ρ,(1 − ρ2)2

2M

)(3)

where ρ = (Nc

/(Nd+Nc)) σ2

s

∑L−1l=0 |hl|2/(σ2

s

∑L−1l=0 |hl|2+

σ2w). The probabilities of false alarm PFA and detection PD,

conditioned on the channel, are given by:

PFA = P (T > λ∣∣H0) =

1

2erfc

(√Mλ

)(4)

PD = P (T > λ∣∣H1) =

1

2erfc

(√M

λ − ρ

1 − ρ2

)(5)

where erfc(.) is the complementary error function.

IV. OUTAGE PROBABILITY ANALYSIS

For some realisations of the wireless channel PD ≥ δ may

not satisfied (where δ is the target probability of detection).

So we introduce a bound on the the outage probability (Pout),

defined as:

Pout = ProbPD < δ ≤ α (6)

where α is a constant specified by the primary network [5].

We will now derive Pout for different channels.

A. Nakagami-m Fading Channel

First we consider that the amplitude for each channel tap

coefficient, |hl|, is modeled as a Nakagami-m random variable

with probability density function (p.d.f.)

f|hl|(t) =2

Γ(m)

(m

Ωl

)m

t2m−1exp(−mt2/Ωl) (7)

where Ωl = E[|hl|2] is a controlling spread parameter for the

l-th tap and m is the Nakagami-m fading parameter for the

l-tap. Note that when m = 1, (7) is is a Rayleigh distribution.

The instantaneous SNR for a frequency-selective channel is

γ = σ2s

∑L−1l=0 |hl|2

/σ2

w. By substituting (5) into (6) then the

outage probability becomes

Pout = Prob

(1

2erfc

(√M

λ − ρ

1 − ρ2

)≤ δ

). (8)

If we re-write ρ in terms of γ, ρ = κγγ+1 and where κ =

(Nc

/(Nd + Nc

), then we get

Pout = Prob

(1

2erfc

(√M

λ − κγγ+1

1 −(

κγγ+1

)2

)≤ δ

). (9)

Equation (9) can be simplified to

Pout = Prob(T∆ ≤ 0) (10)

where

T∆ = α1γ2 + α2γ + α3

67

and

α1 =−κ2

√M

erfcinv(2δ) − λ +

(κ +

1√M

erfcinv(2δ)

)

α2 = − 2λ +(κ +

2√M

erfcinv(2δ))

α3 = − λ +1√M

erfcinv(2δ),

where erfcinv is the inverse complementary error function.

Now T∆ = α1(γ − γ1)(γ − γ2), γ2 > γ1, and it is possible to

show that γ1 ≤ 0 and γ2 > 0. Now since γ ≥ 0, then we can

say

Pout = Prob(γ ≤ γ2) (11)

where

γ2 =(1 +

2β

κ+√

4νβ + 1)/(

2(κν − β

κ− 1)

)(12)

with ν = 1√M

erfcinv(2δ) and β = ν − λ. Re-writing (11) in

terms of the channel coefficients

Pout = Prob(L−1∑

l=0

|hl|2 ≤ σ2wγ2

σ2s

). (13)

The p.d.f. of TNak =∑L−1

l=0 |hl|2 can be approximated by a

Gamma distribution, fTNak(t), with a shape parameter kNak

and a scale parameter θNak:

fTNak(t) =

1

Γ(kNak)θkNak

Nak

exp(−t/θNak)tkNak−1, t > 0

(14)

where kNak = µ2TNak

/σ2TNak

and θNak = σ2TNak

/µTNak. Note

that Γ(.) a Gamma function and µTNakand σ2

TNakare respec-

tively the mean and the variance of TNak: µTNak= E[TNak] =∑L−1

l=0 Ωl and σ2TNak

= E[T 2Nak] − E

2[TNak] =∑L−1

l=0 Ω2l /m

where E[|hl|4] = Ω2l [1 + 1/m]. Fig. 1 illustrates the plots

of the theoretical p.d.f., given in (14), and the Monte-Carlo

simulation of the p.d.f. of TNak. It is clear that (14) is an

excellent approximation to the p.d.f. of TNak. Thus the outage

probability can be written as

Pout−Nak = Gamcdf(σ2

wγ2

σ2s

, kNak, θNak

)(15)

where Gamcdf(., ., .) is the cumulative Gamma distribution

for fT (t) in (14). To ensure Pout−Nak = α, then

γ2 =σ2

s

σ2w

Gaminv(α, kNak, θNak), (16)

where Gaminv(., ., .) denotes the inverse function of the

Gamma cumulative distribution. Substituting (16) into (12),

and after some mathematics, then the threshold, λ(α, δ), that

meets the outage probability, α, and the target probability of

detection, δ, is:

λNak(α, δ) =ν

(1 − 1

2β21

)+

β2

β1

[1

−√

ν2

4β21β2

2

+1

4β22

− ν

β1β2

](17)

0 0.5 1 1.5 2 2.5 30

0.01

0.02

0.03

0.04

0.05

0.06

t

Pro

babilit

yden

sity

funct

ion

(p.d

.f.)

Approximate (Nakagami-m)Simulation (Nakagami-m)Approximate (Rician)Simulation (Rician)

Figure 1. Plots of the approximate p.d.f.’s(fTNak

(t) and fTRic(t))

andthe simulated p.d.f. of TNak and TRic: (i) Nakagami-m fading channel L=7and m=3, (ii) Rician fading with L=7 and K=3. We used 106 Monte-Carloruns.

where β1 = −(γ2 + 1)/κ and β2 = κνγ2 − (γ2 + 0.5). The

PFA that satisfies the α and the δ, can be found by substituting

(17) into (4), and is

PFA−Nak = P (T > λ/H0) =1

2erfc(√

M λNak(α, δ)).

(18)

B. Rician Fading Channel

Here we consider that the amplitude for each channel tap

coefficient, |hl|, is modeled as a Rician random variable with

p.d.f.

f|hl|(t) =t

σ2l

exp

(− t2 + a2

l

2σ2l

)I0

(alt

σ2l

)(19)

where I0(.) is the modified zeroth-order Bessel function. Note

that hl can be expressed as

hl = al + xl + jyl

where al and xl +jyl represent respectively the dominant path

and the other scattered paths; and xl and yl are N (0, σ2l ).

The Rician fading factor of the l-th tap is defined as Kl =

|al|2/

2σ2l . Note that when Kl = 0 then (19) is a special case

of a Nakagami-m distribution with m = 1. The outage prob-

ability is similar to (13) and the p.d.f. of TRic =∑L−1

l=0 |hl|2can be approximated by a Gamma distribution, fTRic

(t), with

a shape parameter kRic and a scale parameter θRic. But in this

case,

µTRic= E[TRic] =

L−1∑

l=0

a2l + 2

L−1∑

l=0

σ2l (20)

and

σ2TRic

= E[T 2Ric] − E

2[TRic] = 4L−1∑

l=0

σ4l + 4

L−1∑

l=0

a2l σ

2l . (21)

We can see from Fig. 1 that the plots of the theoretical p.d.f.

(fTRic(t)) and the Monte-Carlo simulation of the p.d.f. of TRic

68

0 200 400 600 800 1000 1200 1400 1600 18000.45

0.5

0.55

0.6

0.65

0.7

0.75

(a)

(b)

(c)

(d)

Listening time, M

Ach

ieva

ble

Thro

ugh

put

(bits/

sec/

Hz)

(a) Nakagami-m (α = 0.15)(b) Rician (α = 0.15)(c) Nakagami-m (α = 0.25)(d) Rician (α = 0.25)

Figure 2. The achievable throughput, C , versus listening time, M , overNakagami-m and Rician fading channels for different values of outageprobability constant (α). In all cases, L = 2, m = 1 and K = 1.

are identical. The outage probability is

Pout−Ric = Gamcdf(σ2

wγ2

σ2s

, kRic, θRic

)(22)

and to achieve Pout−Ric = α, then

γ2 =σ2

s

σ2w

Gaminv(α, kRic, θRic). (23)

The threshold, λRic(α, δ), and the PFA−Ric that meet the

outage probability, α, and the target probability of detection,

δ, are similar to (17) and (18) but now using γ2 in (23).

V. SECONDARY USER’S THROUGHPUT

The achievable throughput, C, for a secondary user, as

defined in [10], can be written as follows:

C(M) =(1 − M/Nf )(1 − PFA)C0P0

+ (1 − M/Nf )(1 − PD)C1P1 (24)

where Nf is the total length of the secondary user frame, P0

and P1 are the probabilities that the primary user is absent

and present respectively, and C0 and C1 are the throughputs

of the secondary network when it operates in the absence

and the presence of a primary user respectively. Here C0 =log2(1 + SNRs) and C1 = log2(1 + SNRs

SNRp). SNRs and SNRp

are respectively the signal to noise ratio of the secondary

user’s transmitter and the primary user’s transmitter at the

secondary network’s receiver. Notice that C0 > C1, and thus

the second term in (24) can be neglected [10]. The objective

of sensing-throughput tradeoff under outage constraints is to

identify the optimal sensing duration M for each frame such

that the achievable throughput of the secondary network is

maximized and the outage probability, α, and also δ, are met

. Thus the optimization problem is given by [10]

maxM

C(M) = (1 − M/Nf)(1 − PFA)C0P0.

VI. SIMULATION AND DISCUSSION

The parameters that have been used in the simulation are

as follows: SNRs= 20dB, SNRp =10log10

∑L−1

l=0E|hl|2σ2

s

σ2w

=−10dB, parameter δ is chosen to be 0.9, α = 0.15,

P0 = 0.7, Nc= 8, Nd = 32, and Nf =100(Nc + Nd)

samples. For the Rician channel, we assume that K1 =K2 = ... = KL = K . For the frequency-selective case, the

channels taps h = [h0 h1 ... hL−1]T are generated according

to (7) and (19) with an exponential power delay profile

E|hl|2 = E1exp(−0.2l) where E1 a parameter to guarantee∑L−1l=0 E|hl|2 = 1. The plots for simulation and theory

are virtually identical and so only those for the theory are

presented.

Result 1: Nakagami-m and Rician fading channel with

different values of α (Figure 2).

We can clearly see from this plot that as the outage proba-

bility increases the achievable throughput increases for both

Nakagami and Rician fading channels (theory - see (16), (17),

(18), (23) and (24)). Also we can see that the Rician channel

gives better throughput because of the dominant path element.

Result 2: Nakagami-m fading channel with different

values of m (Figure 3).

This figure shows the achievable throughput versus the listen-

ing time under an outage constraint (theory - see (16), (17),

(18) and (24)). We can see from the figure that the achievable

throughput degrades as m decreases and the worst case is

when m = 1 (Rayleigh fading channel) as expected.

Result 3: Rician fading channel with different values of

K (Figure 4).

It is easily seen that the achievable throughput increases with

K (theory - see (17), (18), (23) and (24)). Also, we can see

from Fig. 3 for m = 1 and from Fig. 3 for K = 0 that the

achievable throughputs are both identical.

Result 4: Nakagami-m fading channel with different

values of L (Figure 5).

Here we see that by increasing the number of multipaths (L)

we initially get a considerable improvement in the throughput

(L = 1, 3, 5) and then this gain levels off for large values of

L (L = 23, 25).

Result 5: Rician fading channel with different values of

L (Figure 6).

Lastly, as in the Nakagami fading channel, the achievable

throughput improves initially and then this improvement levels

off with increasing L.

VII. CONCLUSION

The sensing-throughput under outage constraints has been

evaluated in fading environments. The threshold expression for

a certain value of outage probability was obtained theoretically

over Nakagami-m and Rician fading channels. The results

have shown that the throughput improves with increasing both

the Nakagami-m fading parameter and the Rician factor under

outage constraints. Finally, the throughput improves as the

number of multipaths increases for both Nakagami-m and

Rician fading channels.

69

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0.55

0.6

0.65

0.7

0.75

0.8

Listening time, M

Ach

ieva

ble

Thro

ugh

put

(bits/

sec/

Hz)

m = 1m = 2m = 3

Figure 3. The achievable throughput, C , versus listening time, M , overNakagami-m fading channels for different values of m. In all cases, L = 2and α = 0.15.

0 200 400 600 800 1000 1200 1400 1600 18000.45

0.5

0.55

0.6

0.65

0.7

0.75

Listening time, M

Ach

ieva

ble

Thro

ugh

put

(bits/

sec/

Hz)

K = 0K = 2K = 3

Figure 4. The achievable throughput, C , versus listening time, M , overRician fading channels for different values of K . In all cases, L = 2 andα = 0.15.

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