1

Spectrum Monitoring for Cognitive Radios inRayleigh Fading Channel

Erfan Soltanmohammadi, Mahdi Orooji, Mort Naraghi-Pour

Department of Electrical and Computer Engineering

Louisiana State University, Baton Rouge, LA 70803

{e-mail: esolta1, morooj1, naraghi@lsu.edu}.

Abstract—In-band spectrum sensing requires that the sec-ondary users (SU) periodically suspend their communicationin order to determine whether the primary user (PU) hasstarted to utilize the channel or not. In contrast, in spectrummonitoring the SU can detect the emergence of the PU fromits own receiver statistics such as receiver error count (REC).Previously it is shown that in AWGN channels, a hybrid spec-trum sensing/spectrum monitoring system significantly improveschannel utilization of the SUs and detection delay for the PUs.In this paper we investigate the problem of spectrum monitoringin the presence of fading where the SU employs diversitycombining to mitigate the channel fading effects. We show thata decision statistic based on the REC alone does not providea good performance. Next we introduce new decision statisticsbased on the REC and the combiner coefficients. Simulationresults are presented that show significant improvement in systemperformance.

Index Terms—Spectrum sensing, spectrum monitoring, chan-nel utilization, detection delay, fading channel.

I. INTRODUCTION

Dynamic spectrum access (DSA) allows the secondary

(unlicensed) users (SU) to utilize the licensed spectral bands

that are not in use by the incumbent primary users (PU).

Cognitive radio (CR), viewed as the enabling technology for

DSA, utilizes spectrum sensing (SS) to determine whether a

given frequency band is vacant of the PU signal [1]–[3]. Since

during their own communication the SUs do not sense the

channel, SS requires that the SU periodically suspend their

transmission in order to detect whether the PU has emerged

in the band.

In order to protect the PU against undue interference

from the SUs, stringent requirements are imposed on the

performance measures of SS such as detection probability and

maximum detection delay (see for example [4]). Detection

probability can be improved by increasing the duration of

the sensing periods and detection delay can be reduced by

decreasing the duration of the SU’s transmission periods. Both

approaches, however, result in reduced channel utilization in

the secondary network.

There is an intricate tradeoff between protection of the PU

and the quality of service (QoS) of the SU, referred to as

sensing-throughput tradeoff [5]. In [6], Tang et al. evaluate

the effect of PU traffic on the SU throughput. In [7], Akin etal. assume statistical QoS and maximize the throughput for the

SU. Spectrum monitoring using receiver statistics to detect the

emergence of the PU during the SU’s reception is introduced

by Boyd et al. and evaluated in [8] and [9]. In [10] we propose

a decision statistic based on the receiver error count (REC) and

the decision of a CRC (cyclic redundancy check) code and

show that the proposed algorithm will significantly increase

the throughput of the SU subject to a maximum tolerable PU

detection delay.

Spectrum monitoring (SM) using receiver statistics is an

effective approach provided that the changes in the receiver

statistics are mainly due to the emergence of the PU. for

example in the case of AWGN channels. This approach,

however, may not be suitable in the presence of fading as

the changes in the receiver statistics may be due to fading

rather than interference from the PU signal. In this paper we

investigate the problem of SM in Rayleigh fading channels.

We assume that the SU uses diversity combining in order to

mitigate the channel fading effects. We first show that SM

based on receiver error count alone does not perform well.

Next we introduce a new decision statistic which employs the

REC, the decision of CRC, and the combiner statistics. We

evaluate the performance of this new decision statistic in terms

of detection and false alarm probabilities, channel utilization

and detection delay.

The rest of this paper is organized as follows. The system

model and problem formulation are presented in Section II.

The decision statistic using REC and combiner statistics is

introduced and analyzed in Section III. Sections IV and V

contain the numerical results and the conclusions, respectively.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The SU starts with a spectrum sensing interval (SSI) of

duration Ts, in which the SU senses the channel. If at the end

of an SSI the channel is found to be occupied, another SSI

begins1 and this continues until the SU finds the channel to

be vacant. At this time a spectrum monitoring interval (SMI)

begins during which the SU transmits a maximum of KM

packets. After the reception of each packet the SU computes

a decision statistic (described below) in order to detect whether

the PU has emerged or not. If it is decided that the PU

has emerged, the SU terminates the SMI and enters the the

spectrum sensing phase. Otherwise the channel is deemed to

be vacant and the SU continues its packet transmission. To

allow for periodic sensing of the channel the SU terminates

1We should point out that the results presented here will not change if theSU moves to another channel once it finds the current channel to be occupied.

978-1-4673-3/12/$31.00 ©2013 IEEE978-1-4673-3/12/$31.00 ©2013 IEEE

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an SMI after the transmission of (at most) KM packets.

We assume that SU packet length is N bits and the packet

transmission time is denoted by Tp.

In today’s communication systems forward error correction

(FEC) is widely used to combat channel errors. We assume

that a CRC code is also used to determine whether the

packet is correctly decoded or not. At the SU receiver, the

received packet is demodulated and decoded. The decoded

packet is validated by CRC and encoded using a replica of

the transmitter’s encoder. The encoder’s output is compared

with the output of the demodulator to calculate REC, which is

denoted by e. Note that the actual number of errors in a packet,

denoted by k, is not necessarily available in the receiver. In

particular, it is only available if the packet is decoded correctly.

In this case, k = e. Let Hη denote the hypothesis of interest

where η = 0 and 1 correspond to the absence and the presence

of the PU signal, respectively. Also denote by Cc and Cnc the

events that the CRC is checked (i.e., CRC decides that packet

is correctly decoded), and CRC is not checked (CRC decides

that the packet is not correctly decoded), respectively. The

decision statistic to be used during the SMI is defined by,

T (DS) =

{ ({e ≥ μ(DS)} ∩ Cv

) ∪ Cnv, Decide H1

Otherwise, Decide H0(1)

where μ(DS) is the REC threshold which is assumed to be no

greater than, t(FEC), the maximum number of errors in a packet

that FEC is capable of correcting. The decision statistic in (1)

exclaim the appearance of the PU providing that the CRC

is not validated, or it is validated and REC is greater than

a threshold. As (1) indicates, the decision statistic in not a

function of the actual number of errors.

The most commonly used CRC in technical standards

are 16- and 32-bits CRCs, for example CCITT-16, CRC-

32-Castagnoli and CRC-32-IEEE [11]–[13]. It is shown in

[14] that the probability of failure for an χ−bit CRC is

approximately 2−χ for large packets (e.g. N > 100). For 16-

and 32- bits CRC, this probability is around 1.5 × 10−5 and

2.3 × 10−10, respectively. In view of this, we disregard the

event of a CRC failure.

If the packet is decoded correctly, then the CRC will

correctly identify this event (Cv) and in this case e = k. On

the other hand, if the decoder fails, then either the CRC will

identify this event (Cnv) or the CRC fails to identify the decoder

failure (Cv). In the latter case (which has a probability ≤ 2−χ

for an χ-bit CRC) the proposed SM scheme may fail for the

current packet. Consequently the SMI may be terminated when

PU is not present (resulting in loss of SU throughput) or it

may be continued when PU is present (resulting in increased

detection delay). It can be verified that the discrepancy (due

to disregarding the CRC failure) between our computation and

the actual value of PU’s detection delay is less than 2−χ×Tpseconds and for the SU channel utilization it is less than

2−χ × Ts

Ts+KM×Tp< 2−χ.

If Cnv occurs, then the received packet is not correctly

decoded. So, Cnv implies k ≥ t(FEC) ≥ μ(DS). On the other

hand, ignoring the event of CRC failure, Cv implies that the

packet is correctly decoded. So, for (1) we have

p(({e ≥ μ(DS)} ∩ Cv

) ∪ Cnv

)(2)

= p(({k ≥ μ(DS)} ∩ Cv

) ∪ ({k ≥ μ(DS)} ∩ Cnv

))= p({k ≥ μ(DS)})

Therefore, (1) is equivalent to,

T (DS) ≡ kH1

≷H0

μ(DS). (3)

To combat the effects of fading, we assume that the SU

receiver is equipped with L ≥ 1 identical antenna branches

and that the L branches experience identically distributed,

uncorrelated flat fading. The nth received symbol at the lthbranch of the SU is given by

rl,n = snhl + vl,n + ηul,n, l = 1, 2, · · · , L, (4)

where {sn} is the sequence of SU’s transmitted symbols,

{vl,n}Ll=1 denote L independent, identically distributed (i.i.d.)

circularly symmetric Gaussian noise processes with zero mean

and variance Ev, and for k �= l, {vk,n} and {vl,n} are

independent, and {ul,n} denotes the sequence of primary user

symbols at the lth branch of the SU receiver. We assume that

the PU symbols {ul,n} have undergone independent flat fading

which is not explicitly shown but is included in the symbols

{ul,n}. Finally, {hl}Ll=1, which denote the channel fading

coefficients, are i.i.d. circularly symmetric Gaussian random

variables with mean zero and variance 1, i.e., hl ∼ CN (0, 1).Let αl = |hl| and let θl = ∠hl.

It is well known that Maximal Ratio Combining (MRC) is

the optimum diversity technique in the sense of maximizing

the output SNR of the combiner [15]. With MRC the output

of the combiner is given by,

rn �L∑

l=1

wlrl,n (5)

where wl = h∗l , l = 1, 2, · · · , L are the combiner coefficients.

Due to space limitation in this paper we consider binary signal-

ing. The case of higher order modulation schemes requires a

more detailed discussion and is treated elsewhere. With block

fading, where the fading coefficient remains unchanged during

a packet transmission time2 all the bits in a packet have the

same error probability.

If the channel coefficients are perfectly estimated then the

probability of bit error is only a function of the signal-to-noise

ratio (SNR). It is shown in [16] that, if the modulation scheme

of the PU is a constant modulus scheme such as PSK, then

after undergoing Rayleigh fading, the PU sequence’s {ul,n},for l = 1, 2, · · · , L are i.i.d. zero-mean circularly symmetric

Gaussian random variables with variance Eu. This model is

also accurate if the PU uses orthogonal frequency division

multiplexing (OFDM) [17]. For other modulation schemes

with a large constellation this assumption is approximately true

[16]. Furthermore, with independent fading on each branch,

2Although not shown here, for systems with transmission rates of severalMbps and mobile velocities below 70 Km per hour, the correlation betweenchannel coefficients corresponding to the first and last bits of a packet willremain above 98%.

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the sequences on different branches are independent [18]. It

follows that the received symbols in (4) are Gaussian [16].

The SNR for branch l under Hη is given by,

γ(l)η � |hl|2EbEv + ηEu (6)

where Eb is energy per bit of the SU transmitted signal. Note

that for a given packet this SNR is fixed.

Let γη, η = 0, 1 denote the SNR at the output of the com-

biner and let pγη (x) denote its probability density function.

The probability of observing k errors in a packet of length Ncan now be written as

pe(k|Hη) =

∫ ∞

0

(N

k

)pb(x)

k(1− pb(x))N−kpγη(x)dx (7)

where pb(γη) is the bit error probability for the SNR of γη.

For MRC, pγη (x) is given by [18],

pγη(x) =1

(L− 1)!γLb,ηx(L−1)e

−xγb,η (8)

From (3), the probability of false alarm and the probability

of detection in SMI are given by pf = p({k ≥ μ(DS)} | H0)and pd = p({k ≥ μ(DS)} | H1), respectively, and can be com-

puted from (7) and (8). The receiver operating characteristic

(ROC) curves (pd vs. pf ) from the above analysis as well as

from simulation are shown in Section IV.

A. Channel Estimation

The probability in (7) (and consequently, pf and pd) is

derived assuming that the combiner coefficient, wl, is de-

rived from precise knowledge of the channel coefficient hl.However, in practice the channel coefficients have to be

estimated and there is always an error between the estimated

channel coefficients and their actual values. In general, channel

estimation error is caused by two distinct channel impairments

[19]. One is due to the decorrelation of the pilots from the

signal due to distinct distortions that the channel imparts on

them because of their separation in time or frequency. The

second is due to noise. It can be seen that the first phenomenon

affects the channel estimation in the same manner whether the

PU is present (H1) or not (H0). The estimation error due to

noise, however, will be different as the SU experiences more

noise when PU is present.

Denote by hl = αlejθl the estimated channel coefficient

corresponding to hl. As in [19], [20] we assume the channel

estimation error �l � hl−hl to be independent of the channel

coefficient hl and circularly symmetric Gaussian. Given the

hypothesis Hη, the complex correlation coefficient �η between

hl and hl and its magnitude denoted ρη are given by [20]

�η � E[hlh∗l |Hη]√

E[|hl|2]E[|hl|2Hη]= �Rη + j�Iη (9)

ρ2η � |�η|2 = (�Rη )2 + (�Iη)

2, η = 0, 1.

where here and subsequently, superscripts R and I represent

the real and imaginary parts, respectively. From the assump-

tions on {hl}Ll=1, we conclude that {�l}Ll=1 are i.i.d. and

that var(�l|Hη) = var(h)(1 − ρ2η). Moreover, the estimated

channel coefficients {hl}Ll=1 are also i.i.d. circularly symmetric

Gaussian random variables and conditioned on Hη,

hl|Hη ∼ CN (0, 2− ρ2η) for l = 1, 2, · · · , L, (10)

where X |Λ ∼ CN (m,σ2) denotes the conditional distribution

of X given Λ.

In the case of imperfect channel estimation, it is shown in

[20] that the probability of observing a bit in error is identical

to the case of perfect channel estimation with effective SNR,

γ(eff)η � (�Rη )

2γη/(1 + γη(1− ρ2η)). Consequently, when com-

biner coefficients are not perfectly estimated, the performance

of the SM using TDS will be equivalent to a system with a

lower SNR.

III. THE DECISION STATISTICS USING ERROR COUNTS

AND COMBINER STATISTICS

As discussed previously, TDS cannot determine whether an

increase in the number of errors in a packet is a result of

fading or the interference from the PU. So an alternative

decision statistic is needed. To this end we would like to

augment the REC with the channel state information (CSI)

that is available in the SU receiver. In particular the combiner

coefficients can be used to enhance the decision statistic. To

emphasize the fact that the combiner coefficients are obtained

from an estimate of the CSI (rather than the exact values), in

the following the combiner coefficients are denoted by wl and

w = (w1, w2, · · · , wL). We define a new test statistic using

decoder and combiner statistics (DCS) as follows.

TDCS � (11)⎧⎨⎩

({p(e, f(w) | H1)

p(e, f(w) | H0)≥ μ

}∩ Cv

)∪ Cnv, Decide H1

Otherwise, Decide H0

where f(w) is a function of the combiner coefficients to be

determined. Fig. 1 demonstrates the proposed system model.

In the following we study the choice of the function f(.) for

MRC. Similar to the approach from (1) to (3), one can show

that

TDCS ≡ p(k, f(w) | H1)

p(k, f(w) | H0)

H1

≷H0

μ . (12)

Fig. 1. Proposed model using demodulator and combiner statistics.

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In the case of imperfect channel estimation, the combiner

coefficients are given by wl = h∗l . To evaluate the performance

of the decision statistic in (12) we first find the joint probability

of observing k errors and an estimated channel fading vector

h � (h1, h2, · · · , hL) given Hη ,

p(k, h|Hη) = p(k|h, Hη)p(h|Hη) (13)

From (10) and the fact that hl’s are i.i.d., we get

p(h|Hη) =1[

2π(1− ρ2η/2)]L exp

(−

∑Ll=1 |hl|2

2(1− ρ2η/2)

)(14)

Letting

ψ �Re

(∑Ll=1 hlh

∗l

)√∑L

l=1 |hl|2, (15)

p(k|h, Hη) =

∞∫−∞

p(k|ψ, h,Hη)p(ψ|h, Hη)dψ (16)

=

∞∫−∞

(N

k

)[P (E|ψ, h,Hη)]

k[1− P (E|ψ, h,Hη)]N−k

× p(ψ|hη, Hη) dψ

where P (E|ψ, h,Hη) is the bit error probability given ψ, hand Hη and is given by, [20]

P (E|ψ, h,Hη) = Q(ψ√2γη

)(17)

Considering the distribution of hl|Hη from (10) and after some

manipulations one can verify that,

ψ|h, Hη ∼ N⎛⎝

√∑Ll=1 |hl|2

2− ρ2η,

1− ρ2η2(2− ρ2η)

⎞⎠ (18)

By substituting (17) and (18) into (16), we get,

p(k|h, Hη) =

∞∫−∞

(N

k

)Qk(ψ

√2γη)(1−Q(ψ

√2γη))

N−k

×√

2− ρ2ηπ(1 − ρ2η)

exp

(− (A − (2 − ρ2η)ψ)2

(1− ρ2η)(2− ρ2η)

)dψ (19)

where A �√∑L

l=1 |hl|2.

From (14) and (19), it is evident that p(k, h|Hη) depends

only on A and not the values of individual hl’s. All combi-

nations of the estimated channel coefficients h1, h2, · · · , hLwhich result in the same value for A are observed with equal

probability at the SU. Consequently, in the case of MRC,

instead of (k, h) it is sufficient to use the pair (k, A) in the

decision statistic.

Thus we let f(w) � A and define our decision statistic by

TDCS(k, f(w)) � p(k, A|H1)

p(k, A|H0)(20)

Using TDCS(k, f(w)) in (1) instead of T(DS)(k), the 2D

space of k ∈ N, 0 ≤ k ≤ N , and A ∈ R+ is split into

two decision regions, Ω0 and Ω1 associated with H0 and

H1, respectively. Fig. 2 demonstrates three examples of these

decision regions when γ0 = 6 dB, γ1 = 0 dB, ρ0 = 0.95,

ρ1 = 0.85, N = 256 and L = 2 antennas are employed

in the MRC combiner. Fig. 2-(a), (b), and (c), respectively,

show the decision regions for the false alarm probabilities of

pf = 0.05, 0.1, and 0.2 and the corresponding detection

probabilities, pd = 0.86, 0.91, and 0.96. Note that when

the SU experiences large fades (small A), even when a large

number of errors per packet are observed, they are expected

to be caused by fading. Therefore, as demonstrated in Fig. 2,

in this case even for some large values of k, a decision is

made in favor of H0. On the other hand when fading is small

(large A), the SU receiver expects to observe a few errors per

packet (caused by fading). As a result, in this case even for

small values of k a decision is made in favor of H1. This is

how the inclusion of A in the decision statistic improves the

performance of SM over fading channels. The probabilities of

false alarm, pf , and detection, pd, are given by,

pf �∑∫

(k,A)∈Ω1

p(k, A|H0) dA (21)

pd �∑∫

(k,A)∈Ω1

p(k, A|H1) dA (22)

Fig. 2. The decision regions for TDSC when L = 2, γ0 = 6 dB, γ1 = 0dB, ρ0 = 0.95, ρ1 = 0.85, and N = 256. (a): pf = 0.05, (b): pf = 0.1,and (c): pf = 0.2.

IV. NUMERICAL RESULTS

In this section we present numerical results from simulation

and analysis to asses the effectiveness of the proposed decision

statistic for SM in fading channels. Simulation results are

obtained by running at least 104 independent trials, and

trapezoidal numerical integration approach is used to calculate

the integrals. The SS method we have used is energy detector

with the probabilities of detection and false alarm equal to

pd and pf , respectively. As mentioned previously, the length

of the SS interval is identical to the length of a packet, and

the transmission rate is assumed to be 2 Mbps. Finally, Jakes’

model [21] is employed to simulate Rayleigh fading channels.

Fig. 3 shows the ROC curves obtained from TDS(k) and

TDCS(k, A). We observe the result of incorporating REC and

combiner coefficients in the decision statistic in comparison to

the case of REC alone. As expected, TDCS(k, A) significantly

outperforms TDS(k). For example for probability of false alarm

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Fig. 3. Comparison from simulation between TDCS(k, A) and TDS(k) forN = 1024, γ0 = 2 dB, γ1 = −2 dB, ρ0 = 0.95, ρ1 = 0.85, and L = 4.

pf = 0.1, the probability of detection for TDCS is .97 whereas

it is around .62 for TDS(k).Channel utilization and detection delay are two performance

measures used to evaluate the proficiency of the SM algo-

rithm. Channel utilization is the portion of time that (under

hypothesis H0) the SU communicates over the channel, and

detection delay is defined as the average time it takes to

detect the presence of the primary user after its emergence

in the channel. For AWGN channels closed form formulas

for channel utilization and detection delay is derived in [22]

by employing two Markov chain models. However, due to

the correlation in time of the channel coefficients, [18], the

Markov chain model is not applicable here. So, we evaluate

these performance measures from extensive simulations.

Figs. 4 and 5 demonstrate channel utilization and detection

delay versus the probability of detection, for N = 1024,

L = 2, γ0 = 2 dB, γ1 = −2 dB, ρ0 = 0.95, ρ1 = 0.85,

maximum Doppler frequency of fm = 90Hz and pf = 0.1 for

different values of KM . Channel utilization is a decreasing

function of pf owing to the fact that the portion of time that

the SU has a chance to access the channel decreases with

pf . Since probability of detection is a monotone increasing

function of probability of false alarm, channel utilization is

also a decreasing function of pd. Detection delay is also a

decreasing function of pd.

Channel utilization increases with the duration of the SMI

(KM ). This is due to the fact that, for a fixed SSI, as KM

increases, the fraction of time that the SU is able to transmit

also increases resulting in increased throughput for the SU.

However, increasingKM will also increase the detection delay.

The reason is that for equal probabilities of false alarm for

SM and SS, SS has a higher probability of detection. Since

increasing KM (for fixed SSI) reduces the fraction of time the

SU spends in SS, detection delay increases with KM .

Figs. 3, 4 and 5 also show that TDCS outperforms TDS, i.e.,

for fixed SNR and probability of false alarm it has higher

probability of detection and for fixed probability of detection

TDCS has higher channel utilization and less detection delay

than TDS. Although the graphs here are plotted vs. pd, it should

be noted that a comparison based on pd alone is unfair. This

is due to the fact that to achieve the same value of pd (for the

same value of false alarm probability pf ), the scheme using

TDS requires a much higher SNR than that using TDCS. In the

following, we only consider the decision statistic TDCS.

Fig. 4. Channel utilization versus the probability of detection for TDCS andTDS when N = 1024, L = 2, γ0 = 2 dB, γ1 = −2 dB, ρ0 = 0.95,ρ1 = 0.85, fm = 90Hz, pf = 0.1 and KM=5, 10, 25, and 50.

Fig. 5. Detection delay versus the probability of detection for TDCS and TDS

when N = 1024, L = 2, γ0 = 2 dB, γ1 = −2 dB, ρ0 = 0.95, ρ1 = 0.85,pf = 0.1 and fm = 90Hz and KM=10, 25, and 50.

In Fig. 6 we show the ROC curves of TDCS for different

number of branches L. For L = 1, there is no diversity, and

A = |h|. As L increases to 3, the performance improves.

However, for L = 5 the performance starts to degrade and for

L = 25 the ROC is close to the chance line, i.e., pf = pd.

This behavior is due to the fact that as L increases the SNR at

the output of the combiner improves and the REC is reduced.

For very large values of L the emergence of the PU does not

cause a significant change in the SNR or the REC. Therefore

it is difficult for the decision statistic to detect the emergence

of the PU for such cases.

Fig. 7 plots channel utilization versus detection delay of

TDCS for different values of KM and L when N = 256, γ0 = 4

6

Fig. 6. ROC for spectrum monitoring using TDCS when N = 1024, γ0 = 6dB, γ1 = 0 dB, ρ0 = 0.9 and ρ1 = 0.8 for L = 1, 2, 3, 5, 25.

dB, γ1 = −1 dB, ρ0 = 0.9 and ρ1 = 0.8. In particular, for

L = 1 and KM = 10 a channel utilization of 90% can be

achieved with a detection delay of 4 packet times. For L = 4and KM = 10, a channel utilization of 90% can be achieved

with a detection delay of only 2 packet times.

Fig. 7. Channel utilization versus detection delay for TDCS, N = 256, γ0 =4 dB, γ1 = −1 dB, ρ0 = 0.9, ρ1 = 0.8, KM = 5, 10 and L = 1, 4, 10.

V. CONCLUSION

In this paper we investigate the problem of spectrum

monitoring over Rayleigh fading channels. It is assumed

that the secondary user (SU) utilizes diversity combining to

mitigate the effects of fading. We introduce a decision statistic

based on the receiver error count, the decision of the cyclic

redundancy check and the combiner coefficients and evaluate

its performance. Numerical results show that the a hybrid

spectrum sensing/spectrum monitoring scheme achieves high

channel utilization for the secondary users and low detection

delay for the primary users even in the presence of Rayleigh

fading effects.

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