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Sensitivity of Spectrum Sensing Techniques to RF impairmentsJonathan Verlant-Chenet Julien Renard Jean-Michel Dricot Philippe De Doncker Francois Horlin

Universit e Libre de Bruxelles - OPERA Dpt., Avenue F.D. Roosevelt 50, B-1050 Brussels, Belgium jverlant@ulb.ac.be

Abstract Cognitive radios are devices capable of sensing a large rangeof frequencies in order to detect the presence of primary networks andreuse their bands when they are not occupied. Due to the large spectrumto be sensed and the high power signal dynamics, low-cost implementationsof the analog front-ends leads to imperfections. Two of them are studiedin this paper: IQ imbalance and sampling clock offset (SCO). Based on amathematical system model, we study analytically the impact of the twoimperfections on the sensing performance of the energy detector and of thecyclostationarity detector. We show that the IQ imbalance does not impactthe performance of the two detectors, and that the SCO only impacts sig-

nicantly the performance of the cyclostationarity detector.

I. INTRODUCTION

In the eld of wireless communications, each device uses acertain frequency band in order to transmit and receive infor-mation. Actors involved make a request to federal agencies tobe able to legally communicate over a limited band. This leadsto a shortage of available spectrum and slows down the devel-opment of other innovative applications. However, it has beennoticed that, in practice, the spectrum use is less than 10% [1],which leads directly to the interesting idea to reuse the spectrumby a secondary network when it is not requested by the primaryuser ([2] and [3]). Such devices are called cognitive radios be-cause they are aware of the spectrum availability that surroundsit. First, they must investigate a certain range of frequencies inorder to discover potential opportunities (that is, free frequencybands). Then, the devices communicate in the licensed bands,while avoiding interference with the primary networks.

A standard terminal consists of an analog front-end (for con-tinuous signal processing as frequency change, ltering and am-plication) and a digital processor (allowing demodulation andsynchronization algorithms). The design of the analog front-endis critical in the case of cognitive radios. Indeed, to analyze alarge number of frequencies, they have to work on a large spec-trum composed of several bands. In addition, they must be ableto detect signals from close and distant devices together. Cogni-tive radios must therefore cope with high power signal dynamics([4] and [5]). These two constraints lead to different imperfec-tions in the implementation of the analog front-end, such as theIQ imbalance caused by different physical analog paths on theI and Q branches if a direct conversion receiver is used and thesampling clock offset (SCO) caused by the non-ideal local os-cillators.

In this article, we will study the impact of IQ imbalance and

SCO on the sensing performance. In order to do this, we rstintroduce a new system model which includes the two imperfec-tions. Then, we summarize the sensing algorithms in the case of the energy detector and the cyclostationarity detector. An an-alytical study of the impact of each imperfection is performedand the results are supported with Matlab c simulations.

I I . S YSTEM MODEL

During the sensing phase, the cognitive radio has to decidebetween two hypotheses: H 0 : there is a communication signal in the bandwidth; H 1 : no signal is present in the bandwidth.

Fig. 1 illustrates the overall system when a linearly modulatedcommunication signal is present in the bandwidth ( H 1).

cos(t)

sin(t)

Re

Im

InQAM(Tsymb )

Transmitter

u(t)

nTs(1+)

nTs(1+) j

yR(t)

yI(t)

Sensing

(1+) cos(t+)

(1) cos(t)

yn

f(t)

f(t)

Receivery(t)

nRF(t)

cRF(t)

Fig. 1. Bandpass architecture with front-end imperfections.

At the transmitter, independent QAM modulated symbols I nof variance 2I are transmitted at the rate 1/T symb . The sequenceis low-pass ltered by u(t) in order to remove the out-of-bandcomponents (typically a halfroot Nyquist lter is used). Theresulting baseband signal, given by:

s(t) =n

I n u(t nT symb ), (1)

has a variance equal to 2s = 2I

|u(t)|2dt . The radio-

frequency signal sRF (t) is obtained after up-conversion to thecarrier pulsation (corresponding to the carrier frequency f )and transmitted through a frequency selective channel cRF (t).Additive white Gaussian noise n RF (t), of one-sided power spec-tral density (PSD) equal to N 0 , corrupts the received signal.

At the receiver, the RF signal is down-converted to the base-band domain for complex operation and low-pass ltered byf (t) in order to limit the observation to the desired bandwidthB f . The baseband signal y(t) can be expressed as:

y(t) = s(t) + n(t) (2)

where n(t) is the baseband equivalent noise of variance2n = 2 N 0B f . After sampling at the rate 1/T s (ofter a multiple

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of the symbol rate 1/T symb ), the obtained received sequence ynis used for the spectrum sensing.

When there is only noise in the system ( H 0 ), the baseband

signal y(t) is given by:y(t) = n(t) (3)

Because the implementation of the analog front-end is dif-cult, non-idealities are introduced during the RF-to-basebandconversion. First, the IQ imbalance is due to the difference of physical length between the I and Q branches of the receiver.It is represented by the errors and , respectively changingthe amplitude and the phase of the local oscillators. Second, theSCO is a difference between the transmitter and receiver symbolrate reference T symb . It is represented by the error changing thesampling frequency.

III. S ENSING ALGORITHMSThe two sensing algorithms considered in this paper rely on

time averages dened as:

x(t) = lim t

t2

t2x(t)dt (4)

where t is the observation window, nite in practice.The energy detector estimates the variance 2y of the received

signal y(t) given by the integral of the PSD over the frequencybandwidth. If the signal is ergodic, the variance can be estimatedbased on time averages [6]:

2y = y(t)y(t) (5)

The cyclostationarity detector estimates the spectral correla-tion density (SCD) dened as the Fourier transform of the cyclicautocorrelation function [7]. If the signal is cyclo-ergodic, theSCD at cycle frequency c (equal to the inverse of T symb in oursystem) can be evaluated based on time averages [7]:

S cy (f ) = limT 1T

Y T (t, f +c2

)Y T (t, f c2

) (6)

where Y T (t, f ) = t + T 2

t T 2y(u)e j 2ct du is the Fourier transform

of y(t) on an interval T centered on t . In practice, algorithmswill work with nite T (inverse of the frequency resolution f )and t (temporal window).

To compare the performance of the two detectors, we illus-trate the PSD (in case of the energy detector) and the SCD (incase of the cyclostationarity detector) of the received basebandsignal when the channel is ideal (frequency at). As we can seeon Fig. 2, the PSD is a stronger metric than the SCD explainingwhy the energy detector usually outperforms the cyclostationar-ity detector.

This is conrmed by evaluating the probability of misdetec-tion ( pMD ) and the probability of false alarm ( pFA ), dened as:

pMD = Prob {decision is H 0 |H 1} (7) pFA = Prob {decision is H 1 |H 0} (8)

Fig. 3 illustrates the probability of misdetection as a function of the signal-to-noise ratio (SNR) for a xed probability of falsealarm equal to 0.1. Clearly the energy detector outperforms thecyclostationarity detector because it collects more signal energy.However we have to keep in mind that it suffers from the noiseuncertainty that makes it impractical at low SNR [8].

Fig. 2. Power spectral density versus spectral correlation density.

Fig. 3. Probability of misdetection with respect to the SNR ( pFA = 0 .1 ).

IV. I MPACT OF IQ I MBALANCE ON PERFORMANCE

When there is IQ imbalance in the system, the baseband re-ceived signal y(t) is expressed as a linear combination of theideal signal y(t) and its conjugate y(t) as follows [9]:

y(t) = y (t) + y(t) (9)

with:

= cos ( ) j sin ( ) (10) = cos( ) + j sin( ) (11)

A. Impact on energy detector

The energy detector evaluates the signal variance:

2y = E [y(t)y(t)] (12)

= E [y (t) + y(t)][y (t) + y(t)] (13)

= | |2 + | |2 2y + 2 Re E [y(t)y(t)] (14)

where only the second term depends on the hypotheses. If thereis only the noise in the system, the expectation in the secondterm is equal to:

E [y(t)y(t)] = E [n(t)n(t)] (15)

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which is null because the real and imaginary parts of the noiseare uncorrelated. If there is only the signal in the system, theexpectation in the second term is equal to:

E [s(t)s(t)] =n n

E [I n I n ]u(t nT )u(t n T ) (16)

which is null because the real and imaginary parts of the sym-bols are uncorrelated.

Therefore, the estimated metric becomes:

2y = | |2 + | |2 2n if H 0 (17)

2y = | |2 + | |2 2s + 2n if H 1 (18)

and it is possible to derive the probability of misdetection andthe probability of false alarm.

Since the metric is affected by the same power gain | |2 + | |2 under the two hypotheses, the probability of false alarm and the probability of misdetection are unchangeddue to the IQ imbalance if the detector threshold is multipliedby the same power gain.

B. Impact on cyclostationarity detector

Since the cyclostationarity detector evaluates the SCD of thereceived signal, let us develop its expression in presence of IQimbalance to nd out how the detector performances will be af-fected. The expression (6) of the SCD becomes:

S cy (f ) = limT 1T

Y T t, f +c2

Y

T t, f c2

(19)

where:

Y T (t, f ) = t + T 2

t T 2y (u) e j 2f u du (20)

With the properties of linearity of the Fourier transform, we ob-tain Y T (t, f ) = Y T (t, f ) + Y T (t, f ) since we knowthat if g(t) has the Fourier transform G(f ), then g(t) has theFourier transform G( f ). The estimated SCD becomes:

S cy (f ) = | |2S cy (f ) + | |

2S cy ( f ) + 2 Re S cyy (f )

(21)

where:S cyy (f ) limT

1T

Y T t, f +c2

Y T t, f +c2

(22)

The rst two terms in the expression of S cy (f ) represent theSCD of the ideal signal y(t) and the mirrored SCD. The thirdterm depends on the hypotheses.

Let us rst assume that there is only noise in the system:y(t) = n(t). The rst two terms are the SCD of Gaussian noise,and are therefore null by denition [7]. The term S cyy (f ) repre-sents the spectral correlation between the points c2 f when f isvarying. The only value of f where this spectral correlation canbe different from zero is f = 0 (correlation between the same

two points in the spectrum). We get:

S cyy (0) = limT 1T

Y T t,c2

Y T t,c2

(23)

= limT

1T

Y RT t,c2

2 Y I T t,

c2

2(24)

where Y RT t,c2 and Y

I T t,

c2 are respectively the real and

imaginary parts of Y T t, c2In the last expression, we omitted the term:

2 j Y RT t, c2Y I T t, c2

(25)

because it disappears when we later take the real part of thewhole expression. Since the two terms have the same variance,they will compensate for each other, eventually canceling thewhole integral. As a conclusion, the three terms in (21) are nullwhen there is only noise in the system, so that the SCD S cy (f )is equal to zero.

Let us secondly assume that y(t) only contains the signal:y (t) = s (t). The rst two terms of (21) are different from zerofor certain values of c by denition [7]. In the third term, theonly values of f where the spectral correlation S cyy (f ) can bedifferent from zero are f = 0 (correlation between the sametwo points in the spectrum) and f = 12T symb (since two pointsseparated by a cyclic frequency c = 1T symb are correlated). Theterm vanishes for the case f = 0 for the same reasons as forthe noise. The term also vanishes for the case f = 12T symb be-cause it is the correlation between the frequencies 0 and 1/T symbwhere the signal is null. Thus, the third term is null and we get:S cy (f ) = | |

2 S cs (f )+ | |2 S cs ( f ) where S cs (f ) is the SCD

of the signal s(t).The resulting values of the metric are:

S cy (f ) = 0 if H 0 (26)

S cy (f ) = | |2 S cs (f ) + | |2 S cs ( f ) if H 1 (27)

Fig. 4 illustrates the expression (27) for the ideal channel andfor a two-paths time dispersive channel (the second path beingdelayed by 3 times the symbol period and attenuated by a fac-tor 0.1). When the channel is ideal, the SCD is only multipliedby the power gain . When the channel is time dispersive, theform of the SCD may be signicantly modied. However, themetric used for detection is the integral of the SCD which isalso only affected by the power gain , even when the channelis time dispersive. Therefore, the probability of false alarm andthe probability of misdetection are unchanged due to the IQ im-balance if the detector threshold is multiplied by the same powergain.

In conclusion, the IQ imbalance has no impact on the perfor-mance of both detectors.

V. S AMPLING CLOCK OFFSET (SCO)

In the baseband equivalent architecture, the SCO is modeledby a factor 1 + that multiplies the sampling rate T s .

A. Impact on energy detector

Since the variance of the received signal samples does not de-pend on the sampling rate (it only depends on the receiver lterbandwidth), the metric of the energy detector is not modied:

2y = 2n if H 0 (28)

2y = 2s + 2n if H 1 (29)

The probability of false alarm and the probability of misdetec-tion are therefore also unchanged in the presence of SCO.

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Fig. 4. SCD module for the two possible situations in the case of IQ imbalance.

B. Impact on cyclostationarity detector

Since the cyclostationarity detector estimates the SCD of thereceived signal, let us develop its expression in the presence of SCO to nd out how the detector performances will be affected.In order to assess the impact of the SCO correctly, we need totake the nite observation window into account. We assumetherefore that the received signal y(t) results from the multipli-cation of an innite signal x(t) with a nite rectangular windowwT (t) dened as:

wT (t) = 1 T

2 t T

20 else (30)

As we know, a product in the temporal domain corresponds to aconvolution in the frequency domain:

Y T (t, f ) = X (t, f ) W T (f ) (31)

=

X (t, f s) W T (s) ds (32)

where W T (f ) is the Fourier transform of wT (t) given byW T (f ) = T sin( T f )T f . The principle is illustrated in Fig. 5.

We rst derive the SCD in the ideal case, then we add theSCO effect. By using the denition (6), we obtain:

S cy (f ) =

S cx f

s + s2

W T (s) W T (s ) dsds (33)

where c = c (s s ) and S cx (f ) is the SCD of the innitesignal x(t).

The term S cx f s + s2 is different from zero when c = cor when c = 0 . Therefore, we have:

S cx f s + s

2= S cx f

s + s2

(s s )

+ S cx f s + s

2(s s c) (34)

Fig. 5. Windowing of x(t).

and:

S cy (f ) =

S cx (f s) | W T (s) |

2ds

+

S 0x f + c2 s W T (s) W

T (s c) ds (35)

= S cx (f ) | W T (f ) |2

+

S 0x f +

c2

s W T (s) W T (s c) ds (36)

The rst term corresponds to the convolution between theSCD of the innite signal x(t) and the square modulus of thefrequency response of the rectangular window wT (t). SinceT T symb and thus 1T c, the product W T (s) W

T (s c)in the second term is negligible because the value of W T (f )tends to zero when f 1T .

If we want to interpret the effect of the SCO, we have to con-sider the following SCD:

S c(1+ )y (f ) =

S cx f

s + s2

W T (s) W T (s ) dsds (37)

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where c = c (1 + ). The term S cx f s + s2 is differentfrom zero when c = c or when c = 0 . Therefore, we have:

S c(1+ )y (f ) (38)

=

S cx f s +

c2

W T (s) W T (s c) ds (39)

+

S 0x f s +

c2

(1 + ) W T (s) W T [s c (1 + )] ds

(40)

= S cx f +c2

[W T (f ) W T (f c)] (41)

+

S 0x f s +

c2

(1 + ) W T (s) W T [s c (1 + )] ds

(42)

In conclusion, the SCD will decrease when or T is grow-ing. We know that the longer the sampling vector, the betterSCD estimate. But now we also know that if T is growing, theSCO effect will increase. It automatically implies a tradeoff thatwe have to make in practice between computing a precise SCDand avoiding too much SCO effect on it. If we draw the SCD,we see on Fig. 6 that there is a loss of performance (identical inboth AWGN and dispersive channel) when T is growing (as ex-pected). Thus, the channel nature does not change the effect of the SCO. We also observe that performance is decreasing when is increasing (see Fig. 7). In both two graphs, the pMD growsas is increasing (meaning that the performance is decreasing).

If we increase the window length, the performance loss is moreimportant (see Fig. 7b).In conclusion, the SCO does not affect the energy detector

performance but the cyclostationarity detector is less efcientwhen and T are growing.

Fig. 6. SCD module for the two possible situations in the case of SCO.

VI. C ONCLUSION

This paper evaluates the impact of two analog front-end im-

perfections, IQ imbalance and sample clock offset (SCO), onthe performance of the spectrum sensing. We compare the sen-sitivity of the energy detector and of the cyclostationarity de-tector. It is shown that IQ imbalance does not cause any per-formance degradation so that the low-cost direct conversion ar-chitecture incurring IQ imbalance can be recommended for the

Fig. 7. Probability of misdetection in the case of SCO ( and T are varying). Tis 10 times bigger in b) than in a).

spectrum sensing. While the energy detector is not sensitive toSCO, the cyclostationarity detector suffers from a high perfor-mance degradation when there is SCO in the system. Thereforehigh quality of the local oscillators should be integrated on thedevice when the cyclostationarity detector is used for spectrumsensing.

R EFERENCES[1] Federal Communications Commission, Facilitating opportunities for ex-

ible, efcient, and reliable spectrum use employing cognitive radio tech-nologies, in FCC-03-322 , 2003.

[2] F. K. Jondral T. A. Weiss, Spectrum pooling: An innovative strategy forthe enhancement of spectrum efciency, IEEE Radio Communications ,vol. 42, no. 3, pp. S8S14, March 2004.

[3] I. F. Akyildiz W. Y. Lee M. C. Vuran S. Mohanty, Next generation /

dynamic spectrum access / cognitive radio wireless networks: A survey,Computer Networks Journal (Elsevier) , vol. 50, pp. 21272159, September2006.

[4] R. W. Brodersen D. Cabric, S. M. Mishra, Implementation issues in spec-trum sensing for cognitive radios, in Asilomar Conference on Signals,Systems, and Computers , 2004.

[5] B. Natarajan H. Zamat, Use of dedicated broadband sensing receiver incognitive radio, in IEEE Proceedings of ICC , June 2008.

[6] Lee D. Davisson Robert M. Gray, An Introduction to Statistical SignalProcessing , Cambridge University Press, 2004.

[7] William A. Gardner, StatisticalSpectral Analysis: A Non-ProbabilisticThe-ory , Prentice Hall Information and System Sciences Series, 1988.

[8] C.M. Spooner W.A. Gardner, Signal interception: performance advantagesof cyclic-feature detectors, IEEE Transactions on Communications , vol.40, no. 1, pp. 149159, January 1992.

[9] Andr e Bourdoux Francois Horlin, Digital Compensation for Analog Front- Ends: A New Approach to Wireless Transceiver Design , Wiley, 2008.