Cognitive Radios: Discriminant Analysis for Automatic Signal Detection in Measured Power Spectra

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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT 1

Cognitive Radios: Discriminant Analysis forAutomatic Signal Detection in Measured

Power SpectraLee Gonzales-Fuentes, Kurt Barbé, Wendy Van Moer, and Niclas Bjorsell

Abstract—Signal detection of primary users for cognitive ra-dios enables spectrum use agility. In normal operation conditions,the sensed spectrum is nonflat, i.e. the power spectrum is not con-stant. A novel method proposes the segmentation of the measuredspectra into regions where the flatness condition is approximatelyvalid. As a result, an automatic detection of the significantspectral components together with an estimate of the magnitudeof the spectral component and a measure of the quality ofclassification becomes available. In this paper, we optimize themethodology for signal detection in cognitive radios such that theprobability that a spectral component was incorrectly classifiedis iteratively reduced. Simulation and measurement results showthe advantages of the presented technique in different types ofspectra.

Index Terms—Cognitive radio, discriminant analysis, powerspectrum, rice distribution, signal detection, spectral component,spectrum sensing, statistics.

I. Introduction

THE efficient use of a natural resource like the radioelectromagnetic spectrum can be achieved by means

of cognitive radio. Cognitive radio is an intelligent wirelesscommunication system that is cognizant of its radio frequencyenvironment, learns from it and adapts its transmission orreception parameters to the different variations of its surround-ings. It dynamically monitors the unused spectrum, exploitingit by unlicensed users or transferring it when requested by li-censed users such that interference between users is prevented[1]. This sensing feature of cognitive radio is fundamental foran efficient management of the spectrum. Some methods aresuggested for this purpose, i.e. auxiliary beacons, geo-locationdatabase, and spectrum sensing. Spectrum sensing enables thecognitive radio to adapt to the environment by identifyingunused portions of the spectrum so that it can transmit in thesespectrum holes. Cognitive radio should therefore determine if a

Manuscript received September 5, 2012; revised November 28, 2012;accepted February 4, 2013. This work was supported in part by the ResearchProject of BelV “Cognitive Radios for Nuclear Power Plants,” and thePost-Doctoral Fellowship of the Research Foundation Flanders (FWO). TheAssociate Editor coordinating the review process was Dr. Matteo Pastorino.

L. Gonzales-Fuentes and W. Van Moer are with the Department of Fun-damental Electricity and Instrumentation (ELEC, M2ESA), Vrije UniversiteitBrussel, Brussels B-1050, Belgium, and also with the University of Gavle,Gavle 801 76, Sweden (e-mail: [email protected]).

K. Barbé is with the Department of Fundamental Electricity and Instrumen-tation (ELEC, M2ESA), Vrije Universiteit Brussel, Brussels B-1050, Belgium.

N. Bjorsell is with the University of Gavle, Gavle 801 76, Sweden.Digital Object Identifier 10.1109/TIM.2013.2265607

signal from a user is locally present in a determined frequencyband. Hence, the spectrum sensing problem can be reducedto the signal detection of a primary transmitter. The simplestapproach is to visually analyze the signal spectrum. At highsignal-to-noise ratios (SNR), a signal component can be easilydistinguished since it sticks out of the noise floor. However,some factors can turn this into a difficult task: low SNR, fadingand multipath in wireless communication, and noise poweruncertainty [2], [3].

Some spectrum sensing techniques are already available.The energy detection (ED) method [4] does not need anyinformation of the signal to be detected, but requires agood estimate of the noise power [5]. Hence, it is veryvulnerable to noise uncertainty. Some improvement can beachieved by means of energy and autocorrelation statistics,where no knowledge of the signal and noise are requiredbut an increment on the channel gain is needed [6]. Toovercome these shortcomings, test statistics-based methodswere developed: the covariance (CV) method [7], [8] andthe maximum-minimum eigenvalue (MME) detection [9] areblind algorithms insensitive to noise. MME assumes an infinitenumber of samples and requires knowledge of the numberof primary users [10], [11]. A latest method [12] is basedon random Vandermonde matrices (RVM), which presents abetter performance than the previous methods even for a finitenumber of measurement samples. These sensing techniquespresent some specific applications and limitations [13], andgenerally require information either on the noise power or thesignal characteristics.

The problem of discerning signal from noise in an observedspectrum is also tackled by harmonic or periodic componentsdetection methods. Most of these methods are parametric andassume a statistical model of the signal [6]–[12], [14]–[18].The multitaper method (MTM) [19] is a nonparametric spec-tral estimator that uses an orthonormal sequence of Slepiantapers or windows. MTM performs signal classification in acomputationally feasible way, particularly for small samplemeasurements [20]. However, its complexity increases as thenumber of tapers increases. Filter banks have been used to sim-plify MTM [21]. Recent studies indicate that the quality of themethod is controlled by the number of tapers, threshold, andsignal and noise power [22]. However, aside of the detectionof the spectral lines, no information on either the probabilityof misclassification of the frequency lines or the estimate of

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2 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT

the magnitude of the every detected signal line is computedby MTM and other parametric techniques in one approach.

Recently, an automatic harmonic detection method [23]based on a statistical test has been developed. This techniqueautomatically classifies spectral lines as signal or noise linesand has three major advantages over the existing methods.

1) The method requires no user interaction and minimalpostulated noise assumptions.

2) The probability of wrongly detecting a frequency lineas a noise or signal line is obtained.

3) The estimate of the magnitude of every detected signalline is computed for a validation of the classification.

The method presented in [23] is shown to be optimal forflat spectra only [24]. However, it can be extended towardsnonflat spectra for signal detection in cognitive radios [25].This is obtained by segmenting the spectrum and applying thedetection method to each segment. The segmentation clearlyhas some advantages.

1) The presence or the absence of a signal can be detectedregardless of the spectrum shape.

2) Reduction in the misclassification probabilities isachieved when the frequency lines with high probabil-ities of misclassification are regrouped to the correctgroup.

3) The width of the segment can be updated to minimizethe misclassification probabilities.

The quality of the segmentation algorithm depends on theproper selection of the boundaries of every segment. In thispaper, the probabilities of misclassification results will be usedto update the boundaries of the original segments such thatthe automatic signal detection is recomputed and the mis-classification probabilities are reduced. Hence, the detectionalgorithm can become useful for cognitive radio applicationsdue to its superior advantages compared to other existingmethods.

This paper is organized as follows. Section II provides aconcise description of the automatic method as well as itsadvantages and disadvantages. The segmentation technique fornonflat spectra is explained in Section III. In Section IV, thesegmentation technique is used for signal detection. Section Villustrates the proposed algorithms on simulation examples.Section VI assesses the performance of the method on realmeasurement examples. Section VII proposes an update forthe segment bounds. Conclusions are given in Section VIII.

II. Discriminant Analysis Method

In this section, the discriminant analysis algorithm describedin [23] is summarized since it forms the basis of the ex-tended method that is proposed in this paper. Furthermore,the advantages and the disadvantages of this technique areelaborated on.

A. Signal Assumptions

Let x(t) be a continuous time signal such that

x(t) = g(t) + n(t) (1)

where g(t) is a periodic multisine signal with K arbitrary tones,and n(t) is a noise process such that its power spectral densitySn(jω) and its variance σ2

n exist.The spectral content of the signal can be obtained by means

of a digital spectrum analyzer. The signal x(t) is digitizedand the resulting signal is xd(n) with n = 0, . . . , N − 1. Themeasured amplitude spectrum Ax(k) is given by

Ax(k) = |Xd(k)| = |Gd(k) + Nd(k)| (2)

where Xd(k) is the discrete Fourier coefficient of the signalxd(n) at frequency bin k. Gd(k) and Nd(k) are the discreteFourier coefficients at frequency bin k of the signal and noise,respectively. The variable Nd(k) is complex circular Gaussiandistributed with zero mean and variance Sn(jωk). Thus, Ax(k) isdistributed according to the Rice probability density function

Rice

(|Gd(k)| , 1

2Sn(jωk)

). (3)

A random variable Z is Rice distributed Z ∼ Rice(v, σ2), ifZ =

√X2 + Y 2 with X ∼ N(v cos θ, σ2) and Y ∼ N(v sin θ, σ2)

are two independent normal distributions where v sin θ andv cos θ are the means and σ2 is the variance of the signal,whereas θ is a real number.

B. Automatic Detection

The automatic detection algorithm can be divided into threemajor parts: the discriminant analysis, the estimation of themagnitudes of the signal and noise power, and the probabilisticvalidation of the detected spectral lines.

1) Discriminant Analysis: The main philosophy of dis-criminant analysis is to partition the data in two groups suchthat the groups are maximally separated under the constraintthat the variance within every group is as small as possible.Expressing this objective into a statistical testing frameworkresults in Fisher’s quadratic discriminant [19], [26]

T 2 =

(AI

x − AJx

)2

σ2I (|I| − 1) + σ2

J (|J | − 1)(|I| + |J | − 2) (4)

where AIx and AJ

x represent the mean measured amplitude ofthe classified signal lines and noise lines, respectively. Thevariables |I| and |J| represent the respective number of classi-fied signal and noise lines. Finally, σ2

I and σ2J are the respective

variances of the measured amplitudes of the classified signaland noise lines. The objective of the discriminant analysis isto maximize (4). Therefore, the set of frequency bins of thesignal lines I and the noise lines J should be chosen in such away that the numerator or distance between the group means ismaximized, and the denominator or distance within the groupvariances is minimized. A binary grid search is used to cometo the correct discrimination height.

2) Estimation of Signal and Noise Power Magnitudes: Inthis paragraph, the method of moments (MoM) estimatorfor the noise power and signal component magnitudes isderived [23].


GONZALES-FUENTES et al.: COGNITIVE RADIOS: DISCRIMINANT ANALYSIS FOR AUTOMATIC SIGNAL DETECTION 3

Assuming that the noise spectrum is white, the estimate ofthe noise power for noise frequency lines k ∈ J is

Sn(jωk) =1

|J |∑k∈J

(Ax(k))2. (5)

The maximum likelihood for the signal component is notavailable. The MoM estimator in [27] is used to estimate thesignal magnitude for k ∈ I∣∣Gd(k)

∣∣ =√

A2x(k) − Sn(jωk). (6)

3) Probabilistic Validation of the Detected Spectral Lines:In order to assess the quality of the classification, the prob-ability of false classification is computed by studying theprobability distribution of the amplitude measurements Ax(k).To formally introduce the probability of misclassification, wedenote A∗(k) to be the random variable describing a newamplitude measurement at frequency k which follows a Ricedistribution with parameters (|Gd(k)| , (1/2)Sn(jωk)) for signallines and (0, (1/2)Sn(jωk)) for noise lines.

Hence, P(A∗(k) > Ax(k) |k /∈ I ) is the probability thatA∗(k) can be larger than the measurement Ax(k), even when kis a noise line. P(A∗(k) < Ax(k) |k∈ I ) is the probability thatA∗(k) can be smaller than the measurement Ax(k), even whenk is a signal line. Based on the nature of k, the values in (5)and (6) are used to estimate the probabilities as follows:

π(k) =

{1 − FRice(0, 1

2 Sn(jωk))(Ax(k))FRice(|Gd (k)|, 1

2 Sn(jωk))(Ax(k))(7)

where FRice denotes the cumulative distribution function ofthe Rice distribution.

Note that the discriminant analysis method described in thissection has the following advantages.

1) It is fully automatic, with no user interaction.2) It estimates the magnitude of the spectral component.3) It provides a user-friendly and simple validation.However, the presented technique only works under the

assumption that the considered power spectrum of both signaland noise is flat, which cannot be assumed in practical appli-cations, e.g. normal operation conditions of cognitive radios.

III. Segmentation Algorithm

In this paragraph, we propose an extension of the discrim-inant analysis method (Section II) to nonflat spectra usingvisual analysis. This can be done by partitioning the spectrainto small segments in which the power spectrum of the noisecan be said to be approximately flat. To asses this, we need todetect the frequency lines that are purely noise contributions.Doing so, the width of the region where the flatness conditionholds can be determined.

A. Initial Detection Method for Signal and Noise Components:

By a simple visual inspection, one can already have arough idea of which parts of the spectrum contain signal,and which contain noise. A signal line typically has largermagnitude than its neighboring noise frequency lines. In theproposed detection algorithm, we consider a frequency line

Fig. 1. Local maximum and minimum detection. Gray: amplitude measure-ment. Cross: local maximum. Circle: local minimum.

to be a potential signal line if its magnitude is larger thanthe magnitude of its neighboring frequency lines by at leastthe user-defined value δG, whereas a potential noise line isdetermined when the magnitude of the analyzed frequencyline is lower than the magnitude of its neighboring frequencylines by at least δG. To implement the above idea, we apply thefollowing equations to obtain these maximum and minimummagnitude values:

Ax(l) < Ax(k) − δG (8)

Ax(l) > Ax(k) − δG (9)

where k < l. Every Ax(k) satisfying (8) is a local maximum,and is denoted as Amax

x (k). Every Ax(k) satisfying (9) is alocal minimum and is denoted as Amin

x (k). The result ofthis algorithm is illustrated in Fig. 1. The crosses representthe frequency lines k with magnitude Amax

x (k), these linesare possibly signal contributions. The circles represent thefrequency lines k with magnitude Amin

x (k), these lines aredefinitely noise contributions. The remaining frequency linescan be either noise or signal contributions.

B. Segmentation Width

Based on the previously detected noise contributions withmagnitude Amin

x (k), the segments where the spectrum is locallyflat are determined. Let kupper and klower be the frequency lineswith maximal and minimal magnitudes such that they satisfy(10) and (11), respectively

Aminx (l) < Amin

x (kupper) − δSG, (10)

Aminx (l) > Amin

x (klower) − δSG. (11)

The frequency lines kupper and klower are the positionsat which the magnitude of the noise power presents abruptchanges and hence potential changes of the noise coloring.Consequently, these frequency lines determine the bounds ofa segment within which the noise power does not significantlydeviate and can be safely assumed as flat.



The order in which these frequency lines appear definethe right bound of a segment, while the left bound of thenext segment is the right bound of the previous segment.An exception occurs for the frequency lines located at thebeginning and end of the spectrum, for which the left andright bounds are 1 and N , respectively. For instance, having imaximal values kupper and j minimal values klower, the boundsof the different segments can be determined as follows:

1−kupper(1), kupper(1) − klower(2), klower(2) − kupper(3), . . . ,

kupper(i) − klower(j), klower(j) − N.

An illustrative example is presented in Fig. 2. The diamondsrepresent the segment bounds at frequency lines kupper withmaximal magnitude Amin

x (kupper), and the squares representthe segment bounds at frequency lines klower with minimalmagnitude Amin

x (klower). Clearly, the main difficulty is to selectthe proper values for δG and δSG. Although these values canbe selected arbitrarily by the user, some interesting rules-of-thumb are proposed:

1) For SNR values higher than –5 dB. At high SNR, theRice distribution of the amplitude of a disturbed signalcan be approximated by a Gaussian distribution. Visualanalysis indicates that most Amax

x (k) and Aminx (k) are

found to lie around 2 standard deviations of the meanof the amplitude spectrum. Thus, δG would reflect the95% confidence interval for Gaussian random variables.A starting value for δG is given by

δG = |μ + 2σ| (12)

where μ and σ are the mean and the standard deviationof the magnitude of the discrete Fourier coefficient of thesignal Ax(k), respectively. Magnitude values that exceedat least δG, assures an initial detection of a sufficientnumber of signal and noise lines. A starting value for δSG

can be given by the standard deviation of the detectedAmin

x (k) values.2) For SNR values lower than –5 dB. The previous defini-

tions for δG and δSG can result in larger values that donot always detect enough spectral lines to make feasiblefinding the bounds of the segments. However, thesevalues can be used as initial values, and then decreasedthem in such a way that the chosen δG allows detectinga sufficient number of initial spectral lines. For δSG,one can try lower or higher values than the standarddeviation of the Amin

x (k) values. A trade-off between theδG and δSG is present, where increments on δG allowdecrements on δSG.

IV. Signal Detection

In this section, the segmentation technique is used forsignal detection. The idea is to apply the method described inSection II, as detailed in [23], to each of the detected segments.

The segmentation algorithm is applied to the measuredamplitude spectrum as shown in Figs. 1 and 2, where aninitial detection of spectral components allows the detectionof the boundaries of the segments. Once the segments are

Fig. 2. Detection of the bounds of the segments. Gray: amplitude mea-surement. Dashed vertical line: bound. Diamond: bound at frequency linewith maximal magnitude. Square: bound at the frequency line with minimalmagnitude.

Fig. 3. Polynomial fitting curve for an SNR of –10 dB. Gray: amplitudemeasurement. Bold dark curve: discrimination curve. Black circle: center ofdiscriminant height. Solid gray line: discrimination height.

determined, each segment receives a different discriminationheight as shown in Fig. 3. The gray curve is the amplitudespectrum. The horizontal lines represent the discriminationheights. Within each segment, the frequency lines with mea-sured amplitude below the discrimination height are classifiedas noise lines, while the frequency lines with measured am-plitude above the discrimination height are classified as signallines. The dark black curve is a smooth discrimination curveover the full frequency band of interest. This curve is obtainedby a polynomial fitted to the centers of every discriminationheight (black circles over the different segments). Data fittingusing a polynomial regression model is used for this purpose[28].

Given N data points, the discrimination height h is afunction modeled as a linear combination of the variablek, namely frequency line. Regression estimates the modelparameters α and the presence of some uncontrolled errorsε. A p-order polynomial model can be synthesized as

h(k) = p−1i=0 αi.(k)i + ε(k). (13)



The estimate of the polynomial coefficients is given by

α = (XT X)−1XT h. (14)

where α is an p-by-1 vector of the model parameters, X is aN-by-p vector of frequency lines, and h is a 1-by-p vector ofdiscrimination heights. Using least squares method is possibleto fit the model to the data and estimate the polynomialcoefficients. Once the coefficients are found, the polynomialis evaluated over the full frequency range.

The degree of the polynomial is chosen by the user, lowervalues are always desired (p ≤ 7) to avoid overfitting andapproximate better the noise floor shape. The degree can bechosen as the “maximum degree - 1” that does not exit the con-dition number of the matrix beyond numerical precision, andassure the system of (14) to be sufficiently well conditioned.

The obtained polynomial curve in Fig. 3 clearly shows thatthe extended automatic detection algorithm is able to separatethe signal from the noise lines without any user interaction.

V. Numerical Examples

This section provides some simulation examples where theperformance of the technique will be evaluated for differentSNR ranging from 0 to −21 dB as required by the IEEEstandard for cognitive radios [29].

A. Different Amplitude Under White Noise

The signal satisfies the representation given in (1), whereg(t) is a multisine with different amplitude spectrum andeight tones arbitrarily chosen, while n(t) is a zero-mean whiteGaussian noise sequence.

In Fig. 4, the segmentation technique is used for signaldetection of a disturbed signal presenting an SNR of −15 dB.The δG and δSG are computed as described in Section III-B,as a result δG = 15.12 and δSG = 0.97. Since not enoughinitial values are found to run the algorithm, δG is reduced.Therefore, the initial detection of signal and noise spectrallines is performed using (8) and (9) with δG = 8. The reductionof δG allows increments of δSG. Next, the bound segmentsare found with (10) and (11) when δSG = 1.5 is chosen. Thesignal detection is performed as described in Section II, anda different discriminant height is assigned to each segment.Polynomial fitting is performed as described in Section IV,resulting in a discrimination curve of degree p = 7.

Next, the estimators (5) and (6) are computed to estimatethe noise power and signal magnitude. The magnitude estimateGd(k) is ranging from −10.55 to −3.32 dBm, and correspondsto the visual observations. However, the noise estimate Sn(jωk)presents values of −68 dBm, which seems to correspond toan underestimate.

Thereafter, the validation process was performed. The mis-classification probabilities are calculated with (7). The methoddetects 108 signal lines, in which seven lines were correctlyclassified and received misclassification probabilities of 0,while the other 101 lines received probabilities around 0.07.The remaining frequency lines are classified as noise lines andpresent misclassification probabilities ranged from 0 to 0.63.

The frequency lines with magnitudes close to the dis-crimination height are susceptible to be misclassified, and

Fig. 4. Automatic signal detection for an SNR of −15 dB. Gray: amplitudemeasurement. Bold dark curve: discrimination curve. Black circle: Center ofdiscriminant height. Solid gray line: discrimination height.

therefore, to present higher misclassification probabilities. Thissuggests that the classification process was not correct forthose frequency lines, and consequently, that the segmentbounds were not chosen properly. Given that, these frequencylines are moved to the other group and the validation processis recomputed with (7). The misclassification probabilitiesfor the signal lines ranged from 0 to 0.43, while the noiselines received misclassification probabilities ranged from 0to 0.49. Regrouping of the frequency lines balances themisclassification probabilities at the expense of worsening themisclassification probability of one of the groups, particularlythe group of signal lines.

For instance, a detected noise line at a frequency of 0.6197rad/sample presents a misclassification probability of 0.56 asseen in Fig. 5. This suggests that this frequency line waswrongly classified as a noise line, and hence, it is moved to thesignal line group receiving a misclassification probability of 0,as illustrated in Fig. 6. This demonstrates that this frequencyline was a misdetected signal line. An interesting observa-tion is that, even when the method wrongly classifies somefrequency lines either as a signal or noise line, the validationprocess is able to recognize this by assigning considerably highmisclassification probability values to those frequency lines.

B. Uniform Signal Amplitude Under Colored Noise

Following the representation given in (1), the signal g(t) isa multisine with uniform amplitude spectrum and eight tones.The noise sequence n(t) is a zero-mean colored noise, whichis obtained using a Butterworth low pass filter at a normalizedfrequency of 0.5. In Fig. 7, the segmentation technique forsignal detection of the disturbed signal with an SNR of−10 dB is performed as described in Section III. FromSection III-B, one obtains δG = 14.43 and δSG = 16.02.The standard deviation of the Amin

x (k) values is large due tothe shape of the spectra, where frequency lines higher thanthe cut-off frequency present very low magnitude values incontrast with the magnitudes of frequency lines lower thanthe cut-off frequency. Therefore, δSG needs to be reduced. Theinitial detection of signal and noise spectral lines is performedcomputing (8) and (9) using δSG = 14.43, which was obtained



Fig. 5. Probabilistic of misclassification for detected spectral lines in thefrequency band [0.5, 0.7] rad/sample. Upper graph: segmented spectra. Middlegraph: probability of falsely classified noise lines. Lower graph: probabilityof falsely classified signal lines. Solid black rectangle: falsely detected noiseline with its corresponding probability of misclassification.

from (12). The bound detection is performed with (10) and(11) when δSG = 1 is chosen. Once the segments are defined,the signal detection is performed as described in Section II.Polynomial fitting is performed as described in Section IV,resulting in a discrimination curve of degree p = 2.

Next, the estimators (5) and (6) are computed. The signalmagnitude estimate Gd(k) ranges from −4.82 to −3.11 dBm,which coincides with the visual observations. The noise esti-mate presents values of −85.78 to −10.86 dBm.

The probabilistic validation is performed. The misclassifica-tion probabilities are calculated with (7). The method detectscorrectly eight signal lines that received misclassificationprobabilities with values that range from 1.66 10−14 to 0.The remaining frequency lines are classified as noise linesreceiving misclassification probability values ranging from1.20 10−6 to 0.6321. The frequency lines with misclassificationprobability higher than 0.5 are moved to the other group andthe validation process recomputed with (7).

The misclassification probabilities for the 725 detectedsignal lines are lower than 0.43. For the noise lines, thesevalues are lower than 0.49. Again, the regrouping of thefrequency lines balances the misclassification probabilities atthe expense of worsening the misclassification probability ofthe signal lines.

According to the discriminant analysis philosophy, theobjective is to separate the groups in order to minimizethe misclassification probabilities. When the segmentation

Fig. 6. Probabilistic of misclassification for detected signal lines in thefrequency band [0.5, 0.7] rad/sample. Upper graph: segmented spectra. Middlegraph: probability of falsely classified noise lines. Lower graph: probabilityof falsely classified signals. Solid black rectangle: regrouped noise line withits corresponding probability of misclassification.

algorithm is used for signal detection, the validation stepsuggests that the segmentation was incorrect for some fre-quency lines. However, regrouping the frequency lines altersthe misclassification probabilities. This indicates that the widthof the segment can be manipulated when the values of δG andδSG are updated using the probabilistic validation results. Thiscan improve the performance of the method, and hence, reducethe misclassification probabilities.

VI. Measurement Examples

In the measurement examples, signals with different spec-trum shape are measured. The computer-generated signalsare sent to a signal generator in its time-domain versionusing a carrier frequency fc = 1.5 GHz. The amplitude of thesignal was measured with a signal analyzer using a bandwidthBW = 100 MHz and 100 001 number of sweep points. Themeasured amplitude is sent back to a computer, where thesignal detection method is applied to the data. The examplespresent multisine signals with eight tones disturbed either bywhite or colored noise sequences. This controlled laboratoryexperiment will allow us to evaluate the performance of theproposed technique.

A. Multisine with Different Amplitude Under White Noise

The multisine shown in Fig. 3 is measured. The values of δG

and δSG are computed as described in Section III-B resulting



Fig. 7. Automatic signal detection for a multisine with uniform amplitudeand an SNR of −10 dB. Gray: amplitude measurement. Bold dark curve:discrimination curve. Black circle: center of discriminant height. Solid grayline: discrimination height. Black arrow: signal tones.

in δG = 24.43 and δSG = 1.94. However, a reduction for δG isneeded to assure sufficient initial values. δG = 10 is found sothat (9) and (10) can be computed. The initial detection of sig-nal and noise spectral components is performed. The segmentbounds are found with (11) and (12) using δSG = 1. This is adata-driven method, thus these values will differ from the onesof the simulation examples given the higher amount of data.

Next, the signal detection is applied to every segment asshown in Fig. 8. The horizontal black lines are the differ-ent discrimination heights of every segment, according tothe segmentation and signal detection algorithms. The blackcircles are the center of the discrimination heights to whicha discrimination curve of degree p = 2 is fitted, and it can bedistinguished as a bold dark curve.

The estimators (5) and (6) are computed to estimate thenoise power and signal magnitudes. The signal magnitudeestimate Gd(k) is ranging from −47.99 to −35.81 dBm. Thenoise power estimate Sn(jωk) for the different segments rangesfrom −62.88 to −38.54 dBm.

Finally, the probabilistic validation is performed. The mis-classification probabilities are estimated using (7). In total,4747 lines were classified as noise lines and the probabilitiesranged from 0 to 0.63. The method detects 254 signal lines,from which eight are correctly detected as signal lines present-ing a misclassification probability of 0, while the other 246detected signal lines received probabilities around 0.12. Afterregrouping the frequency lines with high misclassificationprobabilities, the detected signal lines received probabilityvalues of 0.44, whereas the detected noise lines receivedprobability values around 0.50.

B. Uniform Signal Amplitude Under Colored Noise

In this example, the amplitude of the signal is uniform. Thenoise sequence is colored using a shaping Butterworth filterwith a cut-off frequency of 0.75. The values of δG and δSG

are computed as described in Section III-B, as a result δG =23.27 and δSG = 7.44. An initial detection of signal and noisecomponents is performed using δG = 6 to compute (8) and(9). The boundaries of the segments are determined with (10)and (11) using δSG = 4. Next, the signal detection is applied

Fig. 8. Automatic signal detection for a multisine with different amplitudeand an SNR of –10 dB. Gray: amplitude measurement. Bold dark curve:discrimination curve. Black circle: centers of discriminant height. Solid graylines: discrimination height. Black arrow: signal tones.

to every segment of the measured amplitude spectrum andpolynomial fitting results in a discrimination curve of p = 2,as shown in Fig. 9. The same legend from Fig. 8 holds.

Using (5) and (6), the signal magnitude estimate Gd(k)ranges from −47.97 to −35.27 dBm, while the noise estimateSn(jωk) ranges from −96.42 to −41.13 dBm. The validationprocess is performed. The misclassification probabilities arecalculated with (7). The method detects 132 signal lines, fromwhich eight are correctly classified as signal lines and re-ceived misclassification probabilities of 0, while the remainingdetected signal lines received a misclassification probabilityvalues that reach up to 0.15. The remaining frequency lineswere classified as noise lines and received misclassificationprobabilities from 0 to 0.63. After regrouping of frequencylines, the signal lines received probability values of 0.43 andthe noise lines received probability values of 0.49.

In this particular example, the frequency lines higher thanthe cut-off frequency present a decreasing noise floor whichmakes the signal lines more visible. Therefore, the algorithmcan find a good position for the discrimination height cuttingbetween the noise floor and the signal lines. This is acknowl-edged by the lower misclassification probabilities assigned tothese frequency lines.

A measure of the quality of the classification can alsobe given in terms of risk of misclassification, where thequality of the detection and the segmentation algorithm areevaluated by measuring the magnitude of misclassificationprobability of every frequency line in the full band of interest.It became obvious during the validation process that somefrequency lines whose amplitude lies close to the discrimi-nation height present a higher risk of being misclassified ineither group. These frequency lines present amplitudes thatare low enough to be considered as noise lines but alsohigh enough to be considered as signal lines. This duality isreflected in the misclassification probability values assigned tothem. Consequently, the amplitudes of the frequency lines withsufficient distance from the discrimination height will presentlow misclassification probabilities, and therefore, a small riskof being incorrectly classified whether as noise or signal lines.



Fig. 9. Automatic signal detection for a multisine with uniform amplitudeand an SNR of –10 dB. Gray: amplitude measurement. Bold dark curve:discrimination curve. Black circle: center of discriminant height. Solid grayline: discrimination height. Black arrow: signal tones.

Fig. 10. Risk of misclassification for spectral lines of an SNR of −10 dB.Black: high risk of misclassification. Dark gray: considerable risk of misclas-sification. Gray: low risk of misclassification. Light gray: very low risk ofmisclassification.

For this purpose, the spectrum is sliced into different regionsthat cover the different levels of risk. The risk of being falselyclassified as being either noise or signal line is illustrated inFigs. 10 and 11 from the measurement examples showed inFigs. 8 and 9. In a light gray color are the frequency lineswith very low risk π(k) < 10%, in gray color are the frequencylines with low risk 10% < π(k) < 30%, in dark gray color arethe frequency lines with considerable risk 30% < π(k) < 50%,and in black color the frequencies with high risk π(k) > 50%of being incorrectly classified.

VII. Boundaries Update Using Probabilities of

Misclassification

In Section III, the segmentation algorithm is an attemptto define flat regions where the automatic signal detectionmethod can be applied. The boundaries are determined atfrequencies where the power spectrum is not constant, i.e.where an abrupt change is perceived. In a disturbed signal,this can occur at many points in the spectra. The properselection of the boundaries thus depends on the proper se-lection of the values δG and δSG. How large these values

Fig. 11. Risk of misclassification for spectral lines of a multisine withuniform amplitude and an SNR of −10 d0B. Black: high risk of misclas-sification. Dark gray: considerable risk of misclassification. Gray: low risk ofmisclassification. Light gray: very low risk of misclassification.

Fig. 12. Automatic signal detection for an SNR of −10 dB after boundupdate. Gray: amplitude measurement. Bold dark curve: discrimination curve.Black circle: center of discriminant height. Solid gray line: discriminationheight. Black arrow: signal tones.

are, determines the sensitivity of these two parameters tothe changes in the spectrum, and present a direct impactin the correct performance of the automatic signal detectionmethod. Generally, when the values of δG and δSG are toolarge, the method can miss some changing points and whentoo small it can detect many irrelevant changing points suchthat the computation becomes slower. From the simulation andmeasurement examples, the reduction of the misclassificationprobabilities of the detected noise lines becomes a challengesince the objective of the automatic signal detection methodis to minimize the misclassification probabilities [23].

In order to reduce the misclassification probability values,this section proposes an iterative algorithm scanning for somemissed changes within the previously defined segments. Forthis purpose, we can reuse the formulas given in (10) and(11) to search the missed changing points. Since the spectrumwithin the segment is approximately flat, δSG can be definedas suggested in (12). New kupper and klower frequency linespresenting maximal and minimal amplitudes are chosen fromthe Amin

x (k) values found within the initial segment bounds,



Fig. 13. Automatic signal detection for a multisine with uniform amplitudeand an SNR of –10 dB after bound update. Gray: amplitude measurement.Bold dark curve: discrimination curve. Black circle: center of discriminantheight. Solid gray line: discrimination height. Black arrow: signal tones.

Fig. 14. Automatic signal detection for a measured multisine with uniformamplitude and an SNR of −10 dB after bound update. Gray: amplitudemeasurement. Bold dark curve: discrimination curve. Black circle: centerof discriminant height. Solid gray line: discrimination height. Black arrow:signal tones.

as explained in Section III-B. The appearance of these newpositions together with the previous ones generates newersegments and hence different discriminant heights. The medianof the misclassification of the detected noise lines is computed.The process is repeated while a reduction on the misclas-sification probabilities is accomplished or stopped when theopposite occurs. When no Amin

x (k) values are found betweenthe bounds, the algorithm then reduces δG until the median ofthe misclassification probabilities is minimized.

Using the misclassification probabilities from the simulationexample shown in Fig. 3, an update of the boundaries ofthe segments is performed and the signal detection methodis applied to each new segment as seen in Fig. 12. The samelegend from Fig. 3 holds. In total, 19 segments are obtainedfrom which 12 were the initial segments and 7 are the newsegments. The median of the misclassification probabilitiesfor noise lines after the update was reduced to 0.39. A newdiscrimination curve of degree p = 7 is fitted to the centers ofthe discrimination heights.

The signal detection is performed after updating the bound-aries of the simulation example shown in Fig. 5, and this canbe seen in Fig. 13. In total, 14 segments are obtained fromwhich 4 are initial segments and 10 are the new segments. Themedian of the misclassification for noise lines initially reached0.63, and after the update of the boundaries it reduced to 0.49.A discrimination curve of degree p = 4 is found.

In the measurement example shown in Fig. 9, the misclas-sification probabilities presented in Section VI-B were used toperform the update of the boundaries as seen in Fig. 14. Thenumber of segments increased from 6 to 28. The median ofthe misclassification probabilities decreased from 0.63 to 0.51.A discrimination curve of degree p = 3 is needed. DecreasingδG did not give better results in most cases.

In all the three cases, it is interesting to point out that in thepresence of more segments both the discrimination heights andespecially the discrimination curve adapt better to the shape ofthe noise floor. This is visually evident when the polynomialcurve is found slightly above the noise floor, leaving only thesignal lines raise above it.

VIII. Conclusion

This paper proposed an extension of [23] to signal detectionfor nonflat spectra using a segmentation method [25] and aniterative algorithm to update the segment boundaries such thatthe automatic signal detection method [23] can be used forspectrum sensing in cognitive radio and the misclassificationprobabilities are minimized. The only prior information neededis that within a frequency band, the discrimination methodseparates two groups: a signal group and noise group. Theadvantage of the method over previous methods is minimaluser interaction and simplicity compared to the MTM. Thismakes it feasible in practical applications for cognitive radios.No knowledge on the disturbing noise power is necessary,which contrasts with the ED method. Furthermore, one doesnot need to specify in advance the number of primary userstransmitting in the band of interest as required by the MMEmethod. Hence, the proposed method is a fully blind spectrumsensing technique.

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Lee Gonzales-Fuentes was born in Arequipa, Peru,in 1985. She received the B.Sc. degree. in electronicengineering with a major in telecommunicationsfrom the Universidad Nacional de San Agustın(UNSA), Arequipa, Peru in 2008, and the M.Sc.degree in electronics and telecommunications fromthe University of Gavle (HIG), Gavle, Sweden, in2012.

In March 2012, she joined the Department ofElectricity and Instrumentation (ELEC) as a Ph.D.student in the domain of pre and postprocessing

of measured data. Her current research interests include measurements andmodeling techniques for high-frequency applications, wireless network design,telemetry, and digital signal processing.

In May 2012, Ms. Gonzales-Fuentes received the 2012 Graduate FellowshipAward from the IEEE Instrumentation and Measurement Society.

Kurt Barbe (M’09) received the M.Sc. degree inmathematics (option statistics) and Ph.D. degree inelectrical engineering from the Vrije UniversiteitBrussel (VUB), Brussel, Belgium, in 2005 and 2009,respectively.

He is currently an Assistant Professor with theDepartment of Fundamental Electricity and Instru-mentation at VUB, Brussel, Belgium. He is alsoa Post-Doctoral Research Fellow with the Flem-ish Research Foundation and the VUB Coordinatorof the Flemish interuniversity training network for

methodology and statistics initiated by the Flemish minister of innovation. Hiscurrent research interests include the field of system identification, time seriesanalysis, and signal processing for biomedical applications. Since 2010, hehas been an Associate Editor for IEEE Transactions on Instrumentation

and Measurement.Dr. Barbe is the recipient of the 2011 Outstanding Young Engineer Award

from the IEEE Instrumentation and Measurement Society.

Wendy Van Moer was born in Belgium in 1974 andreceived the M.E. and Ph.D. degrees in engineeringfrom the Vrije Universiteit Brussel (VUB), Brussels,Belgium, in 1997 and 2001, respectively.

She is currently with the Department of ElectricalMeasurement (ELEC), VUB, Brussels, Belgium, anda Visiting Professor with the Department of Elec-tronics, Mathematics and Natural Sciences, Univer-sity of Gavle, Gavle, Sweden. In 2010, she becamethe Head of the Research Group Medical Measure-ments and Signal Analysis (M2ESA). Her current

research interests include nonlinear measurement and modeling techniquesfor medical and high-frequency applications.

She has published over 100 related conference and peer-reviewed journalarticles. She was the recipient of the 2006 Outstanding Young Engineer Awardfrom the IEEE Instrumentation and Measurement Society. Since 2007, she hasbeen an Associate Editor for the IEEE Transaction on International

Instrumentation. She was also the recipient of 2010 and 2011 OutstandingAssociate Editor Recognition. In 2010, she became an Associate Editor forthe IEEE Transactions on Microwave Theory and Techiniques. Shewas one of the Technical Program Co-Chairs of the 2010 and 2015 IEEEInternational Instrumentation and Measurement Conference and a Co-GuestEditor of the I2MTC 2010 Special Issue. In 2012, she was elected as amember of the administrative committee of the IEEE Instrumentation andMeasurement Society for a four-year term. She has been elected as Vice-President of publications of this society for the term 2013–2014.

Niclas Björsell was born in Falun, Sweden, in1964. He received his B.Sc. in electrical engineer-ing and the Lic. Ph. in automatic control fromUppsala University, Uppsala, Sweden, in 1994 and1998, respectively. He received the Ph.D. degreein telecommunications from the Royal Institute ofTechnology, Stockholm, Sweden, in 2007.

He has several years of experience from researchand development projects that fostered collabora-tions between industry and the academy. For morethan 15 years, he has held positions in the academy

and in industry. Between 2006 and 2009, he worked as the Head of Divisionof Electronics, Department of Technology and Built Environment, Universityof Gavle, Gavle, Sweden. He is currently an Associate Professor with theUniversity of Gavle, Gavle, Sweden and Guest Professor with the VrijeUniversiteit Brussel, Brussel, Belgium. He has published more than 50 papersin peer-reviewed journals and conferences.

He is currently an Associate Editor of IEEE Instrumentation and Measure-ment Society and voting member of the IEEE Instrumentation and Measure-ment Society, TC-10. His current research interests include radio frequencymeasurement technology, analog-to-digital conversion, and cognitive radio.

Cognitive Radios: Discriminant Analysis for Automatic Signal Detection in Measured Power Spectra

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