On the performance of Grassmann-covariance-matrix-based spectrumsensing for cognitive radio

PALLAVIRAM SURE

Department of ECE, M S Ramaiah University of Applied Sciences, Bangalore, India

e-mail: pallavisruti@gmail.com

MS received 25 October 2020; revised 15 July 2021; accepted 12 August 2021

Abstract. Cognitive radio assures efficient utilization of spectral resources by encouraging opportunistic

spectral access. Its inevitable task of spectrum sensing has been widely addressed in the literature through the

covariance-matrix-computation-based approaches. Unlike these blind approaches, recently a Grassmann

covariance matrix (GCM)-based approach has been devised that requires a priori signal covariance matrix. To

alleviate this impractical limitation, this paper proposes a spectrum sensing approach based on the computation

of a modified projection distance metric on the Grassmann manifold. Particularly, the test statistic is derived

using two GCMs estimated from the received signal frame and the threshold is calculated using Gaussian noise

statistics of the null hypothesis pertaining to the detection problem. Simulations on the signals received in

Rayleigh environments show that the proposed approach renders better probability of detection (Pd) compared

with Covariance Absolute Value (CAV), Akaike Information Criterion (AIC)-based and Maximum Minimum

Eigenvalue (MME) approaches. Performance of the proposed approach is at par with the case when signal

covariance matrix is known a priori. Experiments are conducted using ADALM PLUTO software-defined radio

(SDR) measurements in the ultra-high-frequency (UHF) television (TV) band. Using these measured signals it is

verified that the proposed approach renders Pd ¼ 1 at an SNR of –15 dB compared with the CAV, AIC and

MME approaches, which render Pd ¼ 1 at an SNR of –5 dB.

Keywords. Grassmann covariance matrix; Binet–Cauchy distance; projection distance; blind spectrum

sensing; covariance absolute value; maximum minimum eigenvalue.

1. Introduction

Available licensed radio spectral resources are becom-

ing increasingly insufficient to handle the exponentially

growing demand for wireless services. On the other

hand, some of the licensed radio spectral bands such as

the ultra-high-frequency (UHF) television (TV) band

are heavily underutilized [1]. Cognitive radio emanated

as a promising technology to cater to the needs of

growing wireless services and to improve the spectral

utilization. Owing to its opportunistic spectral access,

the cognitive radio technology ushers the IEEE 802.22

wireless regional area networks (WRAN) [2] where

digital television (DTV) and the wireless microphones

form the licensed primary users (PUs). A cognitive

radio acts as a secondary user (SU) that accesses the

radio channel only when the PU is absent. The SU

vacates the channel as soon as any PU activity gets

detected in the radio channel so as to not cause any

interference to the PU transmissions [3]. To be com-

petent in these crucial functionalities, the SU has to

periodically perform spectrum sensing and detect the

channel’s occupancy.

Spectrum sensing techniques are broadly classified into

two types: Wide-band Spectrum Sensing (WSS) and Nar-

row-band Spectrum Sensing (NSS). WSS techniques scan a

large bandwidth and identify the vacant bands (spectrum

holes), which are in turn used for single or multiple SU

transmissions [4]. Contrarily, NSS techniques detect the

presence or absence of the licensed PU in the given band.

The NSS is usually cast as a binary hypothesis testing

problem. Corresponding test criterion and the threshold are

employed for the detection of channel occupancy. Con-

ventionally, NSS is performed using Energy Detection

(ED) approach [5] where the energy of the received signal

frame is compared to a threshold based on the estimated

noise power. In general, NSS techniques can be classified

as blind and non-blind techniques. Non-blind approaches

such as Matched Filtering [6]-based and the cyclostationary

[7]-based sensing require the knowledge of PU signal pat-

terns and cyclic frequencies, respectively.

Typically, the blind approaches are based on either the

covariance matrix computations as in the Covariance

Absolute Value (CAV) approach [8] or the eigenvalue

computations as in the Maximum Minimum Eigenvalue

(MME) approach [9]. The test criterion of CAV assumes

Sådhanå (2021) 46:222 � Indian Academy of Sciences

https://doi.org/10.1007/s12046-021-01719-9Sadhana(0123456789().,-volV)FT3](0123456789().,-volV)

that the received signal is correlated, and hence compares

the diagonal entries of the covariance matrix to the non-

diagonal entries. The test criterion in the MME approach is

the ratio of the maximum to minimum eigenvalues. Many

other eigenvalue-based approaches have been discussed in

[9], which are variants of the MME approach. The thresh-

olds of signal detection in the CAV and MME approaches

are calculated by fixing the probability of false alarm.

However, the Information Theoretic Criterion (ITC)-based

approaches do not require any fixed probability of false

alarm. Two such approaches employ Akaike Information

Criterion (AIC) and Minimum Description Length (MDL)

criterion for signal detection [10], where the AIC (or MDL)

is computed using all the eigenvalues of the covariance

matrix. The corresponding test criterion compares the first

two AIC (or MDL) values to perform NSS. All these

approaches are suited for SUs with either single or multiple

antennas, and are also employable in a cognitive radio

network (CRN) for cooperative sensing [11].

Recently, mathematical notions of Grassmann manifolds

[12] opened up new possibilities in various science and

engineering applications. Basically, Grassmann manifold is

a space of subspaces that are embedded in a higher

dimensional vector space. Grassmann discriminant analysis

[13] encourages the data to be processed on the Grassmann

manifold. Grassmannian optimization finds its application

in the problem of low-rank matrix completion [14].

Grassmann kernels can also be studied for their replace-

ment in various applications such as traffic state estimation

[15]. Customized deep neural networks have also been

developed for Grassmannian data, referred to as Grassmann

Networks [16], which preserve the manifold geometry.

In wireless communication applications, Grassmannian

learning paradigms have been explored for automatic

recognition of space–time constellations in the design of

intelligent multiple input multiple output (MIMO) com-

munication systems [17]. A detailed survey on several other

applications of Grassmann learning is illustrated in [18].

The topological structure of Grassmann manifolds, and

corresponding distance measures that relate different points

on the manifold, can be employed in NSS to detect the

presence or absence of the PU in DTV bands [19]. In

particular, the dominant eigenspace, termed as Grassmann

covariance matrix (GCM) [20], is computed from the

covariance matrix of the received signal and also the PU’s

signal. A modified Binet–Cauchy distance measure has

been defined as a test statistic and the corresponding

threshold has been derived for a fixed probability of false

alarm, assuming that the covariance matrix of the PU’s

signal is available a priori.

In this paper, the distance measures on the Grassmann

manifold are further explored in the context of NSS.

Particularly, a modified projection distance measure is

proposed as a new test statistic and is analyzed for its

detection capability. Subsequently, we propose a GCM-

based modified NSS approach that relaxes the assumption

on the a priori availability of the PU’s signal covariance

matrix and validate its performance with the proposed test

statistic. The threshold is derived based on the probability

density function (pdf) of the test statistic under the null

hypothesis. Simulations are conducted for NSS using the

ED, CAV, MME, AIC, GCM [19] and the proposed GCM

approaches. The performances of these approaches are

evaluated using probability of detection (Pd) versus the

signal to noise ratio (SNR) and the Pd versus probability

of false alarm plots.

Experiments are conducted by receiving a UHF TV band

signal through the ADALM PLUTO software-defined radio

(SDR), which is transmitted by a distant transmitter

ADALM PLUTO SDR. The signal is captured with a suf-

ficiently good SNR and different noise conditions are

simulated using it. The performances of the previously

discussed NSS approaches have been compared using the

Pd versus SNR plots. By varying the transmit gain of the

ADALM PLUTO SDR, many frames of the received sig-

nals are extracted in the UHF TV band. The results of NSS

approaches on these signals demonstrate that the proposed

approach can yield a Pd performance comparable to that of

the [19], even without the a priori knowledge of the PU’s

signal covariance matrix.

The rest of the paper is organized as follows. Section 2

illustrates the adopted system model and the binary

hypothesis testing problem for signal detection. Further, the

necessary details of Grassmann manifolds are also dis-

cussed. A summary on a few of the existing NSS approa-

ches and the details of the proposed approach are discussed

in section 3 along with the analysis on proposed test

statistic. The simulation and the experimental results are

presented and discussed in section 4 and finally, the paper is

concluded in section 5.

Some of the notations adopted in the rest of the paper are

as follows. Bold faced capital letters represent matrices,

while bold faced small letters indicate vectors. The symbols

:ð ÞT and :ð ÞH represent transpose and Hermitian transpose.

gðxÞ� CN l; r2ð Þ indicates that g(x) is a complex Guassian

random variable with mean l and variance r2. E xf gdenotes the expectation of the random variable x and Pr(.)

represents the probability of an event. The ijth entry of a

matrix A is notated as Aij.

2. System model and Grasmann manifolds

2.1 System model

Consider a SU with a single sensing antenna. Let the dis-

crete time base-band signal received by the SU in a given

time frame be x(n) with a length N and the Nyquist sam-

pling interval be TNy ¼ 1fNy, where fNy is the Nyquist rate.

The signal detection forms a binary hypothesis testing

problem [8], represented as

222 Page 2 of 11 Sådhanå (2021) 46:222

H0 : xðnÞ ¼ eðnÞ n ¼ 0; 1; :::;N � 1

H1 : xðnÞ ¼ rðnÞ þ eðnÞ n ¼ 0; 1; :::;N � 1ð1Þ

with rðnÞ ¼ rðtÞgt¼nTNy; n ¼ 0; 1; :::; ðN � 1Þ being the

Nyquist sampled received signal that had undergone fading

due to the wireless channel and e(n) representing the inde-

pendent, identically distributed (i.i.d.) additive white Gaus-

sian noise (AWGN) samples, eðnÞ� CN 0; r2e� �

. Note that

r(n) can also represent the combined signal received from

multiple PUs. Usually r(n) is correlated owing to the filteringmechanism of the dispersive fading channel, filters and

amplifiers in the radio frequency (RF) chain and the over-

sampling of the signal [8]. For instance, to sense a typical

UHF TV band with 6 MHz bandwidth, the sampling rate is

close to 6 MHz while the actual PU such as the wireless

microphone uses a maximum bandwidth of 200 kHz. Thus,

the received signal is oversampled and hence correlated.

In (1) the null hypothesis H0 represents the absence of

PU’s signal, while H1 indicates the presence of the PU.

Employing a suitable NSS approach, the SU performs

signal detection as shown in figure 1. Signal detection is

accomplished via the decision criterion

TNSS ?H1

H0

kNSS ð2Þ

where TNSS is the test statistic employed in the considered

NSS approach, calculated using the received signal x(n), n 20;N � 1½ � and kNSS is the corresponding threshold. The

detection criterion (2) can be derived using the likelihood

ratio test such as the Neyman–Pearson test [21], by fixing the

probability of false alarm. In general, the performance of any

NSS approach can be assessed using Pd ¼PrðH1=PUpresentÞ versus SNR, for a predefined probability

of false alarm Pfa ¼ PrðH1=PUabsentÞ. Note that Pd

increases with SNR and also with the value of Pfa. An NSS

approach is a valid one if the plot of Pd versus Pfa lies above

the Pd ¼ Pfa line. Hence, both these plots are used as per-

formance measures to compare different NSS approaches.

2.2 Grassmann manifolds

The Grassmann manifold Gðn; kÞ; n� k[ 0 is the space of

all k-dimensional linear subspaces embedded in an n-di-mensional real or complex Euclidean space. Typically, any

element on the Grassmann manifold is represented by an

orthonormal matrix Y 2 Cn�k such that YTY ¼ Ik. Thus,Grassmann manifold [12] is the collection of all such Y,mathematically written as

Gðn; kÞ ¼ span Yð Þ : Y 2 Cn�k;YTY ¼ Ik� � ð3Þ

Note that an element in Gðn; kÞ is invariant to rotations. Thisimplies thatY andYD represent the same point on Gðn; kÞ forany D 2 OðkÞ, the set of all k � k orthonormal matrices. To

characterize the discrepancy between two subspaces (or

points)Y;Z ofGðn; kÞ, the notion of distance between the twoelements is required. Typically, this distance can be expres-

sed in terms of principal angles hi; i ¼ 1; 2; :::; k between

Y;Z [12] and these angles can be computed by performing

singular value decomposition (SVD) as in

YTZ ¼ U cosHð ÞVT ð4Þwhere U, V have the k left and right singular vectors,

respectively, and diagonal entries of cosH ¼diag cos h1; :::; cos hkð Þ are known as canonical correlations.

Some of the distance measures used on Grassmann mani-

folds are Binet–Cauchy distance dBC, projection distance dPand the Chordal distance dC [12], defined as

dBC ¼ 1�Yki¼1

cos2 hi

!1=2

ð5Þ

dP ¼ k �Xki¼1

cos2 hi

!1=2

ð6Þ

dC ¼ffiffiffi2

pk �

Xki¼1

cos hi

!1=2

ð7Þ

Note that the principal angles hi can also be defined as

cos hi ¼ maxui2Y;vi2Z

uTi vi ð8Þ

with ui and vi as the ith columns ofU andV, respectively [18].

The principal angles can be viewed as the minimal angles

between all possible bases of the subspaces of Y and Z.

3. Various NSS approaches and the proposedapproach

In this section, the NSS approaches introduced in section 1

are briefly discussed based on the system model in (1)

followed by the proposed approach.

3.1 Various NSS approaches

The received N-length signal frame xðnÞ; n 2 0;N � 1½ � isdivided into Nss subsegments, each of length L. EachFigure 1. Problem statement: NSS at the SU.

Sådhanå (2021) 46:222 Page 3 of 11 222

subsegment is arranged as xi; i 2 1;Nss½ � of dimension

L� 1. In all the NSS approaches discussed in section 1,

first L� L sample covariance matrix of the received signal

is obtained as

Rxx ¼ 1

Nss

XNss

i¼1

xixHi ð9Þ

3.1.1 CAV approach The CAV approach uses (9) to

calculate the detection criterion [8] as

PLp¼1

PLq¼1

Rxx;pq

�� ��PLp¼1

Rxx;pp

�� �� � kCAV ð10Þ

where Rxx;pq is the pqth element of Rxx and kCAV is the corre-

sponding threshold.To determinekCAV , themeanandvariance

of Rxx;pq are derived forH0 andH1 hypothesis. Using central

limit theorem, the numerator and denominator in the left hand

sideof (10) are found asGaussian distributions [8].ThekCAV isthen evaluated using the definition of Pfa as

kCAV ¼1þ ðL� 1Þ

ffiffiffiffiffiffiffi2

Nssp

q1� Q�1 Pfa

� � ffiffiffiffiffi2Nss

q ð11Þ

with QðaÞ ¼ 1ffiffiffiffi2p

pR1a

e�a2=2da. Observe that by fixing Pfa, the

threshold in (11) can be computed independent of the noise

or signal statistics.

3.1.2 MME approach In the MME approach, the

eigenvalues of (9) are first computed. The ratio of

maximum to minimum eigenvalues is used as the test

statistic and the corresponding decision criterion becomes

amax

amin

� kMME ð12Þ

with amax and amin being the maximum and minimum eigen-

values, respectively. The threshold kMME is derived in [9] as

kMME ¼ffiffiffiffiffiffiffiNss

p þ ffiffiffiL

pffiffiffiffiffiffiffiNss

p � ffiffiffiL

p� 2

1þ BF�11 1� Pfa

� �� � ð13Þ

where F1ðxÞ is the cumulative distribution function of the

Tracy–Widom distribution of order 1 and B ¼ffiffiffiffiffiNss

p þ ffiffiL

pð Þ�2=3

NssLð Þ1=6 .

Similar to that of the CAV approach, the threshold in (13) is

calculated for a fixed Pfa and is independent of the noise or

signal statistics.

3.1.3 AIC approach In the AIC approach, the Leigenvalues of (9) are used to calculate the AIC values

given by

AIC kð Þ ¼ h� 2 log

QLi¼kþ1

a1= L�kð Þi

1L�k

PLi¼kþ1

ai

0BBB@

1CCCA

Nss L�kð Þ

ð14Þ

where h ¼ 2k 2L� kð Þ þ 2, and ai; i 2 1; L½ � are the eigen-

values arranged in decreasing order. The detection proceeds

as

AICð1Þ?H1

H0

AICð2Þ ð15Þ

Note that the detection in (15) is independent of Pfa and no

a priori signal or noise statistics are required for its

computation.

3.1.4 GCM approach This approach assumes the a

priori availability of the PU signal covariance matrix Rss

for the case rðnÞ ¼ sðnÞ, where s(n) is the PU transmitted

signal. By SVD on Rss its r\L dominant singular vectors

are grouped to form an element in the complex space,

namely the GCM �S 2 CL�r, which represents a point on the

Grassmann manifold GðL; rÞ. Similarly, from the SVD of

(9), the GCM �X 2 CL�r is obtained by grouping r dominant

singular vectors of Rxx. The principal angles between �S and�X, obtained using (4), are used to obtain the test statistic q,a modified form of dBC given as

q ¼Yri¼1

cos2hi

!1=2

ð16Þ

The corresponding threshold is obtained using the pdf of qunder H0 as [19]

kGCM ¼ Q�1 Pfa

� �rq0 þ lq0 ð17Þ

where rq0 ; lq0 are, respectively, the standard deviation and

mean of the test statistic q under H0.

Note that this approach assumes that Rss is available a

priori. However, in a typical wireless fading scenario, the

system model (1) has rðnÞ ¼ sðnÞ~hðnÞ where h(n) is the

time varying channel impulse response. Hence, to employ

the GCM approach, h(n) should also be known in practice

or Rrr should be known.

3.2 Proposed approach

Unlike CAV, MME and AIC approaches the GCM

approach requires the a priori availability of Rrr 6¼ Rss,

which is a serious practical limitation. To overcome this

limitation, we propose a modified NSS approach based on

the GCM computation illustrated here. The proposed

approach exploits the fact that Rxx is a noisy version of the

unknown Rrr.

222 Page 4 of 11 Sådhanå (2021) 46:222

Consider the hypothesis H1. Starting from the actual

definition of Rxx, it can be observed that

Rxx ¼ E xixHi

� � ð18Þ

¼ E ri þ eið Þ ri þ eið ÞH� � ð19Þ

¼ E rirHi

� �þ E eieHi

� � ð20Þ

¼ Rrr þ r2eIL ð21Þwhere the third step follows from the fact that ri and ei areuncorrelated [22] and IL is the identity matrix of dimension

L. In a multi-path fading environment we can express

ri ¼ Hsi, where si is the L� 1 subsegment of the PU signal

and H represents the channel matrix [9], following which

Rrr ¼ HRssH�1. The requirement on the estimation of H

confirms that the assumption on a priori availability of Rrr

is impractical. However, (21) indicates that Rxx consists of

a noisy Rrr. Hence, rather than assuming the a priori

availability of Rrr , we propose to estimate it as

R̂rr ¼ Rxx � r2eIL ð22Þwhere noise variance r2e is assumed to be known, as in ED

approach [5]. Further, exploiting the Hermitian Toeplitz

property of Rrr , smoothing is performed to improve its

estimation, as illustrated here. Consider R̂rr of dimension

3� 3, calculated for an N-length received signal. Its

structure is of the form

R̂rr ¼rað0Þ rað1Þ rað2Þr�að1Þ rbð0Þ rbð1Þr�að2Þ r�bð1Þ rcð0Þ

264

375 ð23Þ

Note that the entries in (23) should satisfy Hermitian

Toeplitz property so that R̂rr becomes a valid covariance

matrix. Thus, we compute

rð0Þ ¼ rað0Þ þ rbð0Þ þ rcð0Þ3

ð24Þ

rð1Þ ¼ rað1Þ þ rbð1Þ þ r�að1Þ þ r�bð1Þ� ��4

ð25Þ

rð2Þ ¼ rað2Þ þ r�að2Þ� ��2

ð26Þ

to obtain the modified covariance matrix estimate as

�Rrr ¼rð0Þ rð1Þ rð2Þr�ð1Þ rð0Þ rð1Þr�ð2Þ r�ð1Þ rð0Þ

264

375 ð27Þ

We then propose to extract two GCMs, namely �Sest from

the SVD of �Rrr and �X from the SVD of Rxx, by retaining

r\L dominant singular vectors. As both the GCMs �Sest and

�X represent points on GðL; rÞ, the distance between them

can be measured using the metrics discussed in section 2.

The principal angles /i; i 2 1; r½ � between the two GCMs

are calculated, followed by the computation of the proposed

test statistic

Tprop ¼ 1

r

Xri¼1

cos/i

!ð28Þ

Note that (28) is a modified projection distance unlike the

modified Binet–Cauchy distance employed in [19].

Threshold computation is illustrated here. The pdf of Tprop

under H0 is assumed to be f ðTpropÞ�N mp; r2p

�, which is

later justified from experimental findings. From the

definition

Pfa ¼ PrðH1=PUabsentÞ ð29Þ

¼Z 1

kprop

f ðTpropÞdTprop ð30Þ

where kprop is the required threshold value. Fixing the value

of Pfa and simplifying (30), this threshold can be derived

[21] as

kprop ¼ Q�1 Pfa

� �rp þ mp ð31Þ

where rp;mp need to be obtained from the hypothesis H0.

This GCM-based modified NSS approach is summarized in

table 1. Note that though the proposed approach alleviates

the necessity of a priori Rrr it requires the values of r2e , rpand mp, thus making it a non-blind approach. However,

these values can be easily calculated from the pdf of the test

statistic on the null hypothesis H0. Thus, unlike [19], the

proposed approach does not require a priori Rrr .

Before discussing the simulation and experimental

results of NSS using the proposed approach, the statistical

nature of the proposed test statistic Tprop is worth analyzing.

For this purpose, we simulate a multi-tone analog signal

received from a Rayleigh fading channel and corrupt it with

AWGN of zero mean and known variance. Under the

hypothesis H0 and H1 (SNR of –5 and 0 dB) the plots of

Tprop obtained from 1000 iterations are shown in figure 2,

for different r values.The corresponding pdfs of Tprop under

H0 and H1 (SNR of –5 dB) are also shown in figure 3.

The plots for a fixed r value show that the variance of

Tprop underH0 is more when Rrr is unknown than when it is

known a priori. However, under H1, the variance in both

the cases is almost the same. This can be attributed to the

fact that when signal component is present in x(n), the

estimate of Rrr becomes better. The plots also show that, as

SNR increases, the gap between the mean of Tprop underH1

and that under H0 increases, which implies improved

detection performance at higher SNR values. The plots also

Sådhanå (2021) 46:222 Page 5 of 11 222

show that as r increases from 3 to 4, the distance between

the means of H0 and H1 decreases. It has been observed

during the simulations and the experiments that r ¼ 3 is a

good choice.

The proposed approach requires the estimate of mp; rpunder H0 to compute the threshold in (31). In the derivation

of (31), it is assumed that Tprop is Gaussian distributed on

H0. From figure 3, the simulations show that Tprop underH0

has approximately Gaussian distribution. To further vali-

date this fact, we generate a narrow-band multi-tone signal

of bandwidth less than 200 kHz and transmit the signal

using 550 MHz UHF TV band carrier from ADALM

PLUTO SDR.

The received signal is captured using another ADALM

PLUTO SDR having a bandwidth of 6 MHz. For the case of

known Rrr with r ¼ 3, the pdfs obtained from 1000 itera-

tions are compared with their theoretical distributions as

shown in figure 4. Similar results for the case of unknown

Rrr with r ¼ 3 are in figure 5. It can be observed that in

both the cases Tprop follows Gaussian pdf under H0. The

pdf is estimated using ksdensity function in Matlab. The

theoretical pdf is plotted using Gaussian pdf equation,

estimating the mean and variance of the Gaussian fit on the

Tprop values. Thus, the assumption that Tprop is Gaussian has

been validated.

Figure 2. Values of Tprop for H0 and H1.

Table 1. Proposed NSS approach.

GCM-based modified NSS approach

Input: xðnÞ; n 2 0;N � 1½ �, r2e , rp, mp

1. Construct xi; i 2 1;Nss½ � from the N-length input signal

2. Compute Rxx using (9) and obtain the GCM �X 2 RL�r from the SVD of Rxx.

3. Compute Rrr using (22) and calculate the GCM �Sest 2 RL�r from the SVD of �Rrr

4. Calculate the principal angles /i; i 2 1; r½ � between �Sest and �X.5. Obtain the test statistic Tprop in (28) using /i and the threshold kprop using (31).

Output: Decision using Tprop ?H1

H0

kprop

222 Page 6 of 11 Sådhanå (2021) 46:222

It will be convenient if the Pd performance can be

obtained using analytical equations. For this purpose, the

pdf of Tprop under H1 is necessary. Under both the cases

when Rrr is known and Rrr is unknown, the pdfs have been

identified as Beta pdfs as shown in figures 4 and 5,

respectively. These figures show the theoretical Beta pdfs

of Tprop for different SNR values. As SNR increases the

distance between H1 and H0 pdfs of Tprop increases, indi-

cating that the detection performance should improve.

Using the Beta pdf under H1, the Pd can be calculated as

Pd ¼ PrðH1=PUpresentÞ ð32Þ

¼ PrðTprop [ kpropÞ ð33Þ

¼Z 1

kprop

f Tprop� �

dTprop ð34Þ

¼ 1� Ikpropðp; qÞ ð35Þwhere Ikpropðp; qÞ is the incomplete Beta function with p, q

being the scale parameters of the Beta distribution [23].

Using all these analysis results, we now proceed to explore

the Pd performance of the proposed NSS approach.

4. Results and discussion

In this section, the simulation and experimental results are

discussed in detail. First, an analog received signal is

simulated with multiple sinusoidal frequencies of different

amplitudes; this passes through a Rayleigh fading channel

and is corrupted by a complex AWGN of zero mean and

Figure 3. The pdfs of Tprop for H0 and H1.

Figure 4. Estimated and theoretical pdfs for r ¼ 3, with known Rrr .

Sådhanå (2021) 46:222 Page 7 of 11 222

unit variance. The signal strength of the received signal is

varied with respect to the noise variance to simulate dif-

ferent SNR conditions. The Pd versus SNR curves are

extracted using CAV, AIC, MME, GCM [19] and the

GCM-based modified NSS approaches whose results are

shown in Figure 6. As discussed in section 3, in all the

simulations the GCM [19] approach assumes known Rrr

and the test statistic is q with r ¼ 2. The proposed approach

assumes no a priori knowledge of Rrr and the test statistic is

Tprop with r ¼ 3. Further, Nss ¼ 2000 and L ¼ 50 implying

N ¼ 10; 000 and Pfa ¼ 0:05.The theoretical Pd plots using (35) for both GCM and the

proposed modified GCM approaches are also shown in

figure 6, indicated as (Theo). The results show that the

proposed approach renders better Pd performance than the

CAV, AIC-based and the MME approaches. Further,

without a priori Rrr, the performance of the proposed

approach is at par with that of the GCM approach [19],

which assumes known Rrr. The Pd versus SNR plot with a

noise uncertainty (NU) of 2 dB is also shown in figure 6,

whose performance is close to that without the NU.

Second, an analog signal is generated with multiple

sinusoidal frequencies of different amplitudes and a maxi-

mum bandwidth of less than 200 kHz. It is modulated using

the 550 MHz carrier and is continuously transmitted into

the wireless channel using ADALM PLUTO SDR at a gain

of 0 dB. A distant ADALM PLUTO SDR receiver,

operating at an SNR of 20 dB and a sampling rate of 6

MHz, senses the channel and acquires the signal of length

N ¼ 10; 000. The spectrum of the received signal is shown

in figure 7.

The acquired signal is corrupted with AWGN and then

signal detection is performed using all these different NSS

Figure 5. Estimated and theoretical pdfs for r ¼ 3, with unknown Rrr .

Figure 6. Pd versus SNR plots for different NSS approaches.

222 Page 8 of 11 Sådhanå (2021) 46:222

approaches. The obtained Pd versus SNR plots are shown in

figure 8. These results show that the performance of the

proposed approach is at par with that of GCM approach

[19] even when Rrr is unknown. Further, the Region of

Operating Characteristic (ROC) curves namely the Pd

versus Pfa plots are shown in figure 9 for the GCM [19] and

the proposed approach. At SNR of –10 dB and also –20 dB

the plots lie above the Pd ¼ Pfa line, thus validating their

detection performances. At an SNR of –10 dB, the pro-

posed approach and the GCM approach [19] have negligi-

ble difference in the Pd. At –20 dB SNR, the proposed

approach has slightly lower Pd than the GCM approach

[19].

Third, to validate the performance of the proposed

approach experimentally, a UHF wireless link is set up as

shown in figure 10. The transmitter ADALM PLUTO is

programmed to continuously transmit a multi-frequency

sinusoid of maximum frequency less than 200 kHz, which

is modulated using a 550 MHz carrier signal for a desired

gain value. The receiver ADALM PLUTO is placed at a

distance of 2 m, which receives the 550 MHz signal at a

sampling rate of 6 MHz and has a fixed gain of 20 dB. The

gain of the transmitter is varied from –30 to 0 dB, in steps

of 5 dB.

At each gain setting the receiver captures Miter ¼ 430

frames of the received signal, where each frame x(n) has alength of N ¼ 10; 000. Signal detection is performed on

each frame using CAV, AIC, MME, GCM [19] and the

proposed GCM-based NSS approaches. The probability of

detection for each gain setting is then calculated as

Pd ¼ # TNSS [ kNSSð ÞMiter

ð36Þ

where # TNSS [ kNSSð Þ indicates the number of times

TNSS [ kNSS. The experimental results of Pd versus SNR are

shown in figure 11. These results are obtained with Pfa ¼0:05 and L ¼ 200. In the case of GCM [19] Rrr is pre-

computed, while lq0 , rq0 are estimated from H0. For the

proposed modified GCM approach the Rrr is assumed to be

unknown, while r2e , mp and rp are estimated from H0. The

transmitter is switched off to represent H0 and several

frames of the received signal are acquired, each with aFigure 7. Received spectrum acquired from ADALM PLUTO.

Figure 8. Simulated Pd versus SNR plots for acquired signal

with different NSS approaches.

Figure 9. Pd versus Pfa plots for SNR of –10 and –20 dB.

Figure 10. Setup of UHF wireless link.

Sådhanå (2021) 46:222 Page 9 of 11 222

length of N ¼ 10; 000. From this data, the complex noise

variance r2e is directly estimated; this is in turn used to

compute lTprop;0 and rTprop;0 , the average and standard devi-

ation of the test statistic Tprop, respectively, under H0, using

(28).

Note that though the proposed approach requires the

estimation of r2e similar to the conventional ED approach, it

can perform better than CAV, AIC and MME approaches.

On the other hand, though Rrr is not estimated, its perfor-

mance is at par with that of the GCM-based approach [19]

as seen from the simulated and experimental results.

Therefore, it can be recommended as a suitable approach

for NSS.

5. Conclusion

NSS approaches form a crucial part of cognitive radio sys-

tems, owing to their ability in detecting the presence or

absence of the licensed PU. In this work, a GCM-based

modified NSS approach has been proposed. Utilizing the

sample covariance matrix of the received signal and the

signal relationship, an estimation method is developed to

obtain the signal covariance matrix. A modified projection

distance measure is used as the test statistic and the

threshold is computed using estimated noise variance. The

advantage of the proposed approach lies in the fact that it

does not require the signal covariance matrix to be known a

priori. The performances of the proposed test statistic and

the proposed NSS approach are extracted from simulations

and experiments using signals acquired in the UHF TV band

from the ADALM PLUTO SDR. The results confirm that

the proposed approach has better Pd versus SNR perfor-

mance compared with a few other state-of-the-art such as

CAV, AIC and MME approaches. Further, its performance

is at par with that of the GCM approach; the latter uses

modified Binet–Cauchy distance as the test statistic and

requires the knowledge of a priori signal covariance matrix.

Acknowledgements

A part of this project is funded by Vision Group of Science

and Technology (VGST) Karnataka, India, under the RGS/

F scheme of 2017 bearing the GRD number 732.

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