Overcoming the SNR-Wall for Energy Detection Using the Autocorrelation of Cyclic Prefix Based OFDM Signal

Sandip Karar

Institute of Radio Physics and Electronics, University of Calcutta,

92, Acharya Prafulla Chandra Road, Kolkata 700009, India

Email: sk7632@gmail.com

Abhirup Das Barman Institute of Radio Physics and Electronics,

University of Calcutta, 92, Acharya Prafulla Chandra Road,

Kolkata 700009, India Email: abhirup_rp@yahoo.com

Abstract—Energy detection is one of the basic schemes of spectrum sensing. In case of energy detection because of the uncertainty in estimating the noise variance there exists a fundamental limit of minimum SNR, called the SNR-wall, below which the signal could not be reliably detected. In this paper, we propose a method to overcome this SNR-wall for OFDM signal by exploiting the autocorrelation property of the cyclic prefix both in AWGN and multipath fading channel. Required SNR for signal detection versus noise uncertainty for a given false alarm probability (0.01) and detection probability (0.99) has been found out in case of both energy detector and the autocorrelation based detector for OFDM signal. Our result shows that there is about 23 dB advantage with our proposed method for AWGN channel and about 12 dB advantage for multipath fading channel over conventional energy detector for noise uncertainty parameter U = 5 with ~ 5,000 signal samples.

I. INTRODUCTION Spectrum is a natural resource the use of which is licensed

by government organizations. In conventional fixed allocation policy, the licensed spectrum is intended to be used only by the spectrum owners. This policy leads to low spectrum utilization in many of the frequency bands. With rapid growth of wireless applications now-a-days, the problem of spectrum utilization has become much more critical. As a promising solution cognitive radio (CR) technology has been introduced that reuses the licensed spectrum of the primary user (licensed user) in an opportunistic manner. The IEEE 802.22 is the first international standard for wireless regional area network (WRAN) which aimed at using the cognitive radio technology for sharing the spectrum allocated to TV broadcasting services. The first task of a cognitive radio is to sense and identify the unused spectrum holes. One of the basic sensing schemes is the energy detector [1]. This detector performance is optimal if both signal and noise are white Gaussian and most importantly the exact noise variance is known a priori. But it is impossible to estimate the exact noise variance from a finite number of noise samples. So some uncertainty is always there in the estimate of the noise variance. This uncertainty in noise variance affects the detector performance. When the noise variance is perfectly known the detector can detect the presence of the signal at arbitrary low signal-to-noise ratio (SNR) by increasing the sensing time. But in case when there is an uncertainty in noise variance, there exists a fundamental

limit of minimum SNR, known as the SNR-wall, below which the signal could not be reliably detected no matter how much one can increase the sensing time. Significant research works are going on about the issues on noise uncertainty and SNR-wall in energy detection [2-7]. For IEEE 802.22 standard the detection limit of the SNR is –15 dB which is far below the SNR-wall. In paper [8] the authors have presented a method to reduce the SNR-wall for energy detection by using the cross-correlation of the signals received by multiple receivers. In this paper we propose a method using the autocorrelation property of a cyclic prefix based OFDM signal to overcome the SNR-wall for signal detection. The reason we choose OFDM signal is that many modern technologies in wireless communication such as DVB-T, WiFi, WiMAX, 3GPP Long Term Evolution (LTE) etc. use OFDM signaling. In order to study the detection performance of the OFDM signal we first assume that the channel is AWGN or under slow fading and then we will consider the multipath fading channel. The rest of the paper is organized as follows: Section II gives the analysis of energy detection performance considering an uncertainty in noise variance which gives rise to SNR-wall. In Section III we propose a method to overcome the SNR-wall using the autocorrelation property of an OFDM signal. Simulation results are given in Section IV and conclusions are drawn in Section V.

II. FUNDAMENTAL LIMITS ON ENERGY DETECTION AND SNR-WALL

For energy detector, for a desired false alarm probability , , the threshold can be written as [2]:

, (1)

where √ . The term is the noise variance which is assumed to be i.i.d. zero-mean complex Gaussian. The signal is also assumed to be zero-mean complex Gaussian. Let P be the received signal power at the receiver then the signal-to-noise ratio is given by SNR / . For this threshold the received signal power P (or SNR) can be adjusted to achieve a desired detection probability , . From the expressions of , and , we can write [3]:

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, , (2)

For given target probabilities , and , , the number of received samples and the SNR must satisfy the above equation. From the equation it is clear that as SNR ⟶ 0, N ⟶ ∞. So the signal could be detected for arbitrarily low SNR by increasing the sensing time i.e. increasing the number of samples N. Let us now consider that the noise variance estimated from finite number of noise samples is not perfect. Let be the estimated noise variance. The error in estimating noise variance is bounded by [3]:

1 1 (3) with 0 1 and 0. As shown in [3] the worst case of detection probability occurs for the entire set of noise uncertainty for the threshold 0 = , where U is the peak-to-peak uncertainty and is given by . We can guarantee that , over the entire set of noise uncertainty values by setting this worst case detection probability equal to the desired detection probability , .

, (4)

Using (1) and (4) and approximating (1 + SNR) ≈ 1, we obtain

, , (5)

From the equation it is clear that N ⟶ ∞ as SNR ⟶ U – 1. So the signal could be detected robustly for SNR < U – 1 even by increasing the number of samples N. This lower limit of the SNR is called the SNR-Wall.

III. OFDM SIGNAL AND OVERCOMING THE SNR-WALL

Fig. 1. Block diagram of an OFDM transmitter

The Fig. 1 shows a simplified OFDM transmitter. The block of symbol constellations S(0), S(1), …, S( 1) are first converted into parallel symbols and then they are modulated by different subcarriers by passing those symbols through an IFFT function block. The IFFT yields the sequences s(0), s(1), …, s( 1), where

∑ , 0 (6)

After IFFT, the signal is appended by a sequence of samples at the beginning of each block. These samples, called

the cyclic prefix, consist of the last samples sequence. This yields a new sequence of length . Then it is converted back to serial stream of samples and transmitted. The block of samples , … , 1 , 0 , 1 , 2 … 1

at the output of the transmitter is called an OFDM symbol. Without the loss of generality we can change the coordinate points of the OFDM symbol and represent the transmitted samples by x(0), x(1), …, x( 1). The received samples are given by:

(7) 0, 1, 2, … 1

Where h is the path loss and is the additive white Gaussian noise with zero mean and variance . We take N number of consecutive OFDM symbols so that the received signal samples constitute a vector:

0 , 1 , 2 … , 2 , 1 (8)

Here it is assumed that the receiver is perfectly synchronized i.e. the receiver knows exactly the starting time of each OFDM symbol. Now we define,

∑ (9)

i = 0, 1, 2, … 1

Here denotes autocorrelation at time lag . For large N, by using central limit theorem it can be shown [9]:

| ~ 0, (10a)

| ~ , (10b)

for i = 0, 1, 2, … 1. Here | | is the received power at the receiver. means real value of . Now we define R as the average of over i = 0, 1, 2, … 1 i.e. in the cyclic prefix region.

∑ (11)

Then, for large N,

| ~ 0, (12a)

| ~ , (12b)

We take this Re( ) as the test statistic in this case. Then the target probability of false alarm is given by,

, (13)

And desired probability of detection is given by

,Eliminating from (9) and (10) and approsmall SNR we get

, ,.From the equation it is clear that as SNR ⟶this overcomes the SNR-wall and the signal robustly for any arbitrarily low SNR bnumber of samples N. The result also depeprefix length . The greater the value of of samples required for detection. In case when the receiver is not synchrowhen the receiver does not know the startOFDM symbol we take 1 consecutat the receiver. These samples include at lfull OFDM frames. Generally the first sstarting point of an OFDM frame. Hence threceiver is to estimate that starting point of th

Fig. 2. Positions of N number of OFDM frsensing interval of 1 sam

Let be the estimate of this timing offsecalculated by solving the following maximiz

argmax ∑

So, when the receiver is not synchronized th∑ . This test statistic fodistribution as before.

So far we have considered the channel asus consider the OFDM signal from the primreceived through a multipath fading channel.

Fig. 3. Overlapping of symbols received froin a multipath channel.

Let the discrete-time channel impulse reby , , … , where L is the numcomponent. We assume the cyclic prefix len

(14)

oximating for very

(15) ⟶ 0, N ⟶ ∞. So could be detected

by increasing the ends on the cyclic the less number

onized, that means ting point of each ive signal samples east N number of ample is not the

he first task of the he first frame.

rames within the mples.

et. This can be ation problem:

(16)

he test statistics is ollows the same

s AWGN. Now let mary transmitter is .

om different paths

esponse is given mber of multipath ngth of the OFDM

signal is larger than the length i.e. . The received signgiven by

∑Because of multipath fadingparticular OFDM symbol as different paths tend to overlap Fig. 3. But there is still a distance for some part ofprefix property. We take consecutive samples from the rfollowing:

∑We take Re(T) as the test statisby using the central limit theore

| ~ | ~where ∑ | | is thethese results, for a particular can derive the following in the

where prefix length over one OFDMalso we see as SNR ⟶ 0, N ⟶

IV. SIMULATION

In this section, we compareconventional energy detector. required SNR in dB versus thfor detection for different sets set of false alarm and detectio, 1 10 of a convcan be found from equation (5that for U = 1 any arbitrary pairreached by changing the sensinfor all values of SNR but asymptotic behavior convergiBeyond that limit of SNR, reliably. In Fig. 5, the same gproposed method for OFDM sequation (15). We selespecifications 512 and > 1, the curves do not show apair ( , , , , signal ofby increasing the sensing timeOFDM signal with the samemultipath fading scenario. Wegraph. In this case also there is

of the channel impulse response nal at the secondary receiver is

(17)

g it is difficult to identify a the symbols travelling along

with each other as shown in the correlation among samples at

f the signal because of cyclic any arbitrary K number of received signal and compute the

(18)

stic in this case. It can be shown em for large K: ~ 0, (19a)

, (19b)

e received signal power. From U and ( , , , ) pair we same way:

, ,. (20)

is the ratio of the cyclic M symbol duration. In this case ⟶ ∞. NS AND DISCUSSIONS e the proposed method with the Fig. 4 illustrates the graphs of e number of samples N needed of values of U to reach a given n probabilities ( , 10 ,

ventional energy detector which 5). From the curves it is evident r ( , , , can always be

ng time (i.e. number of samples) for U > 1, the curves show ing toward a minimum limit. the signal cannot be detected graph has been plotted for our signal in AWGN channel using ct the OFDM parameter 60. In this case even for U

asymptotic behavior. So for any f arbitrary SNR can be detected . Fig. 6 shows the same plot for

e specification before but with e use equation (20) to plot this s no asymptotic behavior of the

curves for U > 1 but the required SNR is grAWGN channel for same number of signparticular U and ( , , , pair. Fig. 7 shows the plot of the required Swith respect to peak-to-peak noise uncertaintdifferent values on signal samples N in tdetection with same target probabilities increase in the noise uncertainty the requsignal increases rapidly. On the other handmethod for OFDM signal in AWGN channeldoes not increase so rapidly with the parameter as evident in Fig. 8. For U = 5 thapproximately – 17 dB for same signal samto 6 dB for conventional energy detection. same plot only in case of multipath fadingcase the required SNR is approximately – 6 for same signal samples. From the simulationunder noise uncertainty the performance energy detection deteriorates drastically overcome using our proposed method for OF The performance of this method also depof the cyclic prefix of each OFDM symbthe plot of required SNR of the signal versulength in AWGN channel where the total nsymbol is N = 20 and length of each fraconsidering the noise uncertainty U as parashows the plot of required SNR of the signal is the ratio of the cyclic prefix length over olength for 5,000 signal samples in case of mu

Fig. 4. Required SNR in dB versus no. of samples N , 1 10 for different values noise uncedetection.

Number of samples

reater compared to nal samples for a

SNR of the signal ty parameter U for traditional energy as before. With

uired SNR of the d for our proposed l the required SNR noise uncertainty

he required SNR is mples as compared

Fig. 9 shows the g scenario. In this

dB for U = 5 and n graphs it is clear of the traditional

which can be FDM signal. pends on the length bol. Fig. 10 shows us the cyclic prefix number of OFDM ame is 512

ameter and Fig. 11 versus where ne OFDM symbol ultipath fading.

for , 10 and ertainty U in energy

Fig. 5. Required SNR in dB versus no.and , 1 10 for different proposed method using the autocorrelain AWGN channel.

Fig. 6. Required SNR in dB versus no, 1 10 for different valuesmethod using the autocorrelation of multipath fading channel.

Fig. 7. Required SNR in dB versus noi, 1 10 for different valueenergy detection.

Noise U

Number of

s

Numb

of samples (512×N) for , 10 values of noise uncertainty U in our

ation of OFDM signal (Nc = 60, Ns =512)

. of samples (K) for , 10 and of noise uncertainty U in our proposed OFDM signal (Nc = 60, Ns =512) in

ise uncertainty U for , 10 and s of the received signal samples N in

Uncertainty U

f samples (512 x N)

er of samples

Fig. 8. Required SNR in dB versus noise uncertainty U , 1 10 for different values of the OFDM framethod using the autocorrelation of OFDM signal (AWGN channel.

Fig. 9. Required SNR in dB versus noise uncertainty U , 1 10 for different values of the OFDM framethod using the autocorrelation of OFDM signal (multipath fading channel.

Fig. 10. Required SNR in dB versus cyclic prefix l10 and , 1 10 for different values of noimethod of using the autocorrelation of OFDM signal in the total number of OFDM symbol is N = 20 and 5

Noise Uncertainty U

Cyclic Prefix Length N

Noise Uncertainty U

for , 10 and ames N in our proposed (Nc = 60, Ns =512) in

for , 10 and ames N in our proposed (Nc = 60, Ns =512) in

length for ,ise uncertainty U in the AWGN channel. Here 512.

Fig. 11. Required SNR in dB versus 10 for different values of noise uncautocorrelation of OFDM signal in mnumber of samples is K = 5000;

V. CO

In this work, we show detection can be avoided by property of a cyclic prefix baseand frequency selective fading the conventional energy detechave also been shown and therthe autocorrelation based detedetector for OFDM signal.

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