# Detector based on the energy of filtered noise

## Abstract

This study deals with the detection of unknown signals in white noise. The authors present a new detector, based on the difference of a deterministic function of the energy of the signal and the energy of the same signal, which has been filtered. Unlike usual energy detector (ED), the proposed detector consists in exploiting the behaviour of the energy of filtered white noise, which can be a priori determined since the used filter is known. Thus, if the measured energy differs from an expected value, the detector decides that a signal is present in the band. In order to have the same asymptotic complexity as ED, a simple two-tap filter is used. The theoretical expressions of the probabilities of detection and false alarm are developed, and the optimal threshold is deduced. Simulations show that the proposed detector achieves better performance than ED, in both additive white Gaussian noise and Rayleigh channels. Furthermore, the relevance of the analytical results is proved through simulations.

## 1 Introduction

Energy detector (ED) is undoubtedly the most popular method for signal detection in noise, mainly due to its simplicity. Furthermore, it is a signal-agnostic approach, in the sense that ED does not require prior information about the signal to be detected, unlike match-filter [1], or cyclostationarity-based detectors [2]. Thus, ED is implemented in many applications, such as radar detection [3, 4], or in cognitive radio systems [5-8]. In the latter application, ED may be used by secondary users in order to detect the presence of primary users in licensed bands. Secondary users can then opportunistically access the detected free bands. A large number of other sensing techniques are available in the literature, as described in [5-8]. Among others, usual techniques are matched-filter [1], cyclostationarity detector [9, 10], and methods based on random matrix theory [11, 12]. However, in this paper, we will focus on energy-based detection.

The principle of ED is based on the binary decision, i.e. the energy of the scanned band is compared with a threshold [13, 14]: a signal is supposed to be present (resp. absent) in the band if the energy is higher (resp. lower) than the threshold. However, setting an accurate threshold value requires the prior knowledge of the noise energy or the signal-to-noise ratio (SNR). Therefore, the performance of ED is inherently limited due to the noise uncertainty [15-17]. In order to improve the performance of classical ED, several techniques have been proposed, such as described in [18] and references therein. Moreover, ED can be used in order to reduce the complexity of signal feature-based detectors. Thus, the authors in [19, 20] propose to exploit the benefit of both ED and second order moment-based detectors such as cyclostationarity and correlation detectors. In any case, the idea behind detection is to take advantage of the features of the signal (e.g. the energy, the shape, or redundancy of the signal) to decide if the signal is present or not.

In this paper, we examine the issue of energy detection with another paradigm, as we focus on the properties of the Gaussian white noise at the output of the receiver front-end, whereas the usual paradigm is to focus on the energy of received samples in the presence of ‘useful’ signal. Thus, the power spectral density (PSD) of white noise can be considered as constant over the whole frequency band that is sensed [13, 17]. Therefore, in the absence of signal, it is possible to deduce in advance the energy of the noise after any filtering process downstream of the front-end, since the used filter is known. Such a deterministic behaviour can then be used as a base for a detector. In fact, in the presence of a useful signal in the noise, the measured energy of the filtered received signal differs from the expected energy based on the assumption of white noise only. This allows us to decide if the signal is present in the band or not.

Based on the previous considerations, the basic principle of the proposed detector can be summarised as follows: (i) Measure the energy of the received sampled signal at the output of the front-end, and deduce the corresponding energy of the signal at the output of a predefined filter, based on the assumption that the received signal is only composed of white noise. (ii) Apply the filtering process to the received signal, and measure the actual energy of the filtered signal. (iii) Compare the expected value in (i) to the measured one in (ii): if the difference is larger than a given threshold, decide that the signal is present, if not, decide that the signal is absent. It must be noticed that the amplitude of the frequency response of the used filter must not be constant over the scanned band. Should this condition not meet, the energy of the filtered signal is the same as the original received signal, so no conclusion could be drawn.

In order to limit the complexity of the detector while meeting the previous condition, it is proposed to use the two-tap filter . Thus, the samples of the filtered signal are obtained by a simple linear interpolation of the input samples. This operation does not involve any multiplication, as a division by two is obtained by a binary shift, which limits the complexity of the filtering process. Furthermore, it can be noted that *g* is a low-pass filter, which guarantees that the PSD of the filtered signal is not the same as the received signal. In order to improve the performance of the presented detector, it is proposed to carry out several times the filtering process, in order to obtain different reference measures, which can be compared with the corresponding expected energy values.

It is worth mentioning that, since the proposed algorithm is based on energy measurement, the closest related detector is the usual ED. For this reason, we use the very common assumption that the noise at the output of the front-end is white and Gaussian, such as supposed in [13, 14, 17] and numerous other papers. Furthermore, we will use the ED as a natural reference for performance comparison. Otherwise, the filtering processes involved in the proposed method are similar to a first-order approximation of the derivative of the received signal. Then, the suggested algorithm could be confused with the method in [21, 22], in which the derivative of the spectrum of the received signal is considered, in order to highlight discontinuities in the presence of a useful signal. However, the filtering processes of the proposed technique are carried out in the time domain, and in the frequency domain in [21, 22]. Moreover, the processes are iterative, and then not similar to a derivative in the time domain.

- To the best of the author's knowledge, the proposed detector is original, as the paradigm differs from usual ED since the method is based on the behaviour of the filtered noise in the absence of useful signal. Furthermore, a simple implementation is suggested.
- The theoretical aspects of the detector are analysed as well, as the theoretical false alarm probability, detection probability, and optimal detection threshold expressions are derived.
- The developments are supported by simulations, which show the relevance of the proposed analysis. In addition, a performance comparison with ED reveals that the proposed detector outperforms ED in SNR range of −20−0 dB.
- Other possible applications of the method are discussed.

The remaining of the paper is organised as follows: Section 2 presents the signal model, and Section 3 describes the proposed detector. The analytical expressions of the probabilities of detection, false alarm, as well as the optimal threshold value, are developed in Section 4. Simulations results are presented in Section 5, and other applications of the detector are introduced in Section 6. Finally, Section 7 concludes this paper.

## 2 Signal model and hypothesis

*n*th sample of a received signal at the output of the receiver front-end, including analogue low-pass filter, analogue-to-digital converter, sub-sampling and so on. Thus, can be expressed as

*n*th sample of noise, which is reasonably assumed to be complex and white Gaussian [13, 14, 17]. Under the hypothesis , the received signal can be expressed with the general formulation as follows:

*M*the size of the DFT, then the

*m*th frequency sample, for any , can be expressed as

## 3 Proposed detector

The basic idea behind ED is to suppose that the measured energy , where is the Euclidian norm, is statistically higher under hypothesis than under hypothesis . If is larger than a given threshold, is decided. Then, it remains to set the best possible threshold value, in order to maximise the probability of detection. The proposed detector also uses the energy of the received signal but is based on another paradigm. Under hypothesis , one can deduce the energy of any version of filtered noise , since both the PSD of the noise and the used filter are supposed to be known a priori. As a consequence, it is possible to decide by comparing the expected energy to that of the actual filtered noise. More detailed is given hereafter.

### 3.1 General principle

- A ‘deterministic’ energy value (this value, denoted by , will be defined hereafter), which is obtained from . It is referred to as ‘deterministic’ as is deduced from through a predefined process.
- The energy value of a filtered signal, the samples of which are obtained by means of an iterative filtering process with input .

It will be shown afterward that the predefined ‘deterministic’ process in Fig. 1 only depends on the filter used in the iterative process.

Based on the previous description, suppose hypothesis , then the energy of the noise after the iterative filtering process is deterministic (as the filter is known). This is illustrated in Fig. 2 a, where both the PSD of the noise and the used filter are a priori known. Furthermore, it is close to the energy value , by the construction of the deterministic process. As a consequence, their difference should be close to zero.

In hypothesis , the energy of the filtered signal cannot be a priori determined, since the different features of the signal (shape, frequency, energy etc.) are unknown. This is illustrated in Fig. 2 b, where the cases ‘Signal A’ and ‘Signal B’ lead to very different measures. Therefore, unlike , it is likely that the difference between and the energy of the filtered signal largely differs from zero. This allows us to define a detector based on the energy of filtered noise.

It must be emphasised that any filter featuring a non-constant frequency response can be used in the detector. However, in order to obtain a detector with a complexity similar to ED, we propose to use the filter defined as . The resulting signal samples are simply defined as the linear interpolation of two consecutive input samples. Note that the number of iterative filtering processes allows us to obtain a different equivalent filter with their own cutoff frequencies, such as described in the next section. Furthermore, such a filter only involves additions, since the division by two is obtained by a binary shift. Therefore, we deduce that the complexity of the proposed detector is only twice higher than that of ED, in terms of complex multiplications, which is asymptotically negligible. In the following, we provide the expression and the properties of such a signal obtained from iterative linear interpolations using , and then we describe the hypothesis test.

### 3.2 Iterative linear interpolations

*k*the number of iterations of the filtering process, and the vector containing the output samples . Thus, is the vector containing the

*N*successive samples of the received signal , and is the vector containing the interpolated samples , where

*G*the DFT of

*g*such as previously defined. For convenience, we use the time-continuous version of

*g*. Then, the

*M*-point DFT of such a filter can be expressed as

*k*iterations is

*k*is large

*k*-interpolated noise can be entirely deduced from .

### 3.3 Hypothesis test

## 4 Deriving the false alarm and detection probabilities

### 4.1 False alarm probability

### 4.2 Detection probability

#### 4.2.1 Detection probability for deterministic signal

#### 4.2.2 Detection probability for Gaussian random signal

### 4.3 Threshold value

*M*. This particularity is similar to the ED. The specificity of the proposed detector is that the value also depends on the number of interpolations, which appears in the expression of .

## 5 Simulations results

- :
*N*multiplications and*N*additions. - :
*M*multiplication, since the multiplications by in the filtering processes are ‘costless’ in term of complexity and additions.

Then, since we approximate , hence the computation cost of in (14) is multiplications and additions. The complexity of ED is *N* multiplications and *N* additions, then we deduce that the proposed detector is of the same order of complexity as that of ED.

### 5.1 Validation of theoretical developments

Fig. 3 depicts the value versus the false alarm probability. The results obtained through simulation are compared with those obtained with (34), for different SNR values (−20 to −5 dB). Furthermore, samples are used, and iterations. It can be clearly observed that theoretical results match with simulations, which validates the previous developments.

### 5.2 Receiver operating characteristic (ROC) performance

The performance of the proposed detector is analysed through the ROC and is compared with the ED. Fig. 4 shows the spectrum of the received signal in the absence and in the presence of the noise, for an SNR of −10 dB (in all simulations, the signal energy has been normalised). The considered signal in Fig. 4 is orthogonal frequency-division multiplexing (OFDM) with 1000 subcarriers of 1 kHz each. It can be seen that the signal is concealed in noise.

The first series of simulation shows the ROC performance of the detector, considering the previous OFDM signal, in the absence of channel. In all the simulations, the proposed detector is used with ten iterations. The central frequency of the signal has been randomly (uniformly) chosen in the interval [500, 9500] kHz. Fig. 5 depicts the ROC performance of the proposed detector compared with ED, by using samples. Figs. 5 a–c correspond to SNRs of −10, −15, and −20 dB, respectively. It can be observed that the proposed detector outperforms the ED for SNR = −10 and −15 dB, and both achieves almost the same performance at SNR = −20 dB.

Other series of simulations have been carried out in order to compare the proposed method to ED. Results are presented in Fig. 6. The same observations as previously can be drawn from Fig. 6 a, where : the proposed detector outperforms the ED at both SNR = −15 and −20 dB. Obviously, both detectors achieve better performance in Fig. 6 a than in Fig. 5, due to the larger number of samples.

In Fig. 6 b, we compare the performance of the proposed detector with the ED in the presence of a Rayleigh channel. The SNR is set to −15 dB. It can be observed that the proposed detector still outperforms the ED. Moreover, both of them achieve a slightly weaker performance than in AWGN, due to the presence of the channel. Then, the simulations results show that, with similar computational cost, the proposed detector outperforms ED.

Fig. 6 c shows the ROC performance of the proposed detector for different kinds of signals: sinusoid, OFDM, and chirp. The SNR has been set to −15 dB and the sensing duration corresponds to samples. The detector performs better when applied to sinusoid than OFDM, and when applied to OFDM than chirp. This result is mainly due to the bandwidth of the different signals: the narrowest the signal bandwidth, the better the performance of the detector.

## 6 Other possible applications

### 6.1 Estimation of the parameters and

It has been aforementioned that the multiple interpolations act like a low-pass filter. Under several conditions, it becomes possible to estimate the parameters and in the presence of a useful signal . In fact, it must be pointed out that the largest *k*, the lowest the ‘cutoff frequency’ of . Therefore, according to , and assuming that , it may happen that for a given , we have (when is lower than the cutoff frequency of ), and for , we have (when is much larger than the cutoff frequency of ).

As a consequence of the above remark, it exists and , such as . Note that has no physical sense since it may be not an integer. However, it is possible, mathematically speaking, to extend from integer to real domain. In practice, corresponds to the point where the linearly-interpolated trajectories of and intersect.

- The signal is narrow-band with .
- As a consequence, we suppose that (
*C*is a constant) for any*q*in . - The values have been computed for every
*k*value, in such a way that has been estimated. *N*is large enough to consider that the approximationholds.

### 6.2 Detection of coloured noise

## 7 Conclusion

In this paper, we have presented a new detector of narrowband signals in noise, based on the difference of a deterministic function of the energy of the signal and the energy of the filtered signal. Unlike ED, the proposed detector consists in exploiting the behaviour of the energy of filtered white noise, which can be a priori determined since the used filter is known. Thus, if the measured energy differs from an expected value, it is decided that the signal is present in the band. In order to reduce the complexity of the method, it has been proposed to use a simple two-tap filter. The false alarm and detection probabilities expressions have been derived, as well as the optimal threshold value. Furthermore, theoretical results have been verified through simulations. It has been shown that the new detector outperforms the usual ED. Finally, two other possible applications of the detector have been presented. Future work will consist in investigating these pending issues, and to analyse the inherent limits of the proposed detector due to noise uncertainty.

## 9 Appendix

*m*. It is straightforward to show that

*M*tends to the infinity.

Proof.Let be a function of class on the interval , where are finite numbers. Let and two piecewise linear functions on such as, , we have

Fig. 8 illustrates the above definition of compared with . In this example is the function.

*p*is of class on .

*M*, which correspond to the regularly sampled versions of and

*p*on , respectively. For a sake of simplicity, but without loss of generality, we suppose that any element of is positive. As a consequence, for any , we can write

- (from (49)),
- Since , then , hence it can be straightforwardly shown that

As a consequence of the above remarks, we deduce that, for any , it is ever possible to find , if or if such that . Therefore, it exists a vector containing the elements , , such that (51) holds.

*M*is large, then from (50)

*M*first terms in is equivalent to . The term is the sum of and. The same reasoning leads to

*M*first terms in is equivalent to , and is a constant independent of

*M*. Therefore, we deduce that, when

*M*is large enough