Volume 13, Issue 1 p. 36-45
Research Article
Free Access

Detector based on the energy of filtered noise

Vincent Savaux

Corresponding Author

Vincent Savaux

Network Interfaces Lab, IRT b-com, Rennes, France

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First published: 01 February 2019
Citations: 2

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 urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0001. 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.

The contributions of the paper are the following:
  • 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 urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0002 −20−0urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0003 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

We consider that a sensor is scanning and sampling a frequency band with a sampling time urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0004. Let urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0005 be the 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, urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0006 can be expressed as
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0007(1)
where urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0008 is the received analogue signal, urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0009 is the Dirac delta function, and urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0010 is the (rectangular) observation window, with urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0011. We consider the usual binary hypothesis used in signal detection, which can be written as
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0012(2)
where urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0013 corresponds to the absence of ‘useful’ signal, and urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0014 is the n th sample of noise, which is reasonably assumed to be complex and white Gaussian urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0015 [13, 14, 17]. Under the hypothesis urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0016, the received signal urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0017 can be expressed with the general formulation as follows:
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0018(3)
where urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0019 is the transmitted signal of central frequency urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0020 and bandwidth urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0021, and urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0022 is the multipath propagation channel of length urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0023. It is assumed than the signal is narrowband compared with the sensed band, namely urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0024. The term urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0025 points out a possible frequency offset, due to channel frequency offset, and/or Doppler effect for instance. Moreover, urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0026 is an unknown phase shift. Alternatively, the received signal urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0027 in (3) can be rewritten in the frequency domain by means of the discrete Fourier transform (DFT). Let denote by M the size of the DFT, then the m th frequency sample, for any urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0028, can be expressed as
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0029(4)
where the term urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0030 is due to the rectangular window urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0031. The frequency shift urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0032 is due to the convolution by urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0033. Note that we do not provide any detail on the nature of the signal urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0034, in order to propose a general formulation of the detector, which could be used for any kind of signal. However, we define urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0035 the variance of the signal urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0036. Therefore, the energy of the received signal urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0037 can be defined according to the hypothesis as
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0038(5)
In the following section, we describe the proposed detector.

3 Proposed detector

The basic idea behind ED is to suppose that the measured energy urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0039, where urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0040 is the Euclidian norm, is statistically higher under hypothesis urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0041 than under hypothesis urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0042. If urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0043 is larger than a given threshold, urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0044 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 urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0045, one can deduce the energy of any version of filtered noise urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0046, 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 urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0047 by comparing the expected energy to that of the actual filtered noise. More detailed is given hereafter.

3.1 General principle

Fig. 1 depicts the general principle of the proposed detector, compared with the usual ED. The basic idea of the suggested algorithm is to compare two energy values:
  1. A ‘deterministic’ energy value (this value, denoted by urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0048, will be defined hereafter), which is obtained from urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0049. It is referred to as ‘deterministic’ as urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0050 is deduced from urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0051 through a predefined process.
  2. The energy value of a filtered signal, the samples of which are obtained by means of an iterative filtering process with input urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0052.

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

Details are in the caption following the image

General principle of the proposed detector, compared with ED

Based on the previous description, suppose hypothesis urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0053, then the energy of the noise urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0054 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 urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0055, by the construction of the deterministic process. As a consequence, their difference should be close to zero.

Details are in the caption following the image

Illustration of the proposed detector (a) in the absence of signal, (b) in the presence of signal

(a) Under hypothesis urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0056, the resulting energy is a priori known and (b) Under hypothesis urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0057, the resulting energy cannot be a priori known

In hypothesis urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0058, 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 urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0059, it is likely that the difference between urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0060 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 urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0061. 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 urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0062, and then we describe the hypothesis test.

3.2 Iterative linear interpolations

We denote by k the number of iterations of the filtering process, and urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0063 the vector containing the output samples urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0064. Thus, urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0065 is the urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0066 vector containing the N successive samples of the received signal urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0067, and urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0068 is the urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0069 vector containing the interpolated samples urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0070, where
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0071(6)
More generally, for any urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0072 (in practice, we will limit to urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0073) and urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0074, we have
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0075(7)
The energy of the multi-interpolated signal urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0076 is defined as urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0077, where urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0078 is the mathematical expectation. Under the hypothesis urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0079, the expression of this energy cannot be derived, since both the nature and the features of the signal are unknown (we assume a general case where urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0080 can be of any kind). However, under urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0081, the energy can be developed, by using the fact that the noise samples are zero-mean and independent, as follows:
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0082(8)
Alternatively, we can find urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0083 by using the frequency response of the filter. We denote by 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
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0084(9)
and therefore, the filter corresponding to k iterations is
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0085(10)
From (10), we deduce that
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0086(11)
which provides in passing a demonstration of the Wallis’ integral for even orders. In fact, by using the Vandermonde's identity urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0087, we trivially found
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0088(12)
The asymptotic expansion of the factorial, urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0089 leads to the following approximation of urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0090 under hypothesis urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0091 when k is large
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0092(13)
From (8) and (13), we can deduce that, under the hypothesis urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0093, the energy of the k -interpolated noise can be entirely deduced from urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0094.

3.3 Hypothesis test

Based on the previous developments, the test statistic can be expressed as
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0095(14)
where urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0096 is previously defined. Then, the usual binary decision rule is
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0097(15)
where urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0098 is a threshold to be fixed, as shown in the next section. Under the hypothesis urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0099, urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0100 can be rewritten by using the expression of the multi-interpolated signal in the frequency domain (4). Thus, since the DFT size is urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0101, we obtain
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0102(16)
In addition, we assume that urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0103 is large enough to obtain the following approximation:
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0104(17)
since the noise is zero-mean. Under the hypothesis urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0105, (17) simply becomes
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0106(18)
without any approximation.

4 Deriving the false alarm and detection probabilities

4.1 False alarm probability

In this section, we have urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0107 since the useful signal is absent. By definition of (8), (14) can be rewritten as
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0108(19)
Let urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0109 be the variable defined as
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0110(20)
By using the Lyapunov condition for urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0111, it can be proved that urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0112 has a Gaussian distribution (a detailed proof is provided in the Appendix), the mean urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0113 and the variance urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0114 of which are
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0115(21)
and
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0116(22)
Note that the bounds in the sum have been omitted for more readability. Since the frequency noise samples urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0117 are independent and identically distributed (iid), it can be noticed that
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0118(23)
As a consequence, (22) can be simplified as
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0119(24)
Finally, since urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0120, then urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0121 obeys a Chi distribution with 1 degree of freedom, which can be expressed as
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0122(25)
From (25), we deduce the expression of the false alarm probability urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0123 :
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0124(26)
where urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0125 is a threshold that will be determined.

4.2 Detection probability

A first remark concerns the expression of urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0126 in (17). In fact, since the signal urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0127 is band-limited such as urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0128, then urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0129 can be rewritten as
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0130(27)
where urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0131 is the set of indices corresponding to the spectral support of urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0132, and urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0133 is the energy of the interpolated received signal.

4.2.1 Detection probability for deterministic signal

In a first approach, we assume that the signal is deterministic, i.e. both urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0134 nor urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0135 are deterministic processes. In that case, urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0136 is a constant, and the test urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0137 can be expressed as
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0138(28)
where urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0139. The cumulative distribution function (cdf) of urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0140 can then be derived as
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0141(29)
The probability of detection is expressed from (29) as
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0142(30)

4.2.2 Detection probability for Gaussian random signal

If it is supposed that urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0143 obeys a Gaussian random process, then urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0144 has a Gaussian distribution, the mean and variance of which are
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0145(31)
where urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0146 is the energy of the received signal before any interpolation, and without noise contribution, and
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0147(32)
As a consequence, urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0148 (which is defined as previously) is a sum of two uncorrelated Gaussian variables, then urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0149, and finally, the cdf of urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0150 is
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0151(33)
Finally, the probability of detection is expressed exactly as in (30).

4.3 Threshold value urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0152

We here suppose that the threshold value is set according to the false alarm probability urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0153 (it could be equivalently set according to the detection probability urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0154). Thus, from (26), we deduce that from a desired (target) urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0155 value, we find urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0156 by solving (inverting) (26), i.e.
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0157(34)
We deduce from (34) and (24) that, in addition to the target pfa, the optimal threshold value depends on the noise variance and the number of samples M. This particularity is similar to the ED. The specificity of the proposed detector is that the urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0158 value also depends on the number of interpolations, which appears in the expression of urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0159.

5 Simulations results

The aims of the presented simulations results are (i) to validate the previous analytical results, and (ii) to compare the performance of the proposed detector with that of the usual ED. In fact, both detectors are energy-based methods, therefore agnostic to the received signals, and have similar complexity. To show the latter assertion, from Section 3, we deduce that the proposed detector requires the following operations:
  • urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0160: N multiplications and N additions.
  • urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0161: M multiplication, since the multiplications by urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0162 in the filtering processes are ‘costless’ in term of complexity and urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0163 additions.

Then, since urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0164 we approximate urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0165, hence the computation cost of urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0166 in (14) is urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0167 multiplications and urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0168 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 urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0169 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, urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0170 samples are used, and urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0171 iterations. It can be clearly observed that theoretical results match with simulations, which validates the previous developments.

Details are in the caption following the image

urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0172 versus the false alarm probability. Comparison of the simulations and the analysis (34)

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 urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0173 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.

Details are in the caption following the image

Spectrum urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0174 of the received signal in the absence and in the presence of the noise, SNR = −10 dB. The signal is OFDM

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 urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0175 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.

Details are in the caption following the image

ROC performance of the proposed detector compared with ED, urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0176

(a) SNR = −10 dB, (b) SNR = −15 dB, (c) 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 urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0177: 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.

Details are in the caption following the image

ROC performance of the proposed detector compared with ED, for various parameters

(a) urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0178, SNR = −15 and −20 dB, (b) urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0179, in the presence and the absence of Rayleigh channel, (c) Comparison for different signal kinds, urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0180 in AWGN channel

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 urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0181 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 urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0182 and urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0183

It has been aforementioned that the multiple interpolations act like a low-pass filter. Under several conditions, it becomes possible to estimate the parameters urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0184 and urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0185 in the presence of a useful signal urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0186. In fact, it must be pointed out that the largest k, the lowest the ‘cutoff frequency’ of urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0187. Therefore, according to urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0188, and assuming that urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0189, it may happen that for a given urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0190, we have urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0191 (when urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0192 is lower than the cutoff frequency of urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0193), and for urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0194, we have urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0195 (when urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0196 is much larger than the cutoff frequency of urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0197).

As a consequence of the above remark, it exists urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0198 and urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0199, such as urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0200. Note that urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0201 has no physical sense since it may be not an integer. However, it is possible, mathematically speaking, to extend urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0202 from integer to real domain. In practice, urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0203 corresponds to the point where the linearly-interpolated trajectories of urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0204 and urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0205 intersect.

In order to estimate urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0206 and urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0207, some conditions must be assumed:
  • The signal is narrow-band with urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0208.
  • As a consequence, we suppose that urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0209 (C is a constant) for any q in urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0210.
  • The values urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0211 have been computed for every k value, in such a way that urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0212 has been estimated.
  • N is large enough to consider that the approximation
    urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0213
    holds.
Furthermore, for a clarity purpose, we note urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0214. According to the above assumption and using (27), the equality urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0215 leads to
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0216(35)
where urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0217. Since urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0218 is a strictly decreasing function for any urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0219, then it exists a unique urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0220 which is the solution of (35). Finally, since urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0221 is the (scaled) central frequency of the signal, then it can be estimated as
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0222(36)
Furthermore, if urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0223 is known in advance, then urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0224 can be estimated by
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0225(37)

6.2 Detection of coloured noise

It is possible to take advantage of the nature of low-pass filter linear interpolator to detect coloured noise, and to decide whether it is pink or blue noise. In that case, the detection hypothesis can be rewritten as
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0226(38)
where urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0227 is the coloured sample. The decision test (15) remains, but the detector decides whether the noise is white or coloured. Furthermore, it can detect the kind of colour, by applying the test
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0228(39)
Fig. 7 shows the trajectories of urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0229 in cases of pink, white, and blue noises versus the number of interpolations. Since the white noise corresponds to the case urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0230, the test statistic (39) can be verified from Fig. 7.
Details are in the caption following the image

Trajectories of the energy of interpolated pink, white, and blue noises versus the number of interpolations

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

In this appendix, we prove that, for urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0231, urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0232 in (20) has a Gaussian distribution. Let urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0233 (in (20)) be a random variable, which obeys a Chi-squared distribution with two degrees of freedom. The Lyapunov condition states that: if the urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0234 th moment (with urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0235) of independent variables urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0236 exists, the mean urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0237 and the variance urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0238 are finite, and if
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0239(40)
where
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0240(41)
and
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0241(42)
then urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0242 tends toward a Gaussian distribution, the mean and the variance of which are (21) and (24). In the following, we set urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0243. Note that, due to the symmetry of urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0244, we can limit to the positive index of m. It is straightforward to show that
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0245(43)
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0246(44)
and
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0247(45)
Let define urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0248 the urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0249 vector which contains the elements urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0250, then we can rewrite urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0251 in (40) as
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0252(46)
where urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0253 and urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0254 are the 4-norm and the Euclidian norm, respectively. We now prove that the ratio of the two norm in (46) tends to zero when M tends to the infinity.

Proof.Let urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0255 be a function of class urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0256 on the interval urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0257, where urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0258 are finite numbers. Let urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0259 and urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0260 two piecewise linear functions on urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0261 such as, urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0262, we have

urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0263(47)

Fig. 8 illustrates the above definition of urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0264 compared with urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0265. In this example urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0266 is the urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0267 function.

Details are in the caption following the image

Illustration of urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0268 and urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0269

In the following, we only focus on urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0270, as the developments remain the same for urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0271. We suppose that urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0272 is defined on urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0273 distinct intervals urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0274 such, such that for any urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0275 and urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0276, we have
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0277(48)
where urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0278 and urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0279, since p is of class urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0280 on urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0281.
We define urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0282 and urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0283 two vectors of size M, which correspond to the regularly sampled versions of urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0284 and p on urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0285, respectively. For a sake of simplicity, but without loss of generality, we suppose that any element urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0286 of urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0287 is positive. As a consequence, for any urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0288, we can write
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0289(49)
where urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0290 is a real positive value. By definition of the sampling process, urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0291 can be expressed as
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0292(50)
The first step of the proof is to show that it is possible to find urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0293 such that
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0294(51)
To this end, we analyses the gradient of urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0295. For any urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0296, we define the function urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0297 as
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0298(52)
the derivative of which is
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0299(53)
From (53), we deduce that the numerator of urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0300 is null for urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0301. Moreover, a straightforward analysis shows that
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0302(54)
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0303(55)
Therefore, urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0304 reaches a local minimum at urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0305, and a local maximum at urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0306. It can be noted that
  1. urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0307 (from (49)),
  2. Since urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0308, then urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0309, hence it can be straightforwardly shown that urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0310

As a consequence of the above remarks, we deduce that, for any urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0311, it is ever possible to find urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0312, urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0313 if urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0314 or urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0315 if urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0316 such that urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0317. Therefore, it exists a vector urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0318 containing the elements urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0319, urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0320, such that (51) holds.

From the above result, we can now provide an upper bound of urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0321. In fact, when M is large, then from (50)
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0322(56)
since the sum of the M first terms in urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0323 is equivalent to urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0324. The term urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0325 is the sum of urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0326 and. The same reasoning leads to
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0327(57)
since the sum of the M first terms in urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0328 is equivalent to urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0329, and urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0330 is a constant independent of M. Therefore, we deduce that, when M is large enough
urn:x-wiley:17519675:media:sil2bf00699:sil2bf00699-math-0331(58)
which concludes the proof.