Wavelet Transform-Spectrum Sensing

Abstract

Spectrum sensing is a vital cognitive radio function that protects licensed users from dangerous interference and finds accessible spectrum for better spectrum use. In practice, however, multipath fading, shadowing, and receiver uncertainty frequently degrade detection performance. Communication performance and continuity in cognitive radio networks are heavily dependent on how well the spectrum sensing function is implemented. The significance of selecting the right wavelet system is discussed.

1. Introduction

Spectrum sensing is crucial in cognitive radio technology for effective bandwidth usage. The problem of interference between adjacent spectrums can be efficiently avoided by carefully finding the spectrum boundaries. Increase the data rates in the channel by adding bands and increasing the data rates in the channel. Various attempts have been made. Wavelet Edge detection is a technique for accurately detecting spectral boundaries [1]. The method that is extensively used nonetheless, the system’s efficiency is quite low. Depending on the wavelet type utilized. After analyzing the nature of peaks in the power spectral density of the spectrum, a spectrum sensing technique selects the best wavelet function for the supplied spectrum. For the problem of edge detection, traditional solutions use a specific wavelet function, followed by energy detection or periodicity detection [2, 3]. However, a single wavelet function cannot be employed efficiently for real-time spectrum data with significantly different PSD properties.

1.1 Spectrum sensing

Cognitive Radio (CR) is a new technology that aims to solve the problem of wasteful spectrum use by increasing idle spectrum usage in both time and space. CR has the capacity to dynamically access the spectrum, determining which frequencies are not in use, and reserving them for data transmission and reception. When compared to traditional radio techniques, cognitive radio has a number of advantages, such as the ability to activate several licensed frequency bands to allow an unconstrained secondary user to communicate with another CR in some spectrum policy that defines some CR rules and limitations, and the ability to transmit data simply by changing the operating factor without any changes to the hardware components [4, 5]. Spectrum sensing is a vital step in the evolution of technology Cognitive Radio, in which the major users are detected in that spectrum band to detect spectrum holes and minimize unintended interference. Spectrum sensing can be done in a variety of ways. The following three types of spectrum sensing techniques are based on primary user availability.

Non-cooperative detection (Transmitter detection), cooperative detection, and interference detection are the three types of detection. Wavelet-based spectrum sensing algorithms are utilized for signal edge detection in transmitter detection, where primary users are present to detect spectrum opportunities.

1.2 Wavelet transform

The wavelet theory is used to evaluate signals by breaking them down into their constituents and basic functions. The wavelet transform can characterize the local regularity of signals and is a mathematical tool for analyzing singularities and irregular structures. In order to study the primary users, the wavelet transform approach for spectrum detection in CR is well justified. Wavelets are useful for studying fluctuations in signals and spectrum since they are described by both scale and position. The concept of local regularity is used to convey scale, while a list of domains is used to describe time aspects. The Continuous Wavelet Transform (CWT) is a two-parameter wavelet function signal expansion [6].

When compared to other types of spectrum sensing techniques, this wavelet transform technique takes much less time to detect whether the principal user is consuming the spectrum or not. When the de-noising and compression processes are at their center points, the disintegration is considered complete. Because only the available frequency, i.e. spectrum holes in which the secondary user can communicate, is denoted at each level of the wavelet decomposition.

2. Traditional spectrum sensing methods

Spectrum sensing is essentially a form of energy detection in which the presence or absence of useful data in a specific frequency band is determined. The presence of data indicates a rise in energy from the noise floor, or the presence of a noise-like signal with higher-order periodicity. Traditional techniques work in narrowband and use a set of FIR filters that are tuned for the frequency range in question. While this strategy works well for narrow bandwidths, it becomes inefficient as the dynamic range of Cognitive Radios’ functioning expands. This approach is quite difficult for ultra-wideband cognitive radio systems. However, it is recommended that the reader has first-hand knowledge of traditional methods in order to comprehend the properties of Wavelet-based algorithms so that the discussion is limited to those parts of spectrum sensing that cannot be done using these methods.

2.1 Short-time fourier transform (STFT)

The “Windowed Fourier Transform” is another name for this approach. In general, the Fourier Transform is a time-to-frequency domain transformation that yields time averaged values for distinct frequency components. While this technique is beneficial for examining frequency components, it does not allow us to determine when they occur. i.e. frequency localization in time is impossible. The entire signal is separated into smaller segments (using an appropriate window function) and the Fourier Transform is obtained for these intervals in a short-time Fourier transform.

While this method outperforms Fourier analysis in terms of results, it suffers from limited frequency resolution, large volatility in the predicted power spectrum, and high side lobes/leakages.

2.2 Periodogram

Until better methods were found to replace it, the Periodogram was one of the most widely utilized Spectrum Sensing techniques. The infinite length sequence is trimmed with a rectangular window function in this manner, and the FFT is obtained. The square of the FFT yields a rough Spectral Density plot. The signals are abruptly truncated, which is a fundamental flaw in this strategy. This produces a Dirichlet Kernel in the frequency domain, which is defined by the width of the main lobe and side lobes, as detailed in [7, 8]. As a result, spectral leakage occurs at the discontinuities. Furthermore, time-frequency localization was not possible with this strategy.

2.3 Matched filter approach

This is a test method of detection. This is the quickest way to detect spectrum, but it fails because it requires prior knowledge of the primary user’s modulation type, pulse shaping, and packet format. Coherence requires precise timing and synchronization. However, temporal dispersion and Doppler shifts can occur as a result of channel fading effects, affecting synchronization. In [9] there is a lot of information about how to implement matching filters.

2.4 Cyclo-stationary feature detection

First-order periodicity can be applied to any observable regularity. Because of modulation techniques or in the sent data. The signal gains a specific periodicity as a result of source coding, which can be useful. Only nonlinear time-invariant transformations of the time can be seen. Series [10]. Cyclostationarity is the name for this form of second-order periodicity. The mean, autocorrelation, and other statistical properties, in general, demonstrate a pattern of behavior This can be used to determine whether something is present or not a frequency band’s worth of data. The advantage is that this is the only method that accurately measures spectral occupancy in bands with very low SNR. The borders, on the other hand, cannot be precisely specified. As a result, this technique, in combination with border detection, is frequently utilized in the development of algorithms. Adoum and Jeoti [2] provides a full explanation of how to compute the Cyclic Spectrum Density.

2.5 Multi taper spectrum estimation

In this strategy, the issues that plagued the Periodogram approach are partly mitigated. Multiple orthogonal filters are used to limit spectral leakage and volatility in the calculated power. Consider the signal X(n) = [x(n) x(n - 1) x(n - 2) … x(n - M + 1)] T, which consists of M samples. These data points are used to create an orthogonal basis, and the expansion coefficients are modified using a set of values that represent the way the spectrum tapers. Thomson [11] provides a more thorough formulation. The average of numerous Periodograms with different windows can be deduced as MTSE. As a result, each window shape displays distinct aspects of the spectrum, while the average value smooths out the discontinuous points and reduces spectral leakage. This solution suffers from expensive computations and the fact that it cannot totally solve the Spectrum leakage problem, although better methods can.

2.6 Quadrature mirror filter banks

The entire wideband spectrum is separated into M-bands using this strategy. These are predefined bands in which user activity is monitored using n stages of Quadrature Mirror Filters tuned for the band, with M = 2n. As a result, as with the Matched Filter Method, previous knowledge of the Primary user is required. However, because a set of Filter banks is utilized, the same filters can be used for data reception after spectral gaps have been detected [12]. As a result, they serve a dual purpose. In addition, the tree structure aids in the reduction of computer complexity. The energy detection process starts with only two filters in the first step. Thresholds are set to determine whether or not to move on to the next step. The procedure of analyzing the sub-bands is skipped if the signal energy is larger than the threshold. The difficulty with this strategy is that free spectra will be wasted if the spectrum’s border does not correspond with the designated pass-band of a particular filter. As a result, spectral holes narrower than the pass-band of the last stage filter are not detectable using this technique. Furthermore, the channel fading effects have a significant impact on its performance.

These methodologies, as well as their faults, prompted researchers to develop a domain transformation technique that could analyze both spatial and spectral data at the same time. Wavelets developed as a promising concept with a lot of promise for fixing the challenges described above. To a large extent, they helped with temporal frequency localization. Their ability to respond to function discontinuities (singularity) also enables researchers to use them in a variety of border detecting applications. The mathematics of Wavelet Edge Detection is presented in the next section.

3. Wavelet theory of edge detection

Before getting into the analysis of Edge Detection with Wavelets, it’s a good idea to give a quick overview of how the Wavelet theory came to be. This is followed by the Edge Detection Technique in the following subsection.

Edges in the frequency spectrum are generated when a signal’s frequency changes. This feature was utilized in the detection process. The wavelet transform is commonly used on these sub-bands for detection and estimation of native spectral irregularities/transitions, which carries significant information on Power Spectral Densities (PSDs) and frequency locations [13]. The entire frequency range is divided into various sub-bands, and the wavelet transform is generally used on these sub-bands for detection and estimation of native spectral irregularities/transitions, which carries significant information on PSDs and frequency locations. The wavelet transform has been employed instead of the standard Fourier transform because it provides information about the exact placement of different frequency locations and spectral densities. Furthermore, the Fourier transform can only display the various frequency components, not their location. Edges representing transitions from empty to the occupied band have been sought for in the spectral densities of all sub-bands. The initial stage in wideband spectrum sensing is to determine the frequency position of each channel in the RF spectrum. Regardless of the PSD’s actual shape, sharp variation points (singularities) along the channel’s edges are expected [14]. The difficulty of detecting these anomalies might be thought of as an edge detection problem. Depending on the PSD level of each channel, the discovered spectrum bands have now been categorized as occupied or unoccupied. The existence of an edge indicates that the band contains PU.

PSD of the received signal can be written as:

$$\left(f\right)=\sum \left(f\right)+\left(f\right),f\left[f0,\mathit{fN}\right]$$E1

Where (f) & (f) are the signal and noise PSDs, respectively, within the nth band, and (f) & (f) are the signal and noise PSDs, respectively. The wideband frequency range is denoted by f [f0, fN]. This technique has worked well for ultra-wideband (3–10GHz) CRs with a variety of narrowband incumbents and other users like as WiMAX and Wi-Fi. Pros: Adapts quickly to changing PSD structures. Cons: Characterizing the whole bandwidth was required at higher sampling rates.

4. Types and classification of wavelets

Wavelets are divided into two groups, as seen in Figure 1 [15]. Wavelets, which are characterized by mathematical formulas that are continuous and infinite in nature, are the first category [15]. They are also known as crude wavelets because they must be converted to wavelet filters with a finite number of discrete points before they can be used in any signal processing system. The Mexican hat wavelet (Figure 1) is a good illustration of this sort of wavelet.

Figure 1.

Types of Wavelet.

Wavelets that start off as filters with two points of definition in the initial state are the second category of wavelets [15]. These wavelets form an approximation of a continuous wavelet by interpolating and extrapolating more points from the starting two points. The Daubechies 4 (Db4) wavelet, depicted in Figure 1 [15, 16], is a good example of this type of wavelet.

Wavelets are widely used in a variety of scientific and technical sectors. They each have different features that make them best suited for a specific use. The wavelet and scaling functions of biorthogonal wavelets are symmetric. Because the human vision system is more tolerant of symmetric faults than asymmetric errors, this characteristic makes biorthogonal wavelets suitable for human vision perception [17]. Wavelet transforms such as Haar, symlet, and Daubechies are known to have asymmetric errors.

Shannon wavelets feature a sharp localization and limitless frequency support. As a result, they’re great for identifying events with specified frequencies [14, 18]. Two key characteristics of Haar wavelets make them perfect for edge identification. The first is that Haar wavelets maintain signal energy [19]. The second virtue is that Haar wavelets are exactly reversible without edge effects [20], which is critical for edge detection.

Coiflet wavelets are ideal in numerical analysis because of their high number of vanishing moments and almost interpolating and linear phase low-pass within a given passband [21]. They can also deal with fractals in signal processing because of their almost interpolating and linear phase low-pass within a given passband.

The Daubechies wavelets are characterized by a high degree of regularity, many vanishing moments, and approximate symmetry. Because the wavelets get smoother as the vanishing moments grow, these characteristics are extremely desirable in signal processing and data compression applications [22, 23]. In Section 6, an example of Daubechies applicability for spectrum sensing is given.

The Morlet wavelet is frequently used in signals that are related to the environment, such as seismic vibrations. The Morlet wavelet’s ability to capture both amplitude and phase features of a signal while maintaining the signal’s temporal aspect makes it appealing in this application [24, 25, 26].

Patch and gap events are well-localized in Mexican hat wavelets. They also have a lot in common with MUAPs (motor unit action potentials), hence they can be used in EMG (electromyography) [27, 28].

5. Wavelet-based spectrum sensing

Wavelets are short-duration signals. Wavelets differ from sinusoids in theory because, whereas sinusoids stretch from—to, wavelets have a finite beginning and ending locations. The premise in wavelet-based spectrum sensing is that the CR system receives a signal that spans N spectrum bands and that the CR must detect the PSD (power spectrum density) levels and frequency positions of each band. Figure 2 [29] depicts a spectrum band between f0 and fN, with sub-band frequencies at f0, f1, and fn. Shiann-Shiun et al. [30] defines the nth band in Figure 2 as follows:

Figure 2.

N frequency bands with piecewise smooth PSD.

$$\mathit{Bn}:\left\{f\in \mathit{Bn}:\mathit{fn}-1\le f<\mathit{fn}\right\},n=1,2,\dots ,N.$$E2

PSD of a CR system receiving an input signal:

$$\mathit{Sr}\left(f\right)=\sum _{n=1}^N\alpha 2\mathit{nSn}\left(f\right)+\mathit{S\omega }\left(f\right),f\in \left[f0,\mathit{fN}\right]$$E3

There are three different ways to apply the wavelet approach to spectrum sensing. The continuous wavelet technique, discrete wavelet technique, and discrete wavelet packet technique are examples of these. DWT (Discrete Wavelet Transform) and CWT (Continuous Wavelet Transform) techniques are also used for reduced BER with an increase in SNR in optical wireless systems which are based on On–Off keying modulation (OOK) [31].

5.1 Continuous wavelet transform-based spectrum sensing

The spectrum sensing based on continuous wavelet (CWT) [29] assesses the similarity between a signal and an analysis function. by making use of inner coefficients multi-scale product and multi-scale sum are two methods for obtaining spectrum sensing with continuous wavelets. Taking the multi-scale and wavelet transforms, and then estimating the edges given in [29, 32], the multi-scale product technique includes determining discontinuities in a signal’s PSD:

$$p\left(f\right)=\prod _{j=1}^{J}W\text{'}s=2\mathit{jSr}\left(f\right)$$E4

where j is the scaling function’s upper limit; S_r(f) is the received signal’s PSD; W^′_s = 2ʲ is the first-order derivative at scale s = 2ʲ, and p(f) is the multiscale product.

The spectral boundaries are assumed to be represented by discontinuities in the PSD in this calculation. The energy is calculated for each sub-band to get the spectrum occupancy.

The multi-scale sum technique [33] is based on the idea that various signals at different scales have varying cross scales of information. This means that at different scales, wavelet transformations convey information about the Lipschitz exponent at acute variation spots. As a result, the multi-scale sum technique is employed to preserve signal information at all scales while avoiding attenuation. The multi-scale sum at the jth dyadic scale for a CWT is provided as:

$$\mathit{\text{XjSr}}\left(f\right)=\sum _{j=1}^{J}W2\mathit{jSr}\left(f\right)$$E5

The following are the benefits and drawbacks of the continuous wavelet transform-based spectrum sensing technique:

Advantages

1. CWT allows for enhanced transient localization and oscillatory behavior characterization in a signal.

2. Because of its fine-grained resolution, CWT is frequently used for singularity detection.

3. Because of its narrow sample scales, CWT has excellent fidelity in signal analysis.

Disadvantages

1. Because CWT contains a lot of redundancy and is computationally demanding, it’s frequently utilized for offline analysis.

2. For an evaluated signal, CWT does not offer phase information.

3. A perfect reconstruction of an original signal from CWT coefficients is impossible.

5.2 Discrete Wavelet Transform (DWT) based spectrum sensing

The discrete wavelet transform (DWT) breaks down an input signal x[m] into coarse and fine information. The decomposition, which allows the DWT to examine a signal at multiple frequency bands and resolution, is accomplished by filtering the time-frequency domain signal with successive high pass and lowpass filters [34, 35]. This can be stated mathematically as (Figure 3)

Figure 3.

DWT Filter Bank.

$$y_{\mathrm{low}}\left[k\right]=\sum _{m}x\left[m\right]h_0\left[2k-m\right]$$E6

$$\mathit{\text{ylow}}\left[k\right]=\sum _{m}x\left[m\right]h0\left[2k-m\right]$$E7

y_high[k] is the high pass filter output and y_low[k] is the lowpass filter output. Mathematically, the scaling (c_j) and wavelet (d_j) coefficients are represented in [36, 37] as:

$$c_j\left(k\right)=\sum _m{h}_0\left(m-2k\right)c_{j+1}\left(m\right)$$E8

$$\mathit{dj}\left(k\right)=\sum _{m}h1\left(m-2k\right)\mathit{cj}+1\left(m\right)$$E9

Advantages and disadvantages of Discrete wavelet transform-based spectrum sensing technique:

Advantages

1. By using a subset of coefficients to capture significant aspects of signals, DWT enables the sparse representation of many natural signals.

2. DWT achieves signal compression by providing a high-quality approximation of a signal. This is accomplished by deleting several of its close-to-zero coefficients.

3. Because they have nonredundant orthonormal bases, DWT has flawless reconstruction and is computationally efficient.

Disadvantages

1. Shift sensitivity is a problem with DWT, in which a change in the input signal induces an unanticipated change in the transform coefficients.

2. The DWT representation of signals in image processing suffers from inadequate directionality, which impairs its optimality.

3. DWT lacks phase information, which is critical for describing a function’s amplitude and local behavior.

5.3 Discrete wavelet packet transform (DWPT) based spectrum sensing

The discrete wavelet packet transform (DWPT) works similarly to the discrete wavelet transform, with the exception that the DWPT transform decomposes both the approximation and detail spaces of a signal [38, 39]. The structure of the DWPT is shown in Figure 4 [40]. As shown in Figure 4, DWPT decomposes a signal x[n] into 2 L sub-bands, where L is the decomposition level.

Figure 4.

DWT Structure.

To perform spectrum sensing, the energy in each sub-band is calculated [40] and compared to a threshold to determine if the sub-band is occupied or not by the principal user. In Ref. [41, 42] gives the following formula for calculating the energy in each sub-band:

$$E=\frac{1}{T}{\int }_0^T\left[\sum _{j\ge j0}\sum _k\mathit{cj},\mathit{k\varphi j},k\left(t\right)+\mathit{dj},\mathit{k\psi j}.k\left(t\right)\right]2\mathit{dt}$$E10

$$E=\frac{1}{T}\sum _{j\ge j0}\sum _k\left(\mathit{cj}2,k+d2j,k\right)$$E11

Advantages and disadvantages of Discrete wavelet Packet transform-based spectrum sensing technique.

Advantages

1. Because it decomposes both the high and low-frequency components of an input signal, DWPT has more fidelity than DWT.

2. Without assuming any statistical attribute of the signal, DWPT exhibits good universality in adapting its transform to a signal.

Disadvantages

1. In a DWPT, the high-pass coefficients oscillate around signal singularities.

2. The nonlinear frequency perception phenomena are ignored by DWPT in voice recognition.

Advantages and disadvantages of wavelet-based spectrum sensing technique.

Wavelets, like any other spectrum sensing technology, have advantages and disadvantages. However, when it comes to dynamic frequency management, the benefits of wavelet sensing techniques exceed the drawbacks. The following are some of the benefits of wavelet-based spectrum sensing:

1. Wavelet-based methods are used to accomplish significant data compression. As a result of this compression, the number of sensing measures required is reduced, enhancing estimate speed and lowering the communication power required for transmission. In terms of power source longevity, a reduction in necessary transmission power is a significant benefit for mobile communication devices.

2. Wavelet-based estimates for sharp-featured sources have the advantage of having fewer side-lobes and thus fewer leakages than most older methods.

3. Wavelets have a strong ability to modify the time-frequency window to retain orthogonality, allowing them to identify dynamic changes in statistical parameters in any spectrum.

4. The effect of noise and interference on the signal can be reduced by using a variable time-frequency window design.

5. The tuning capability and flexibility of wavelet bases are used to mitigate channel problems such as inter-symbol interference (ISI) and inter-carrier interference (ICI).

Wavelets do not require a cyclic prefix or guard bands in OFDM applications, making them more efficient than the Fourier transform in terms of spectrum consumption.

The following are some of the drawbacks of wavelet-based spectrum sensing:

1. Frequency selection is not natural.

2. Requires modification to carry signal phase information.

3. Requires the creation of filters that implement the required wavelet to meet very strict criteria.

4. The higher the level of decomposition in the discrete wavelet transform and discrete wavelet packet transform, the more complicated the system becomes.

6. Conclusions

A number of traditional spectrum sensing methods have been studied in this chapter, and their disadvantages have been highlighted. The use of Wavelets’ unique properties in Spectrum Sensing has also been explained. A Wavelet Theory of Edge Detection analysis has also been presented. Wavelet categorization has also been briefly addressed. Finally, wavelet-based spectrum sensing was explored, as well as numerous ways to apply the wavelet technique to spectrum sensing, as well as their benefits and drawbacks. Wavelets-based spectrum sensing approach merits, shortcomings, and applications have also been examined.

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Assume a frequency band of interest, as illustrated in Figure 2, that is further subdivided into N sub-bands, each having its frequency bounds at f0 < fn. The objective is to find any empty areas between two neighboring sub-bands so that a secondary user can use them without interfering with the primary. In the absence of noise, the normalized power spectral density (PSD) in each sub-band is:

$${\int }_{\mathit{Fi}-1}^{\mathit{Fi}}\mathit{Si}\left(f\right)\mathit{df}=\mathit{Fi}-\mathit{Fi}-1$$EA1

Power Spectral density of observed signal r(t):

$$\mathit{Sr}\left(f\right)=\sum _{n=1}^N\alpha 2\mathit{nSn}\left(f\right)+\mathit{S\omega }\left(f\right),f\in \left[f0,\mathit{fN}\right]$$EA2

Where $\alpha 2n$ indicates the signal power density within the nth band and S_n(f) is PSD of each sub-band. PSD inside each sub-band is estimated as:

$$=\frac{1}{\mathit{Fi}-\mathit{Fi}-1}\underset{\mathit{Fi}-1}{\overset{\mathit{Fi}}{\int }}\mathit{Sr}\left(f\right)\mathit{df}$$EA3

The wavelet transform properties are used to identify the frequency boundaries between successive sub-bands. The following is a one-scale dilation of the wavelet functions $\mathit{\psi s}$:

$$\mathit{\psi s}\left(f\right)=\frac{1}{s}\psi \left(\frac{f}{s}\right)$$EA4

The continuous wavelet transform is defined as:

$$\mathit{\text{WsSr}}\left(f\right)=\mathit{\text{Srψs}}\left(f\right)$$EA5

It gives localized information about S_r(f) at fine scales, and we must identify its first and second derivatives to find abnormalities in it. The first derivative is as follows:

$$\mathit{Ws}1\mathit{Sr}\left(f\right)=s\frac{d}{\mathit{df}}\left(\mathit{\text{Srψs}}\right)\left(f\right)$$EA6

The second derivative is:

$$\mathit{Ws}2\mathit{Sr}\left(f\right)=s2\frac{d2}{\mathit{df}2}\left(\mathit{\text{Srψs}}\right)\left(f\right)$$EA7

Different scales propagate edges and discontinuities. With dyadic scales s = 2j −2, 1–1, 2, 3 … j, the CWT is obtained. The multi-scale product of the J CWT gradients is used to follow the propagation of edges and discontinuities over multiple scales:

$$p\left(f\right)=\prod _{j=1}^{J}W\text{'}s=2\mathit{jSr}\left(f\right)$$EA8