The wavelet transform is a powerful mathematical tool for analyzing singu- larities and edges [35]. In wavelet method based spectrum sensing schemes, the spectrum of interest is usually decomposed as a train of consecutive fre- quency sub-bands and wavelet transform is then used to detect irregularities in these bands. An important characteristic of the power spectral density (PSD) is that it is relatively smooth within the sub-bands and possesses irregularities at the edges between two neighboring sub-bands. So, wavelet transform carries information about the locations of these frequencies and the PSD of the sub-bands. Vacant frequency bands can be obtained through the detection of thesingularitiesin the PSD of the signal observed, by per- forming the wavelet transform of its PSD [20].
The process for the wavelet detection methods can be described as fol- lows [35]. First, let us assume that we have a total of B Hz spread across the frequency range [f0, fN] for a wideband wireless system. Further, we
assume that the entire band is divided into N sub-bands where each sub- band is occupied by individual PU and all sub-bands are being simulta- neously monitored. The sensing task involves detecting the locations and PSD within each sub-band. Let us suppose that the sub-bands lie consec- utively within [f0, fN], such that there are frequency boundaries located at
f0 < f1 < · · ·fN. The n-th band may thus be defined by Bn : {f ∈ Bn :
fn−1 ≤ f < fn}, n = 1,2,· · · , N. Under H1, the normalized, unknown
power shape within each band, Bn is denoted by Sn(f) and satisfies the
conditions [35]
Sn(f) = 0, ∀f /∈Bn; (2.2.38)
∫ fn
fn−1
Section 2.2. Local Spectrum Sensing Techniques 23
If it is assumed that the PSD within each band, Bn is smooth and almost
flat but exhibits discontinuities from its neighboring bandsBn−1 andBn+1,
such that irregularities in PSD appears only at the edges of the bands,Sn(f)
may be approximated as Sn(f) = 1, ∀f ∈Bn. 0, ∀f /∈Bn. (2.2.40)
The PSD of the observed time domain signal, x(t), can then be written as
Sx(f) = N ∑ n=1 ¯ α2nSn(f) +Sw(f), f ∈[f0, fN] (2.2.41)
where it is assumed that the noise is additive and white with two sided PSD,Sw(f) = N20,∀f, and ¯αn2 indicates then-th band signal power density.
Furthermore, the corresponding time domain equivalent of (2.2.41) can be written as x(t) = N ∑ n=1 ¯ αnpn(t) +w(t) (2.2.42)
where Sn(f) is the signal spectrum of pn(t) and w(t) is the additive noise
whose PSD isSw(f). Furthermore, if we assume a pulse shaper,ht of band-
width fn−fn−1 , and the center frequency is denoted by fc,n = fn−12+fn,
the spectral shape, Sn(f) is proportional to |F{ht}|2, where F{.} denotes
the Fourier transform (FT). It is desired that x(t) with PSD Sx(f) be used
to estimate {fn}Nn=1−1 and {α¯2n}Nn=1, which characterize the wideband spec-
tral environment under consideration. If we letκ(f) be a wavelet smoothing function, for example, the Gaussian function with a compact support,gvan- ishing moments andgtimes continuously differentiable, the dilation of κ(f) by a scale factors is given by [35]
κs(f) =
1
sκ( f
Section 2.2. Local Spectrum Sensing Techniques 24
where for dyadic scales, s takes values from powers of 2, i.e. s = 2j, j = 1,· · · , J. The continuous wavelet transform (CWT) of Sx(f) in (2.2.41) is
given by
WsSx(f) =Sx∗κs(f) (2.2.44)
where ∗ denotes the convolution operation. It is worth noting here that CWT in (2.2.44) is implemented in the frequency domain and Sx(f) is re-
lated to x(t) via the FT. For the Sx(f) under consideration, the edges and
irregularities at scalesare defined as local sharp variations points of Sx(f)
smoothed by κ(f). Furthermore, since the edges of a function are often indicated in the shapes of its derivatives, by using the CWT, the first and second order derivatives ofSx(f) smoothed by the scaled wavelet, κ(f), can
be written as [35] W′ sSx(f) =s d df(Sx∗κs)(f) =Sx∗(s dκs df )(f) (2.2.45) and Ws′′Sx(f) =s2 d2 df2(Sx∗κs)(f) =Sx∗(s2 d2κs df2 )(f) (2.2.46)
respectively. According to [36], the signal irregularities is characterized by the local extrema of the first derivative and the zero crossings of the second derivative. However, for spectrum purposes, the local maxima of the wavelet modulus are sharp variation points which yields better detection accuracy than local minima points. Therefore, the edges or boundaries corresponding to the spectral content,{fn}Nn=1−1, in the received signal,x(t), of interest can
Section 2.2. Local Spectrum Sensing Techniques 25
with respect tof as
ˆ
fn=maximaf{|Ws′Sx(f)|}, f ∈[f0, fN] (2.2.47)
or from the zero crossing points of (2.2.46) as
ˆ
fn=zerosf{Ws′′Sx(f)}, subject to Ws′′Sx( ˆfn) = 0. (2.2.48)
In searching for the presence of frequency, ˆfn, only those modulus maxima
or zero crossings that propagate to large dyadic scale, sare retained while others are simply regarded and removed as noise [36].
After determining the frequencies present in x(t), i.e. {fn}Nn=1−1, the next
task is to estimate the PSD level, {α¯2
n}Nn=1. The average PSD within the
band Bn,∀ncan be computed as
βn= 1 fn−fn−1 ∫ fn fn−1 Sx(f)df. (2.2.49)
Based on the earlier assumption that the PSD within each band is smooth and almost flat, but exhibiting discontinuities from the neighboring band,
βn is related to the required ¯α2n according to βn≈α¯2n+N0/2. However, in
an empty band, i.e. where the PU is absent, say then′-th band, ¯α2n′ = 0 so
thatβn′ =N0/2 forf ∈Bn′. Therefore, the estimate of spectral density, ¯α2n
denoted as ˆα2n′ can be obtained from Sx as [35]
ˆ
α2n′ =βn−min n′ βn
′, n= 1,· · ·, N (2.2.50)
where {fn} used for computing {βn} in (2.2.49) can be replaced by their
Section 2.2. Local Spectrum Sensing Techniques 26