In functional analysis, the Shannon wavelet (or sinc wavelets) is a decomposition that is defined by signal analysis by ideal bandpass filters. Shannon wavelet may be either of real or complex type. Shannon wavelet is not well-localized (noncompact) in the time domain, but its Fourier transform is band-limited (compact support). Hence Shannon wavelet has poor time localization but has good frequency localization. These characteristics are in stark contrast to those of the Haar wavelet. The Haar and sinc systems are Fourier duals of each other.
Definition Sinc function is the starting point for the definition of the Shannon wavelet.
Scaling function First, we define the scaling function to be the sinc function.
ϕ (Sha) ( t ) := sin π t π t = sinc ( t ) . {\displaystyle \phi ^{\text{(Sha)}}(t):={\frac {\sin \pi t}{\pi t}}=\operatorname {sinc} (t).}
And define the dilated and translated instances to be
ϕ k n ( t ) := 2 n / 2 ϕ (Sha) ( 2 n t − k ) {\displaystyle \phi _{k}^{n}(t):=2^{n/2}\phi ^{\text{(Sha)}}(2^{n}t-k)}
where the parameter n , k {\displaystyle n,k} means the dilation and the translation for the wavelet respectively. Then we can derive the Fourier transform of the scaling function:
Φ (Sha) ( ω ) = 1 2 π Π ( ω 2 π ) = { 1 2 π , if | ω | ≤ π , 0 if otherwise . {\displaystyle \Phi ^{\text{(Sha)}}(\omega )={\frac {1}{2\pi }}\Pi ({\frac {\omega }{2\pi }})={\begin{cases}{\frac {1}{2\pi }},&{\mbox{if }}{|\omega |\leq \pi },\\0&{\mbox{if }}{\mbox{otherwise}}.\\\end{cases}}}
where the (normalised) gate function is defined by
Π ( x ) := { 1 , if | x | ≤ 1 / 2 , 0 if otherwise . {\displaystyle \Pi (x):={\begin{cases}1,&{\mbox{if }}{|x|\leq 1/2},\\0&{\mbox{if }}{\mbox{otherwise}}.\\\end{cases}}}
Also for the dilated and translated instances of scaling function:
Φ k n ( ω ) = 2 − n / 2 2 π e − i ω ( k + 1 ) / 2 n Π ( ω 2 n + 1 π ) {\displaystyle \Phi _{k}^{n}(\omega )={\frac {2^{-n/2}}{2\pi }}e^{-i\omega (k+1)/2^{n}}\Pi ({\frac {\omega }{2^{n+1}\pi }})}
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