The discrete time wavelet transform: its discrete time Fourier transform and filter bank implementation

Viewing the discrete time wavelet transform DTWT[m,n] of a sampled signal s(nT) as a sequence in n, a closed-form expression is derived for its Discrete Time Fourier Transform (DTFT) D^(m)(e^jω) in terms of the DTFTs of the sampled mother wavelet ψ(nT) and sampled signal s(nT). Next, an expression is derived for the output Y^(m)(e^jω) of a filter bank defined by the digital filters a[n] and b[n] and excited by s(nT). The filter a[n] emerged as an ideal low-pass filter, and the filter b[n] turned out to be a time-reversed and complex conjugated version of the sampled mother wavelet ψ(nT)