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 fur its Discrete Time Fourier Transform (DTFT) D-(m) (e(jw)) in terms of the DTFTs of the sampled mother wavelet psi (nT) and sampled signal s(nT). Next, an expression is derived for the output Y-(m) (e(jw)) 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 he a time-reversed and complex conjugated version of the sampled mother wavelet psi (nT).