Wavelets

Abstract

The wavelet transform is believed to avoid the ‘frequency leak’ problem even better. Consider the case of an impulse function (Example B.2): both in the DFT and the DCT transform, it has non-zero amplitudes in all frequencies. Thus, what would take a single number to describe in the time domain, will require several numbers in the frequency domain. The problem is that the DFT has no temporal locality: each of its coefficients provide information about all the time instants. A partial remedy would be the so-called ‘Short Window Fourier Transform’ (SWFT) [RV91): We can divide the time sequence into frames of, say, w consecutive (non-overlapping) samples, and do the w-point DFT in each of these windows. Thus, an impulse function in the time domain will have a restricted ‘frequency leak’. Figure C.1 shows intuitively what happens: In the time domain, each value gives the full information about that instant (but no information about frequencies). The DFT has coefficients that give full information about a given frequency, but it needs all the frequencies to recover the value at a given instant in time. The SWFT is somewhere in between.