To accomplish frequency domain signal processing, one can take the Fourier transform (or Fourier series or DFT) of a signal, multiply some of the Fourier coefficients by zero (or some other constant), then take the inverse Fourier transform. It is possible to completely remove certain components of a signal while leaving others completely unchanged. The same can be done by using wavelet transforms to achieve wavelet-based, wavelet domain signal processing, or filtering. Indeed, it is sometimes possible to remove or separate parts of a signal that overlap in both time and frequency using wavelets, something impossible to do with conventional Fourier-based techniques.

The classical paradigm for transform-based signal processing is illustrated in Figure 1 where the center “box" could be either a linear or nonlinear operation. The “dynamics" of the processing are all contained in the transform and inverse transform operation, which are linear. The transform-domain processing operation has no dynamics; it is an algebraic operation. By dynamics, we mean that a process depends on the present and past, and by algebraic, we mean it depends only on the present. An FIR (finite impulse response) filter such as is part of a filter bank is dynamic. Each output depends on the current and a finite number of past inputs (see (Reference)). The process of operating point-wise on the DWT of a signal is static or algebraic. It does not depend on the past (or future) values, only the present. This structure, which separates the linear, dynamic parts from the nonlinear static parts of the processing, allows practical and theoretical results that are impossible or very difficult using a completely general nonlinear dynamic system.

Linear wavelet-based signal processing consists of the processor block in Figure 1 multiplying the DWT of the signal by some set of constants (perhaps by zero). If undesired signals or noise can be separated from the desired signal in the wavelet transform domain, they can be removed by multiplying their coefficients by zero. This allows a more powerful and flexible processing or filtering than can be achieved using Fourier transforms. The result of this total process is a linear, time-varying processing that is far more versatile than linear, time-invariant processing. The next section gives an example of using the concentrating properties of the DWT to allow a faster calculation of the FFT.