A View of Linear Time Varying Digital Filters

As discussed in Appendix F, linear time-varying (LTV) digital filters may be represented as matrix operators on the linear space of discrete time signals. Using the matrix representation, this appendix provides an interpretation of LTV filters that the author has found conceptually useful. In this interpretation, the input signal is first expanded into a linear combination of orthogonal basis signals. Then the LTV filter can be seen as replacing each basis signal with a new (arbitrary) basis signal. In particular, when the input-basis is taken to be sinusoidal, as in the Discrete Fourier Transform (DFT), one may readily design a time varying filter to emit any prescribed waveform in response to each frequency component of an input signal.