On linear periodically time varying (LPTV) systems with modulated inputs, and their application to smoothing filters

We show that the output of a linear periodically time-varying (LPTV) system varying with frequency ω
_s
, when driven by an input of the form Σ
_k
a[k]δ(t - kTs), where T
_s
= 2π/ω
_s
, is indistinguishable from the output of an appropriately chosen linear time-invariant filter driven by the same input. We extend this result, and give an expression for the transfer function of the equivalent time-invariant filter. We apply this principle to the design of a smoothing filter that follows a digital-to-analog converter. We show that using an LPTV smoothing filter rather than a time-invariant one not only results in better image attenuation, but can also avoid group-delay distortion.