Time-variant Filtering

In a linear time-invariant (stationary) filter the output g(t) is related to the input h(t) by the convolution

$$g(t)=a(t)*h(t)=\int_{-\infty}^\infty a(t-\tau)h(\tau)d\tau$$

where a(t) is the impulse response of the filter. In order to extend the applicability of this simple expression for the response of a non-stationary filter, we could replace $a(t-\tau)$ in the previous equation with the more general expression $a(t,\tau)$, indicating that the impulse response itself is now a function of the input time $\tau$. This expression, however, is too general and gives little insight into what the response of such a time-variant filter would be. Margrave, Margrave (1998) propose to maintain the convolutional nature of the impulse response by adding an explicit time dependence to it, that is, replacing $a(t-\tau)$ with either $a(t-\tau,\tau)$ (convolution) or $a(t-\tau,t)$ (combination). The formal definitions of non-stationary convolution and combination are, respectively Margrave (1998)

$$\begin{eqnarray} g(t)=\int_{-\infty}^\infty a(t-\tau,\tau)h(\tau)d\tau\\ \bar{g}(t)=\int_{-\infty}^\infty a(t-\tau,t)h(\tau)d\tau\end{eqnarray}$$ (1) (2)

These equations are clearly straightforward extensions of the stationary convolution concept. The introduction of the second index accounts for the non-stationarity. Comparing the two equations we see that the difference between non-stationary convolution and combination lies in the way the impulse responses are considered in the convolutional matrix. In non-stationary convolution the filter impulse responses (as a function of input time $\tau$)correspond to the columns of the matrix, whereas in the non-stationary convolution case they correspond (as a function of the output time t) to the rows of the matrix (time reversed).

Just as with stationary filtering, it is convenient to find equivalent expressions in the frequency domain. These expressions are Margrave (1998)

$$\begin{eqnarray} G(f)=\int_{-\infty}^\infty H(F)A(f,f-F)dF\\ \bar{G}(f)=\int_{-\infty}^\infty H(F)A(F,f-F)dF\end{eqnarray}$$ (3) (4)

where f and F are the Fourier duals of t and $\tau$ respectively and H(F) and G(f) are the Fourier transforms of $h(\tau)$ and g(t). It is interesting to notice from these equations that non-stationary convolution in time domain translates into non-stationary combination in the frequency domain and vice versa. This is as opposed to the stationary case in which convolution in time domain corresponds to multiplication in the frequency domain.

Since we now have two time indexes (t representing the filter samples and $\tau$ to keep track of the sample index to which each filter is applied), it is possible to have a third domain of computation, the so-called mixed domain in which the impulse response of the filters in the convolutional matrix are replaced with their corresponding frequency spectra. The equations to apply non-stationary filtering in the mixed domain are slow generalized Fourier transforms given by Margrave (1998)

$$\begin{eqnarray} G(f)=\int_{-\infty}^\infty \alpha(f,\tau)h(\tau)e^{-2\pi if\tau... ...\\ \bar{g}(t)=\int_{-\infty}^\infty \alpha(F,t)H(F)e^{2\pi iFt}dF\end{eqnarray}$$ (5) (6)

where $\alpha(p,v)$ is the so-called non-stationary transfer function which is the basic matrix in which the horizontal axis is time and the vertical axis is frequency (an example of this matrix will be given below). The transfer function is given by

$$\alpha(p,v)=\int_{-\infty}^\infty a(u,v)e^{-2\pi ipu}du$$ (7)

where a(u,v) is the matrix of impulse responses, that is, the matrix with $\tau$ as its horizontal axis and t as its vertical axis (again, an example will be shown below).