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The response of time invariant filters is independent of time and have filter weights which do not change over time. Time invariance (TI) is a nice quality for filters to have when analyzing them mathematically and have many applications in which adaptation is not needed. Time-varying filters (TV) are common in radio receivers such as equalizers, automatic gain control and polyphase filters. For example an equalizer is time-varying because the filter weights are dependent on previous input samples. Understanding TI filters is necessary for having a proper DSP foundation before moving onto TV filters.
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Time Invariant FIR Filters
The output of a finite pulse response (FIR) filter h[n] is represented by convolution,
Figure 1 is an example of a FIR filter.
From(2) it can be seen that x[k] is time reversed, x[-k], and delayed by n samples,
Mathematically either x[n] or h[n] could be time reversed and delayed and the same output would be produced. However for easier analysis the convolution will be implemented by time reversing and delaying x[n].
Delaying x[n] by T samples and substituting into (4) results in
which can be simplified as
From (7) it can be seen that a delay in x[k] by T samples, x[k-T], results in a delay of the output y[n] by T samples, y[n-T] with no other affects. Figure 2 shows how a delay on an input signal results in the same delay in the output when filtered by a time-invariant FIR filter.
Time Invariant IIR Filters
Delaying x[n] by T samples, x[n-T], and substituting into (9) results in
From (10), delaying x[n] by T results in the output being delayed by the same amount, y[n-T], with no other impact. The output y[n] is dependent on the input x[n] and the filter weights ak and bk. The filter weights ak and bk are constant for all time and therefore will not being time-varying. A delay of x[n] therefore corresponds directly to a delay in y[n]. Figure 4 gives an example of time-invariance in an IIR filter.
A decimating FIR filter is a simple case of a time-varying filter. A basic decimation operation is performed by first low-pass filtering and then downsampling, shown in Figure 5.
Combining the low-pass filtering with the downsampling operation will reveal the time-varying nature of the filtering operation and will produce computational efficiencies, referred to as a polyphase decimating filter. The low-pass filter output can be written as
Downsampling by 2 discards every other sample. Mathematically, it allows for the opportunity to select even indices,
or odd indices,
The filter h[n] produces a single output for each input x[n]. However, the downsampling by 2 operation discards all of the odd indices wasting the computation. The filter h[n] will be rearranged into a polyphase structure to avoid wasting the computation and making the decimation more efficient.
The filter output is arranged into even time indices x[2n] and odd time indices x[2n+1].
From (16) it can be seen that the even input samples x[2(n-k)] are filtered by the even filter weights h[2k] while the odd input samples x[2(n-k)+1] are filtered by the odd filter weights h[2k+1]. The time index of the input samples x[n] therefore indicates which set of filter weights will be applied. Figure 6 gives the polyphase partitioning and shows how the input switch for x[n] controls which filter weights will be applied.
The polyphase decimating filter is therefore periodically time-varying as it produces different filter responses based on the time delay of the input signal. Figure 7 gives an example of how the PTV nature of the polyphase decimating filter effects delays on the input signal.
Note that the decimate by 2 polyphase filter is periodically time-varying (PTV) with a period of 2 samples. Delaying the input signal into the filter by 2k synchronizes the input signal to the periodic nature of the cycling filter weights and the filter response becomes time-invariant under these conditions. Similarly, delaying the input by 2k+1 will also be time-invariant. Figure 7 shows that delaying the input signal by 0 and 2 samples results in the same output signal but with a 1 sample delay. Conversely, delaying the input sample by 1 sample produces a different filter response and results in a 1/2 sample delay.
Other Examples of Time Varying Filters
A subset of time-varying filters include adaptive filters whose filter weights are effected by previous samples of the input signal. Such adaptive filters include equalizers and automatic gain control. Another class of time-varying filters are periodically time varying filters which includes polyphase filters.
Delaying the input to a time invariant filter produces the same delay of the output signal with no other modifications. Time-invariant filters have mathematically useful properties for filter analysis and also form the basis for more sophisticated time-varying filters. A polyphase decimating filter is an example of a periodically time-varying filter because the filter response depends on the time delay of the input signal.
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