The digital reconstruction or resampling filters employed in most digital sound processing systems tend to be based on methods equivalent to what is usually called a

Many of the existing systems used in CD players. etc, have a “Time Symmetric” set of coefficient values. For the sake of the examples in the ‘Ringing in the Ears’ pages I’ve chosen the most basic time symmetric behaviour described in the textbooks and used the pattern illustrated in Figure 2. This is based up what is called the Sinc function low-pass filter. This shows the classic, , pattern for a chosen low-pass cut-off frequency of 10 cycles per unit time.

The time symmetry means that for any output instant we have to take into account both ‘past’ and ‘future’ values. i.e. the system has to have some information about what pattern details are approaching as well as those that have just passed by. We do this by treating the values in the middle of the row (shift register) as being for ‘now’, those which have passed along most of the locations as being ‘in the past’ and those which have only just entered as being ‘future’ values. Time symmetry means we then use a coefficient value which is the same for a given time offset into the past as for the same offset into the future. This shows up with the symmetry of the shape illustrated in Figure 2 which is symmetric about the central value.

If we were to supply the filter with a series of input values that consisted of just one non-zero sample value, preceeded and followed by series of zero values, then the output patterns we would see emerging from the filter would sweep out the waveform shown in Figure 2. This is because only one coefficient (time offset) at a time would then ever contribute to the filtered output as any other contributions would be suppressed by all the other samples being zero. This response is then called the ‘Impulse Response’ of the filter system. It is often used to indicate the filter’ response to a transient event.

In reality, the requirement to ‘know the future’ means we have to delay the signal passage through the system. In effect, we start collecting values on a given feature of the waveform before we decide that now is the time to really respond fully to them. If the filter did not do this it would have no way to know what was coming next! The main result is that the filter delays the signal a little. A secondary result may be the pre ringing or time smear which is the topic investigated in these web pages.

Since any real TDF can only have a finite number of memory locations it follows that the shape (and set of coefficient values) can in practice only cover a limited range of times. For the example used to produce the results on these webpages I’ve assumed the filter only provides coefficients spanning a range of 0·5 time units before and after the ‘present moment’. This means that when the filter calculates the filtered output value required for any instant in the output waveform it only considers values in the input data stream within this time range of the instant for which it is currently working out the required output value.