Author on pg 292 claims the windowed-sinc "is not a filter for signals with information encoded in time domain", because its "step response has overshoot and ringing". That phrase, along "encoded in frequency domain", are used in text, but I'm unsure they're ever explicitly defined. My guess:

  • Frequency-domain encoded: periodic processes in a system - something that recurs, persists the duration of the signal, and is sensibly described by Fourier coefficients
  • Time-domain encoded: the opposite - time-local events in a system, such as a sharp rise, temporary change in DC offset, momentary noise injection.

A single square pulse would qualify for latter, as the physical system that generated it almost certainly didn't synch hundreds of sinusoidal sources just right. A noiseless EEG would qualfiy for former, as neuronal excitations recorded as polarity alignments have periodic traits.

Is this accurate? If so, how is a filter's step response relevant in describing time-local events - and are there other "responses" characterizing time-encoded information?

Note: I'm aware of the step response's role in control systems, circuit design; I ask specifically about non-periodic "events" in a signal (if that's what time-encoded means).


The simplest (imho) explanation is this.

Consider a time-domain signal which has narrow pulses or sudden jumps in value or on/off switches at certain (unknown!) time instants. Such a signal is said to carry its information in those specific time instants at which such sudden changes occur.

A PWM (pulse width modulation) signal is an excellent example for such class of signals.

If you consider the frequency domain reflection of those sudden changes in the signal amplitudes, you will see that they have wide band spectrums. They are local in time, therefore spread in frequency.

When you filter such a signal with a typical LPF (low pas filter) or alike, which discards parts of the spectrum away and passes only a subset of it, you may effectively be throwing away part of the information that's carried by the signal too.

For example, the specific instants at which the PWM wave switches on/off becomes blurry, and hence the resolution of the information is reduced.

The author seems to mention this issue with filtering of time-domain encoded information...

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    $\begingroup$ Good example, thanks, and a fairly important caveat with any low-pass filter; a pulse is akin to a jump with a temporary flatline ('no change'), then a fall - and this can describe a large variety of non-pulse events. $\endgroup$ – OverLordGoldDragon Sep 20 '20 at 18:40
  • $\begingroup$ Actually mistook low-pass for 'low-reject'; the wider the pulse, the relatively lesser the higher frequencies. Low frequencies always remain dominant, but high freq's are still pronounced for brief pulses, and "events" more resemble a briefer pulse. So I'd say in general, a low-pass is safer than a high-pass, but a narrow band-reject fares best. $\endgroup$ – OverLordGoldDragon Sep 20 '20 at 23:31
  • $\begingroup$ Any insight on the relevance on step response? In finite context it's but a sufficiently long pulse, so something about the filter's effect on the output after such a pulse is input into the system. $\endgroup$ – OverLordGoldDragon Sep 20 '20 at 23:40

Signals exist in both time- and frequency domain, the only difference is what domain is more important for a specific application. This is also true for non-periodic events (signals) - they too can be shown in frequency domain (see the difference between Fourier Series and Fourier Transform).

Consider two simple examples:

  • You want to design a current protection system. In this case you care about instantaneous values, which means that things like fast rise time and low overshoot are of interest. This is example of time-encoded signal, ie to you it is more important how signal evolves in time than what is its spectrum.

  • You want to design a digital controller. Here you care more about frequency components, eg things like filter sharp roll-off on Nyquist frequency and no pass-band ripple are of interest. In this case it is more important to know what frequency components does the signal contain, than how it evolves over time.

  • $\begingroup$ Strictly speaking, signals exist in time or space domains only. The concept of frequency is a purely a mathematical construct for defining a quantified quality of periodicity in a waveform. We can transform a time or spatial domain signal into its corresponding frequency domain by means of a Fourier transform, which describes the same signal in terms of a mixture of periodic components each with a specific frequency, or better a continuum of frequencies... $\endgroup$ – Fat32 Sep 28 '20 at 20:55

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