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Causality of a LTI transfer function $G(\tau)$ in the continuous time domain, i.e. for $$y(t)=\int G(\tau)x(t-\tau)d\tau$$ is characterized by $$G(\tau < 0) = 0$$ By the way, in the frequency domain, causality is characterized by the Kramers-Kronig relations, which is why I am particularly interested in the Fourier transform $G(\omega)$.

Now I am analyzing the functions $x(t)$ and $y(t)$ in discrete time and for a finite set of equidistant samples (measurements). In order to work in the Fourier domain, I will have to assume that they are periodic functions (or rather series): $$x(t+T)=x(t) \qquad y(t+T)=y(t) \qquad G(t+T)=G(t)$$ Even with such techniques as zero-padding, they are still periodic as seen from the view of FFT.

But what means causality for periodic functions? If $G(t)$ is the response of a periodic Dirac comb, it is not clear if it is the causal response of the Dirac impulse at $t=0$ or the non-causal 'response' at some of the later Dirac impulses at $t=nT$ where $T$ is the measurement duration. Strictly speaking, it is the response to all the Dirac impulses in the comb and it makes no sense to define the notion of 'before or after' the impulse.

Am I missing something? Is there a proper notion of causality for periodic functions and hence on finite domains?

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  • $\begingroup$ Interesting question and appears that the notion of causality is lost in the Discrete Fourier Transform specifically since both time and frequency become periodic and thus extend (unroll) to the equivalent of non-causal periodic functions only. However I believe the Kramers-Kronig relations would still apply (I need to confirm) in that the imaginary and real portions of a system's frequency response are related by the Hilbert Transform for minimum phase systems only -- is that relationship your interest in this? $\endgroup$ – Dan Boschen May 8 at 15:01
  • $\begingroup$ Since I am still learning about control theory, I am not sure if I understand you correctly. Note that I don't want to design a control system, but I want to model a physical system (say, 'open-loop'). Is minimum phase the term that corresponds to causality for the continuum? Then, yes, that's what I am interested in. But while I exactly know what causality means, I don't know what minimum phase means for a (discretized) physical system... $\endgroup$ – oliver May 8 at 16:18
  • $\begingroup$ Note also that I have found something about the discrete form of Kramers-Kronig/Hilbert transform in this paper: researchgate.net/publication/…. But it doesn't seem to explain what the formula means in terms of causality for a discrete series either. $\endgroup$ – oliver May 8 at 16:37
  • $\begingroup$ Yes I am referring to open loop systems - nothing to do specifically with control systems. A discrete physical system will have a response to its output given an impulse at its input; the minimum phase system is the one that has the quickest response time (decays the fastest) compared to any other system that has the same magnitude response in frequency. (And min phase means all the zeroes are inside the unit circle). But what is it you are looking to do ultimately that is related to the Kramers-Kronig relation? As if the system isn’t minimum phase you cannot derive the phase from the amp resp $\endgroup$ – Dan Boschen May 8 at 16:50
  • $\begingroup$ I want to derive a phenomenological transfer function from input measurements $x$ and output measurements $y$ in a linear prediction way (solution of Yule-Walker equations, well at least kind of...). I am worried that what I find describes a non-causal (and hence, unphysical) system. To rule that out, I need a sound understanding of what causality means for the solution space (which is discrete periodic if I compute the auto-/cross-correlations of $x$ and $y$ from FFT). I could also compute $G(\omega)=|y|^2(\omega)/|x|^2(\omega)$, but this would not be causal because it is centered around t=0. $\endgroup$ – oliver May 8 at 18:19
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I am interested in this question and since no-one has answered yet I will offer my further thoughts (up for debate or confirmation):

I agree with the OP's perspective that the Discrete Fourier Transform (DFT) fundamentally represents when cast to domains that extend to $\pm \infty$ periodic sequences in time (and in frequency) and therefore is fundamentally non-causal due to the time periodicity specifically.

However, to utilize these transforms with context of causality, we can consider the cases that best match an approximation of the Continuous Time Fourier Transform for the same causal representation of our sampled system. Such cases are the ones with no time aliasing, which would occur when $x[n]=0$ for $n\ge N/2$ given $N$ samples with $n \in [0, N-1]$. When we can restrict ourselves to causal sequences specifically (meaning when we know the underlying process is causal) then this restriction can be extended to be $x[n]=0$ for $n\ge N$ given $N$ samples with $n \in [0, N-1]$. This means basically that we know the time duration of the data capture exceeds the expected response time of the underlying continuous time process.

Further in frequency for such an equivalently causal time domain waveform the phase would be going increasingly negative versus frequency indicating the delay that is indeed properly being modeled.

It is the counter case, when for a known causal sequence $x[n]$ has significant non-zero values in the upper-half of the time domain samples right up to the $N-1$ boundary that we cannot be reasonably assured we are not observing the effects of time domain aliasing. We are no longer able to uniquely distinguish between causal and non-causal sequences, or more specifically and practically when we know the sequence is causal; we are no longer able to distinguish from underlying continuous-time responses that went beyond the sample duration of the sequence due to time domain aliasing (so specifically in the OP's case there is no way to know if the solution is representing longer response times than the samples are providing).

This is not very different from what I would typically do when evaluating results in the frequency domain: if I observe strong spectral content right up to the Nyquist boundary I am not assured that my sampling rate is high enough (or filtering is not tight enough) as frequency domain aliasing is likely occurring. While if the spectrum rolls off to sufficiently low levels prior to the Nyquist boundary (and if I can assume the system in question would not have higher spectral content at further offsets) then I am reasonably assured that the digital representation accurately represents the continuous time spectrum. This obviously requires knowledge of the system and what the likely spectral content is beyond the sampling rate, but is very analogous to the time domain challenge presented here.

For further details on matching an underlying CTFT with the DFT scroll down to the "Exact CTFT Result from DFT" section of my answer at this post: Why do we have to rearrange a vector and shift the zero point to the first index, in preparation for an FFT?

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  • $\begingroup$ Thanks for your deep thoughts. Especially the aliasing metaphor sounds convincing to me. So what you say is, causality is fuzzy in discrete limited time, i.e. we can assume that a certain response is causal if it decay quickly enough, but it may well be that it has some non-causal content in the time-domain aliasing sense you mentioned. $\endgroup$ – oliver May 9 at 8:52
  • $\begingroup$ @oliver also see this related question that you inspired: dsp.stackexchange.com/questions/67328/…. I wouldn't say fuzzy but that I would agree with you that causality doesn't exist since to me discrete in frequency and limited in time is mathematically equivalent to periodic in time which is NOT causal. However we use discrete representation of what we know is otherwise causal in infinite time--we just lose that information so we need to state it. No different than the spectrum of a time sampled waveform-- $\endgroup$ – Dan Boschen May 9 at 11:20
  • $\begingroup$ The time sampled waveform is periodic in frequency, but if we KNOW what frequency band the waveform was in then we can proceed with that representation but to be complete as far as information goes we would need to state this, it is not self-evident in the waveform samples themselves. So in your case you would need to know the process is causal, state it, and ensure the time capture duration exceeds the time duration of the signal it represents. $\endgroup$ – Dan Boschen May 9 at 11:22
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A DFT places an inherent finite length window on the infinite time domain, and the IDFT places an inherent finite length window on the infinite frequency domain. The transform to both of these windows is a Sinc shaped response in the other domain.

Note that since anything with finite support in one domain is infinite in the other, the inherent window of any finite length DFT or IDFT will always produce a non-zero noise floor, either/or in aliasing, truncation noise, and/or causality, in the other domain.

So, in practice, one needs to make sure that the length of the DFT or IDFT is long enough such that the Sinc ripples are below your desired noise floor at the window boundaries in the other domain, otherwise the finite window length will allow aliasing in the frequency domain, and non-causality and window edge clicking noise in the time domain.

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