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I know that all periodic continuous time signal have discrete spectral representations, but are all discrete spectral representations periodic in continuous time?

Also, can all periodic signals be represented by a fourier series?

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  • $\begingroup$ Yes and yes, subject to some restrictions such as finite power. $\endgroup$ – Dilip Sarwate Feb 26 '14 at 3:31
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    $\begingroup$ No, not all discrete spectral representations are periodic in continuous time. Counterexample: $S(\omega) = \delta(\omega-1) + \delta(\omega-\sqrt{2})$. The resulting signal is not periodic, because $\sqrt(2)/1$ is not rational and the individial periods do not have a least common multiple, which would be the overall period. Also no, the Fourier series cannot represent all periodic continuous time signals. It converges on a subset of those signals. Counterexample: $f(t) = 1$ for $t=0$ and $f(t)=0$ elsewhere. The Fourier series will reconstruct $f(0)=0$. Discrete time is much better behaved. $\endgroup$ – Jazzmaniac Mar 17 '14 at 8:33
  • $\begingroup$ Good example! It's discrete but can't be uniformly sampled in a way that it stays discrete. On the other hand, this should apply to discrete time as well: $\delta(t-1) + \delta(t-2^.5)$ would also be not periodic in frequency $\endgroup$ – Hilmar Sep 9 '15 at 9:14
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    $\begingroup$ @Jazzmaniac : You should make your comment an answer. I realize this is an only question, but I think your comment is the best "answer" here. $\endgroup$ – Peter K. May 16 '16 at 13:45
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All discrete spectral representations that exist strictly on a fixed spacing grid that includes 0 (DC) represent a signal that is periodic in time. Other grids (with different irrational number spacings for instance) can indicate aperiodic waveforms in continuous time.

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This is a 2 parts answer:

  1. Given a discrete representation of the Spectral Data does necessarily represent a periodic function in the time domain?
    Well, it is by definition. Look, the DFT is a fancy name to a scaled Discrete Fourier Series, namely combination of harmonic signals. By the nature they are all an integer multiplication of a base frequency. Now think about a linear combination of finite harmonic signals which all are multiplication of known frequency -> The are all periodic in the lowest frequency. And so is the signal.
  2. Can all periodic signals be represented by a Fourier Series?
    Now you are asking about convergence and there are few requirements about signals to be represented by a Fourier Series which deals with the ability to integrate them over the period. Another issue, when you say Fourier series, remember an infinite number of signals are considered.
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i get into fights occasionally at the USENET newsgroup comp.dsp regarding the inherent nature of the DFT. but i'll repeat it here:

anytime one uniformly samples a continuous function in one domain, it makes it representable as a discrete function in that domain and it causes periodicity in the reciprocal domain. and anytime one makes a function periodic in one domain, it causes it to be discrete (appearing as uniformly sampled) in the reciprocal domain. that is always the case.

the thing that gets me in trouble with some of my peers (but not with the math, i'm quite comfortable with the DFT math) is that i (and not just me) conclude that the DFT transforms one discrete and periodic function (with period $N$) in one domain to another discrete and periodic function (having the same period $N$) in the reciprocal domain. but in both domains, the periodic function is discrete, so it is fully described with $N$ numbers in either domain.

this means that the DFT effectively periodically extends the data passed to it. you pass to the DFT (or FFT) $N$ samples, and the DFT will treat it as if it were one period of a periodic function. the DFT is essentially the same as the DFS.

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  • $\begingroup$ This really is a non-answer to the questions asked which ask whether all continuous-time periodic signals have a representation as a Fourier series? and whether given that a continuous-time signal has a discrete spectral representation, is the signal periodic? The DFT etc has nothing to do with the issue at all. $\endgroup$ – Dilip Sarwate Feb 26 '14 at 19:52
  • $\begingroup$ what is relevant to the question is that uniformly-sampled (discrete) in one domain corresponds to periodic in the reciprocal domain. that is always the case. and, Dilip, of course the DFT has something to do with that, at least as an example, since it is both discrete and period in both domains. $\endgroup$ – robert bristow-johnson Feb 27 '14 at 13:18
  • $\begingroup$ How is something that is both discrete and periodic in both domains an example that helps in understanding whether all periodic continuous-time signals have a representation as a Fourier series? $\endgroup$ – Dilip Sarwate Feb 27 '14 at 14:24
  • $\begingroup$ the Fourier series is discrete: $$ x(t) = \sum_{k=-\infty}^{\infty} c_k e^{j 2 \pi (k/P) t} $$ and adds to something that is periodic: $$x(t)=x(t+P) \ \forall t$$. $\endgroup$ – robert bristow-johnson Feb 27 '14 at 22:18
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Yes, all periodic signals can be represented by the Fourier Series, and the discrete time signals have periodic spectra (you can try this by applying DFT or FFT for a discrete time signal in MATLAB). The reason for this is you are trying to find the coefficients $X(e^{j \;\omega})$ using the formula for a DFT which has the term $\exp{(\dfrac{-j2\pi k n}{N})}$, because of which you can find periodicity in the spectrum.

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