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From paper:

Bradford R., Dobson R., ffitch J. - Sliding is Smoother than jumping

In chapter 6 - Signal Reconstruction, the inverse of the sliding DFT can be achieved by this formula:

$$f_0=\frac{1}{N}\sum_{k=0}^{N-1}F_0(k)e^{-2\pi i k (0/N)} = \frac{1}{N}\sum_{k-=0}^{N-1}F_0(k)$$

In other words, the average of all the frequency bins is equal to the first sample of the window.

But I wonder, if the input signal is a single sine wave, then the DFT of the signal will be the same throughout the whole signal. And then using the above formula, the resynthesized samples are all going to be the same number, when in reality it has to vary from -1 to 1.

Am I understanding this wrong?

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Possible roots for this misinterpretation are three-fold, based on a confusion in Fourier transforms' properties.

if the input signal is a single sine wave, then the DFT of the signal will be the same throughout the whole signal

No, because only the Fourier magnitude might be invariant to shifts, on whole signals (infinite, with a sine). As said by @hotpaw2 before, there is a phase term.

No as well because you have a rectangular window on your sine (or finite length), and the shift property won't apply unless you have exactly an integer number of full periods in you rectangular time frame (check out this answer by @Peter K. on periodicity and discrete signals).

Finally, if you look at chapter 4 for the same paper, you will see that:

But that only gives the contribution to the sample from a single point in the collection of windows that contain the required sample point.

Additional information might be found in The sliding DFT by Jacobsen and Lyons, and Understanding and Implementing the Sliding DFT.

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  • $\begingroup$ I was surprised to see the equation in the right column on the first page of the "Sliding is better Than Jumping" paper. That equation, which often shows up in early papers discussing the Sliding DFT, is incorrect. The correct Sliding DFT equation is presented by Jacobsen in his blog at: dsprelated.com/showarticle/776.php $\endgroup$
    – Richard Lyons
    Dec 13, 2015 at 23:24
  • $\begingroup$ @Richard Lyons Thank you, I believe I gave a quite related link (the second one) $\endgroup$
    – Laurent Duval
    Dec 14, 2015 at 5:44
  • $\begingroup$ Laurent Duval: Thank you for your "Additional information" sentence! Both the correct as well as the incorrect equation for implementing the Sliding DFT appear in many places on the Internet. Eric Jacobsen told me he wrote his blog in an effort to "settle the matter" once and for all. $\endgroup$
    – Richard Lyons
    Dec 15, 2015 at 3:21
  • $\begingroup$ @Richard Lyons Sorry, but how is the equation different? I really can't see the difference between the equation in "Sliding is better Than Jumping" paper and the equation in dsprelated.com/showarticle/776.php. $\endgroup$
    – MujjinGun
    Dec 15, 2015 at 6:19
  • $\begingroup$ Mujjingun: Oh shoot! Please forgive me. When I looked at the first equation in the "Sliding is Smoother" paper my brain registered it as the incorrect version where only the previous output sample is multiplied by the complex exponential. My mistake. The first equation in the "Sliding is Smoother" paper is correct! I do apologize for the confusion. $\endgroup$
    – Richard Lyons
    Dec 15, 2015 at 13:14
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The DFT won't be the same as it is slid along a real-valued sine wave. The magnitude of an FFT result bin might be the same, but its phase will change (rotate) as the DFT is slid. Thus the sum of a rotating phasor and its complex conjugate mirror will vary (sinusoidally in fact).

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