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I know the Hanning window, $w_{Hann}(n) = 1 - \cos \left(2\pi \frac{n}{N} \right)$, is typically applied to discrete Fourier transforms. $N$ being the total number of points in a transient signal that is to be transformed with the DFT. However, I'm trying to have a look at how this window (which is what I use to measure) affects the spectral line shape of the resultant Fourier transform.

For now I simply say my transient signal is a damped oscillator of the form $$f(t) = e^{-t/\tau} \cos(\omega_0 t).$$The Fourier transorm of which is well known to be a Lorentzian profile. However As I measure my data with a Hanning window I want to take the Fourier transform of $$f(t) w_{Hann}(t)$$ This means I need the Hanning window in the time domain, which I assume to be simply $$w_{Hann}(t) = 1 - \cos \left(2\pi \frac{t}{T} \right)$$ where $T$ is the averaging time for my Fourier transform.

Is this a good approach and a sounds assumption I can make with the Hanning window in a time domain rather than for a discrete index of points?

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So what you mean is that you want the continuous-time Hann window instead of the discrete-time window?

$$w_{Hann}(t) = 1 - \cos \left(2\pi \frac{t}{T} \right)$$

is not correct, since it goes to 0 at 0.

Blackman & Tukey give the continuous time window as:

$$D_2(\tau)=\frac 1 2 \left (1+\cos \frac{\pi \tau}{T_m} \right )$$

for $|\tau| < T_m$ and 0 elsewhere.

Wolfram Alpha

https://en.wikipedia.org/wiki/Hann_function

https://archive.org/details/TheMeasurementOfPowerSpectra/page/n58

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  • $\begingroup$ Exactly! This was my concern, could you tell me what $\tau$ and $T_m$ are? Thanks $\endgroup$
    – Q.P.
    Jul 1 '19 at 14:09
  • $\begingroup$ Also if you have a source for this, that would be really helpful $\endgroup$
    – Q.P.
    Jul 1 '19 at 14:21
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    $\begingroup$ @QuantumPenguin $\tau$ is the argument of the function (so time if you're working in the time domain). $T_m$ is the window length $\endgroup$
    – endolith
    Jul 1 '19 at 18:56
  • $\begingroup$ One final question, when you say window length, in the context of time rather than a discrete, you mean the time that the FFT would be averaged for in seconds? $\endgroup$
    – Q.P.
    Jul 2 '19 at 12:05
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    $\begingroup$ @QuantumPenguin Windows can be applied in continuous-time or discrete-time, yes. "how do you define $T_m$ or the window length in the continuous time window?" It's the length of the window (actually half the length, with this definition). If $T_m$ is 5, then the window is 10 long. $\endgroup$
    – endolith
    Jul 2 '19 at 19:13

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