0
$\begingroup$

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?

|improve this question
$\endgroup$
1
$\begingroup$

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

|improve this answer
$\endgroup$
  • $\begingroup$ 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
  • 1
    $\begingroup$ @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
  • 1
    $\begingroup$ @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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.