IFFT of custom frequency array - confusion with the time domain sampling frequency
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From: Fred Marshall on 6 May 2008 12:37 Matt, Answers interspersed below at "***" Fred <mattwankling-google(a)yahoo.co.uk> wrote in message news:bca9bcba-2366-42f9-8464-90e97e356e13(a)k13g2000hse.googlegroups.com... > >> Matt, >> >> The time span is the reciprocal of the frequency resolution. >> So it would be the reciprocal of 0.2Hz or 5 seconds if everything else is >> correct. > > Ok, that's good. Am I right in thinking I'll only take half of the > output of the IFFT (as it should be mirrored), meaning with a > frequency resoultion of 0.2, I can get 2.5 seconds of impulse > response? > ***No. That would be closer to correct for the results of an FFT but not for an IFFT. You should be very careful about throwing out what seems "redundant" because it really isn't. That's because generally: ***The time series is real but not purely even or odd. Thus the frequency series is complex with the real part even and the imaginary part odd around zero/fs and is zero at fs/2. ***If you start with a purely real frequency sequence that is also even then the IFFt will be real and even. That's what might suggest you could "throw away" half of the result. But if you do that then you are effectively increasing the frequency resolution by a factor of 2 to 0.4Hz *and* are changing the frequency response to one that's no longer real and even. It's pretty easy to show that the magnitude response is also not at all the same. ***The time span is 5 seconds if the frequency resolution is 0.2Hz. >> >> However, you say the span in frequency is 0 to 300Hz to get the 1500 >> points >> at .2Hz resolution i.e. 300/1500=0.2. But the sample rate must be at >> least >> 600Hz here - which would require 3,000 points. > > I do have 1500 points (300/0.2), but then I mirror this in the code > (using the fliplr then creating the conjugate values for the second > half), resulting in 3000 points. Is this correct? Giving me a sample > rate of 600Hz? But then I thought this equation was the case: df=fs/N > and N now equals 6000 after mirroring, df still equals 0.2, so fs=1200 > - oh. hmmm.??! > ***Let's get all these numbers straight: Time span 5 seconds. Frequency resolution 0.2Hz. Bandwidth 300Hz. Sample rate 600Hz. N=600/0.2= 3,000 df=fs/N = 600/3000 = 0.2 ***Mind you, "N" refers to the number of complex samples in general so you would have 2*N values for N samples. Is that part of the confusion re: a factor of 2? >> >> There must be samples from 0 to fs-0.2 >> I see code that says fs=44,100. > > fs in the code is just for resampling up to that frequency to create > a .wav file, sorry, the sampling frequency for all the frequency and > IFFT stuff in my code above is fs_lf (sorry for confusion) ***I don't "get" fs-lf = fs . lf has nothing to do with anything we've dealt with above. > >> That would make 220,500 samples. >> There are other issues with just the code it seems.... have you run it at >> all? > > I have run the code, it 'seems' to give me the desired result (when I > analyse as a .wav file) if sampling frequency (fs_lf) is 1200Hz. (from > the equation: df=fs/N ) > >> >> So, for starters you need to create a frequency array that contains both >> the >> positive (0 to 300Hz) part and the negative part which is -300Hz to 0. >> Normally the negative part is the upper half of the array so it goes from >> 300Hz to fs and the value at fs is the same value as at zero - and is not >> included in the array. The frequency array is *periodic* - so this is >> how >> it's represented for a single period in frequency. > > I thought I had done this with the mirroring of the frequency array > (press). Is this not the case? ***If you did it correctly then presumably yes. Just to be clear: If you start with a purely real frequency response then it will mirror around zero - meaning that it mirrors left/right around fs/2. If the array is of even length then there is a single value at fs/2 and if the array is of odd length then there are two equal values around fs/2 with fs/2 midway between those samples. Like this: Even: [1 2 3 4 5 6 5 4 3 2] length = 10 value at fs/2 = 6; value at zero and fs = 1. Odd: [ 1 2 3 4 5 6 6 5 4 3 2] length = 11 value near fs/2 = 6; value at zero and fs = 1. So neither of them fliplr exactly at fs/2 .... > >> >> Also, if the time domain transform is to be purely real then the >> frequency >> domain values need to be real and even around zero and fs or, to be more >> general, need to be: >> - real and even >> - imaginary and odd. >> >> Fred > > Finally, I have plotted the (real) output of the IFFT and it doesn't > come out symmetrical which I thought it did, hence removing the second > half in my code. ***If the real part of the IFFT is not even then that means that the frequency samples weren't purely real. Since there appears to be no reason why your frequency response would have imaginary values to start with, I'm going to assume that they are zero. Then you have only one possibilities: - frequency samples are (real and) even If the frequency samples are (real and) even then the IFFT will be real and even. ***If the frequency samples are complex and properly formed so the real part is even and the imaginary part is odd, then the IFFT should be purely real and need not be even.. > > Thanks for your help. I really appreciate it. > > Matt > |