Window Presum FFT
in [DSP]
|
From: rudykeram on 28 Jan 2010 07:57 Hi everyone, I would really appreciate if someone answers to my question. I was trying to learn about presum windowing, and I found this article http://archive.chipcenter.com/dsp/DSP000315F1.html I am trying to understand the idea. I did a quick matlab simulation on this subject, and it is behaving as it's been explained in this article. So basically, let's say, if we have a hamming window of size 512, which translates to some passband frequency in the frequency domain. If, I wan't to make the passband of may hamming window narrower (let's say by a factor of 2), then this can be accomplished by making the size of my hamming window to be 1024, and then take an 1024-point FFT of this new size hamming window. The alternative, and more practical approach, is the Winodow Presum FFT, in which, similarly, we double the length of our hamming winodw. But now, instead of taking a 1024 FFT, we devide our hamming winodw into 2 chunks: samples ranging from [1:512] sapmles ranging from [513:1024] and then "Sum" these two regions up together. and Now we take the 512-point FFT of the "Sum", which ultimately is almost the same thing as taking the 1024-point FFT of our 1024-size hamming window. I hope that I addressed my question clearly. As, I said, I ran a quick matlab simulation, and it appears to be working. I am just trying to undertand the reasoning behind this, and why it works the way it works?? Any explanation? I would appreciate if someone can explain what is, in fact, happening. I really cannot see too much connection between why breaking up the time window to two lenghts and adding those will result in narrowing the passband of our window!!!! thanks, and I would appreciate in advance. Rudy
From: Jason on 28 Jan 2010 10:39 On Jan 28, 7:57 am, "rudykeram" <rudyke...(a)yahoo.com> wrote: > Hi everyone, > I would really appreciate if someone answers to my question. > I was trying to learn about presum windowing, and I found this article > > http://archive.chipcenter.com/dsp/DSP000315F1.html > > I am trying to understand the idea. I did a quick matlab simulation on > this subject, and it is behaving as it's been explained in this article. > So basically, let's say, if we have a hamming window of size 512, which > translates to some passband frequency in the frequency domain. > If, I wan't to make the passband of may hamming window narrower (let's say > by a factor of 2), then this can be accomplished by making the size of my > hamming window to be 1024, and then take an 1024-point FFT of this new size > hamming window. > The alternative, and more practical approach, is the Winodow Presum FFT, > in which, similarly, we double the length of our hamming winodw. But now, > instead of taking a 1024 FFT, we devide our hamming winodw into 2 chunks: > samples ranging from [1:512] > sapmles ranging from [513:1024] > and then "Sum" these two regions up together. > and Now we take the 512-point FFT of the "Sum", which ultimately is almost > the same thing as taking the 1024-point FFT of our 1024-size hamming > window. > I hope that I addressed my question clearly. As, I said, I ran a quick > matlab simulation, and it appears to be working. > I am just trying to undertand the reasoning behind this, and why it works > the way it works?? > Any explanation? I would appreciate if someone can explain what is, in > fact, happening. > I really cannot see too much connection between why breaking up the time > window to two lenghts and adding those will result in narrowing the > passband of our window!!!! > thanks, and I would appreciate in advance. > > Rudy Simple explanation of window presum, also known as Windowed Overlap- Add (WOLA), and I believe I've also heard it called a polyphase FFT: - A longer window allows you more freedom (i.e. coefficients) to design the frequency response that you want. You can then have resulting FFT bins that have very sharp cutoffs in magnitude with little overlap, reducing spectral leakage. Typical window lengths are 2x or 4x the desired FFT length (dictated by desired bin resolution bandwidth in the final result). you design the window as a lowpass filter whose bandwidth corresponds to the bin bandwidth that you want out of your final FFT. - You take that longer window and pointwise-multiply a chunk of your data by it. This gives you a windowed dataset that is a multiple of the desired FFT length. - You then cut the longer dataset into the 2 or 4 FFT-sized chunks that it contains, stack them up, and add them. This is aliasing in the time domain. Just as decimation in the time domain gives you aliasing in the frequency domain, aliasing in the time domain gives you decimation in the frequency domain. You need to decimate in this way because your window is designed for the bin spacing dictated by your FFT length. If you didn't perform this step, you would end up with 2x or 4x the number of bins that you want with smaller spacing between them, giving you a lot of overlap. - The resulting FFT-sized block then goes into your FFT. The resulting frequency response of each bin is dictated by the window you designed. - Assuming you have more data to process, you would typically take some stride through the data (I've seen the stride length be Nfft/2, Nfft, or even larger, depending upon what you're trying to accomplish; if you're going to use a synthesis filterbank to reconstruct the signal later, you can't stride too far) and repeat the process to get a new STFT result. Jason
From: rudykeram on 28 Jan 2010 12:09 >On Jan 28, 7:57=A0am, "rudykeram" <rudyke...(a)yahoo.com> wrote: >> Hi everyone, >> I would really appreciate if someone answers to my question. >> I was trying to learn about presum windowing, and I found this article >> >> http://archive.chipcenter.com/dsp/DSP000315F1.html >> >> I am trying to understand the idea. I did a quick matlab simulation on >> this subject, and it is behaving as it's been explained in this article. >> So basically, let's say, if we have a hamming window of size 512, which >> translates to some passband frequency in the frequency domain. >> If, I wan't to make the passband of may hamming window narrower (let's sa= >y >> by a factor of 2), then this can be accomplished by making the size of my >> hamming window to be 1024, and then take an 1024-point FFT of this new si= >ze >> hamming window. >> The alternative, and more practical approach, is the Winodow Presum FFT, >> in which, similarly, we double the length of our hamming winodw. But now, >> instead of taking a 1024 FFT, we devide our hamming winodw into 2 chunks: >> samples ranging from [1:512] >> sapmles ranging from [513:1024] >> and then "Sum" these two regions up together. >> and Now we take the 512-point FFT of the "Sum", which ultimately is almos= >t >> the same thing as taking the 1024-point FFT of our 1024-size hamming >> window. >> I hope that I addressed my question clearly. As, I said, I ran a quick >> matlab simulation, and it appears to be working. >> I am just trying to undertand the reasoning behind this, and why it works >> the way it works?? >> Any explanation? I would appreciate if someone can explain what is, in >> fact, happening. >> I really cannot see too much connection between why breaking up the time >> window to two lenghts and adding those will result in narrowing the >> passband of our window!!!! >> thanks, and I would appreciate in advance. >> >> Rudy > >Simple explanation of window presum, also known as Windowed Overlap- >Add (WOLA), and I believe I've also heard it called a polyphase FFT: > >- A longer window allows you more freedom (i.e. coefficients) to >design the frequency response that you want. You can then have >resulting FFT bins that have very sharp cutoffs in magnitude with >little overlap, reducing spectral leakage. Typical window lengths are >2x or 4x the desired FFT length (dictated by desired bin resolution >bandwidth in the final result). you design the window as a lowpass >filter whose bandwidth corresponds to the bin bandwidth that you want >out of your final FFT. > >- You take that longer window and pointwise-multiply a chunk of your >data by it. This gives you a windowed dataset that is a multiple of >the desired FFT length. > >- You then cut the longer dataset into the 2 or 4 FFT-sized chunks >that it contains, stack them up, and add them. This is aliasing in the >time domain. Just as decimation in the time domain gives you aliasing >in the frequency domain, aliasing in the time domain gives you >decimation in the frequency domain. You need to decimate in this way >because your window is designed for the bin spacing dictated by your >FFT length. If you didn't perform this step, you would end up with 2x >or 4x the number of bins that you want with smaller spacing between >them, giving you a lot of overlap. > >- The resulting FFT-sized block then goes into your FFT. The resulting >frequency response of each bin is dictated by the window you designed. > >- Assuming you have more data to process, you would typically take >some stride through the data (I've seen the stride length be Nfft/2, >Nfft, or even larger, depending upon what you're trying to accomplish; >if you're going to use a synthesis filterbank to reconstruct the >signal later, you can't stride too far) and repeat the process to get >a new STFT result. > >Jason > Hi Jason, Thanks a lot for your detailed answer. It makes a lot of sense. Now I completely understand. If I view this the way that you explained with time domain aliasing, it makes sense. thank you very much Rudy
From: dbd on 28 Jan 2010 13:27 On Jan 28, 4:57 am, "rudykeram" <rudyke...(a)yahoo.com> wrote: > Hi everyone, > I would really appreciate if someone answers to my question. > I was trying to learn about presum windowing, ... > > Rudy There is a PhD thesis on the topic out of BYU: Time Aliasing Methods of Spectrum Estimation Dahl, Jason F 1967- http://contentdm.lib.byu.edu/ETD/image/etd157.pdf Dale B. Dalrymple
From: dbd on 28 Jan 2010 14:20 And let's not forget: http://www.embedded.com/design/embeddeddsp/208800863?_requestid=336150 Dale B. Dalrymple
|
Pages: 1 Prev: Skip diving/dumpster diving Next: why first order filter have slope 6db/octave |