Hi Morag and Jack, Thank you for the prompt response. It was great to learn about the Hermitian trick to combine 2 real FFTs - that's very interesting.
So, unless I am wrong, perfect reconstruction / inversion is not possible because the information in the Nyquist bin is thrown away? (I hope I am wrong) Cheers, Nikhil On Fri, 4 Feb 2022 at 05:41, Jack Hickish <[email protected]> wrote: > > > On Fri, 4 Feb 2022 at 07:08, Morag Brown <[email protected]> wrote: > >> Hey Nikhail, >> >> The FFT is a Hermitian function, which means that it has the property: >> >> [image: Screenshot from 2022-02-04 08-34-24.png] >> >> This principle is used in the real wideband FFT to compute 2 real FFTs >> using one complex FFT core - this >> <http://www.hyperdynelabs.com/dspdude/papers/COMPUTING%20THE%20FFT%20OF%20TWO%20REAL%20SIGNALS%20USING%20A%20SINGLE%20FFT.pdf> >> paper >> explains it well. For a detailed explanation on how the CASPER FFT works >> specifically, Ryan Monroe's paper on Improving the Performance and Resource >> Utilization of the CASPER FFT and Polyphase FIlterbank goes into quite a >> lot of detail. >> >> > Hi Mikhil, Morag, > > >> You only get 32 output channels because the CASPER wideband FFT discards >> all negative frequency components as they're just a mirror of positive >> frequency components and aren't needed. >> > > This is completely right, but to address the subtlety Mikhil raised about > N/2 vs N/2+1 chans -- > > The CASPER FFT doesn't give you the "last" FFT bin (the one whose edge is > at the Nyquist frequency). This bin, like the DC bin, contains real-valued > output for real-valued input (since the DC bin coefficients are all 1 and > the Nyquist bin coefficients are +/-1), so I think the CASPER FFT could use > the DC bin that it does output to encode both the DC and Nyquist bin > contents (one in the real part, one in the imag). I'm fairly sure the > CASPER FFT doesn't do this but I also vaguely recall this coming up in > conversation before. I think the issue is not that the FFT internals don't > compute the relevant numbers but is simply that outputting N/2+1 channels > would lead to an annoying input:output data ratio. > > Cheers > Jack > > >> Morag >> >> On Fri, Feb 4, 2022 at 7:10 AM Nikhil Mahajan <[email protected]> >> wrote: >> >>> Dear CASPERites, >>> >>> I am a graduate student at the University of Toronto (working with >>> Marten van Kerkwijk) and I have some raw baseband data collected using >>> PUPPI (Arecibo) - and I am on a quest to invert the polyphase filter bank. >>> I have 32 channels of complex-baseband that I would very much like to >>> combine into a single 100 MHz bandwidth stream. >>> >>> To do this, I would need to understand some of the specifics of the >>> filter bank pipeline (so that I can successfully invert each step). This is >>> my current understanding of what happened to the data I have: >>> >>> 1. Real-valued data sampled at 200 MS/s arrives at the Casper BEE2 board. >>> 2. This goes through a real-input PFB implementation such as >>> `pfb_fir_real` and using a 12-tap, 64-branch polyphase filter (I have the >>> filter coefficients that were used here). This step outputs 64 streams of >>> real-valued data. >>> 3. Then, for the DFT step of the filterbank, the 64 real-valued streams >>> are passed through the `fft_wideband_real` block to get 32 channels of >>> complex-valued data. >>> 4. This is then saved to disk. >>> >>> (I hope someone familiar with PUPPI can correct me here if I am wrong >>> about any of the above) >>> >>> Step 3 is the step I am confused about. `fft_wideband_real` does not >>> appear to be a conventional real-input N-point FFT implementation (Else I >>> would have N/2 + 1 channels instead of just N/2). Some documentation on >>> this block says that it "computes the real-sampled Fast Fourier >>> Transform using the standard Hermitian conjugation trick". What is this >>> standard Hermitian conjugation trick? I am totally unfamiliar with this. >>> Would I be wrong in guessing it uses some sort of trick to convert 64 real >>> numbers to 32 complex numbers and then applies a regular ol' complex-valued >>> FFT on them? >>> >>> Thank you so much! I appreciate any and all guidance this mailing list >>> can provide. >>> >>> Cheers, >>> Nikhil Mahajan >>> >>> -- >>> You received this message because you are subscribed to the Google >>> Groups "[email protected]" group. >>> To unsubscribe from this group and stop receiving emails from it, send >>> an email to [email protected]. >>> To view this discussion on the web visit >>> https://groups.google.com/a/lists.berkeley.edu/d/msgid/casper/CAA39X0%2B3z051d%2B%2BAB0D%3D1LHNHdOSDJp6JEYzzRXatj-ij11_qA%40mail.gmail.com >>> <https://groups.google.com/a/lists.berkeley.edu/d/msgid/casper/CAA39X0%2B3z051d%2B%2BAB0D%3D1LHNHdOSDJp6JEYzzRXatj-ij11_qA%40mail.gmail.com?utm_medium=email&utm_source=footer> >>> . >>> >> -- >> You received this message because you are subscribed to the Google Groups >> "[email protected]" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to [email protected]. >> To view this discussion on the web visit >> https://groups.google.com/a/lists.berkeley.edu/d/msgid/casper/CAGH-0Tc%2BWk1JExurEHYFPM8MarkmTY_TnjFqPpjy2a%3D7zqxvng%40mail.gmail.com >> <https://groups.google.com/a/lists.berkeley.edu/d/msgid/casper/CAGH-0Tc%2BWk1JExurEHYFPM8MarkmTY_TnjFqPpjy2a%3D7zqxvng%40mail.gmail.com?utm_medium=email&utm_source=footer> >> . >> > -- > You received this message because you are subscribed to the Google Groups " > [email protected]" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To view this discussion on the web visit > https://groups.google.com/a/lists.berkeley.edu/d/msgid/casper/CAG1GKSnnp18TS5YfGmS9W5dDHyOf7GsMNjYtWVKkRrbn2GJCmw%40mail.gmail.com > <https://groups.google.com/a/lists.berkeley.edu/d/msgid/casper/CAG1GKSnnp18TS5YfGmS9W5dDHyOf7GsMNjYtWVKkRrbn2GJCmw%40mail.gmail.com?utm_medium=email&utm_source=footer> > . > -- You received this message because you are subscribed to the Google Groups "[email protected]" group. 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