# Synthesis DFT Filter Bank

I have a DFT bank analyzer with N channels, but the the bandwidth of the signal occupies only the first 2 channels, therefore I'll reconstruct only these and not entire full bandwidth. I'd like to know if is mandatory to use an FFT with N points all polyphase components in the synthesis bank and . I would to use only FFT 2 points and only 2 polyphase components(8 taps each) in the synthesis bank. Is it possible?

Thanks to all Stefano

• My application of oversampled DFT FB is to change the location of the channels in the spectrum. In other hands a transponder. The bank has 16 channels, with a sampling frequency of 1.25 GSPS, therefore each channel has a bandwidth about 39 MHz, sampled at 156.250 MHz each (because oversampled). – Stefano Mar 14 '16 at 20:21
• In the analysis bank I have 16 polyphase components with 8 taps each, and IFFT 16 points. My system has to live in the FPGA, so i don't want reconstruct all bandwidth but only those of my interest. Hence, suppose to have a signal that occupies the first 2 channels, I only want reconstruct these. In the synthesis bank I wouldn't to implement entire FFT16, but only FFT2 and only 2 polyphase components, in this way my reconstructed signal is still sampled at 312 MHz (because 2 channels). – Stefano Mar 14 '16 at 20:21
• Your question has beeen answered. Do not hesitate to vote for the useful ones and accept the most suitable – Laurent Duval Feb 9 '17 at 17:13

Any spectrum that is between the two DFT bin centers ends up getting represented, in various proportions, in all the DFT bins (in the rectangular window case, the proportions decay as in a Sinc function, roughly by 1/(i-k)). Those proportions might be needed for your synthesis to be an accurate enough reconstruction.

Using 2 bins alone usually creates a beating effect at a low beat frequency.

• I give you more informations about my filter bank. This has 16 channels oversampled of factor of 2. I don't want to do processing in the middle, but only change the order of channels in frequency. Infact, to process only my bandwidth it will mean to have a rate of 1 channel at outputs of synthesis , rather than main sampling frequency, needed for all channels. – Stefano Mar 14 '16 at 19:40
• @Stefano coould you please add this comments to your questions, and provide more details on why you need to change the order of channels? – Laurent Duval Mar 14 '16 at 20:08

If you only want to reconstruct your signal, as perfectly answered by @hotpaw2, the answer is no in general, especially if you have finnite length windows. But I do not see the point of performing analysis and synthesis without processing in the middle.

If you do perform some processing or detection inbetween, then errors in reconstruction (ripples, leakage) can become negligible, especially if you use some amount of oversampling and use optimized inverses.

An option though: downsample your sample first at a $2/N$ rate, and you can use the DFT filterbank on the full bandwidth, yet at a reduced rate.

[EDIT] After your mail, it took me a while to understand... something. Let me try to rephrase it (and tell me if wrong). You have a 16-channel analysis filter-bank. You keep the first 2 channels, $c_1$ and $c_2$. You have a 16-channel analysis filter-bank, and you want to split it into two polyphase components $d_1$ and $d_2$. From what I understand, both A and S frequency comments won't match.

Yet, you can try to perform an implementation, input a random noise, see what happens, and report the results here.

• I don't understand if I replace full FFT N points in the synthesis bank with only FFT 2 points and 2 polyphase components.. – Stefano Mar 14 '16 at 19:48
• To saving hardware resources I wouldn't implement whole FFT in synthesis bank, because I have to reconstruct only 2 channels. I would to know if is possible with this configuration (FFT2 + 2 polyphase components). – Stefano Mar 15 '16 at 9:53
• My application of oversampled DFT FB is to change the location of the channels in the spectrum. In other hands a transponder. The bank has 16 channels, with a sampling frequency of 1.25 GSPS, therefore each channel has a bandwidth about 39 MHz, sampled at 156.250 MHz each (because oversampled). – Stefano Mar 15 '16 at 19:48
• In the analysis bank I have 16 polyphase components with 8 taps each, and IFFT 16 points. My system has to live in the FPGA, so i don't want reconstruct all bandwidth but only those of my interest. Hence, suppose to have a signal that occupies the first 2 channels, I only want reconstruct these. In the synthesis bank I wouldn't to implement entire FFT16, but only FFT2 and only 2 polyphase components, in this way my reconstructed signal is still sampled at 312 MHz (because 2 channels). For the synthesis filter I thought the same of analysis but downsampled of factor 8, in the case with FFT2 – Stefano Mar 15 '16 at 19:50