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Near Perfect Reconstruction Polyphase Filterbank

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Near Perfect Reconstruction Polyphase Filterbank



03 Aug 2007 (Updated )

a near perfect reconstruction polyphase filterbank with two times oversampling

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This set of files can be used to construct a polyphase filter bank with near perfect reconstruction. The filter bank is implemented using two uniformly modulated DFT filter banks that are frequency shifted by 1/2 channel width. Effectively, two-times oversampling is used.

The number of channels must be a multiple of two. The number of taps per channel can be freely selected. A script is provided for designing the prototype filter, which is based on a 'root-raised-error-function'. The prototype has exactly linear phase.

With 128 taps per channel an average reconstruction error of -300 dB can be achieved. Channel separation is better than 150 dB for neighboring channels, and better than 300 dB for channels that are further apart.

The processing speed is around 200 kSamples/second, for 32 taps / channel, irrespective of the number of channels.
With enough memory, a large number of channels (e.g. 64K) is feasible.

Required Products Filter Design Toolbox
MATLAB release MATLAB 7.4 (R2007a)
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Comments and Ratings (22)
13 Apr 2015 kranthi thatikonda

Thank you Wessel Lubberhuizen

if it compared with integer oversampling and WOLA structures it gives more Reconstruction error.
and ability of the system to separate in the frequency domain the signal under consideration into two or more signals or to compose two or more different signals into a single signal not exceptional if the no. of channels and iterations are more.

23 Sep 2014 Xianhai

Very nice jobs!

01 Mar 2014 Homa Esfahanizadeh  
18 Apr 2013 Yang

Yang (view profile)

Hi Wessel,

Thank you for your perfect job!

I have a question here. You mentioned that this is an oversampling implementation (two-times. For the output of npr_analysis, is it already decimated? Similarly, for the output of npr_synthesis, is it already interpolated?

Thank you very much!

30 Nov 2012 Ken

Ken (view profile)

Hello,Wessel Lubberhuizen!
I am a student of Chinese Academy of Sciences.It is really great share your work.I have run your Matlab code on Near Perfect Reconstruction Polyphase Filterbank, but I don't quit understand the principle on this algorithm.Would you please share the paper about the algorithm.
Your response will be appreciated!

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04 Jun 2012 Kanishk

Hello Mr Wessel,
I studied your code and I must commend you for your wonderful work.
However, I am not able to understand how it works. All the scripts and references I have read, speak about two analysis IDFT filter banks where the input signal is fed with and without delay of M/2 (M = number of polyphase components).
In your code, you introduced a frequency shift for the lower filter bank signal and inverted the sign for every alternate column. Could you perhaps provide your reference. I would very much appreciate it.

Also, the rref function was genius :)

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20 Feb 2012 Gilberto Romero

Thanks a lot!

22 Jun 2011 Sanjay Sindhi

Hi all, Can anybody share code for Modified DFT (MDFT) filter bank?

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12 Apr 2011 Sanjay Sindhi

@Wessel: Hi, could you please give references for your code? I mean the paper or text book you referred for two times oversampled DFT filter bank structure and its theory. It would be a great help Mr. Wessel.

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05 Mar 2011 Wessel Lubberhuizen

The Root Raised Error Function (RREF) is defined as sqrt(1-erf(x)), where erf(x) is the well known error function. It is used to shape the frequency response of the subband filters in such a way that neighboring filters are power complementary. It is quite similar to a Root Raised Cosine Function. The RREF is a smooth, rapidly decreasing function. The RREF gives better results than the RRCF when approximated by a finite number filter coefficients.

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31 Oct 2010 Arash

Arash (view profile)

Hi. would you please explain to me what is Root Raised Error Function meaning? I searched it a lot but I didn`t underestand it. would you please explain it for me or give me some reference?

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03 Feb 2010 Rajkumar Ramasamy  
22 Nov 2009 Vlad

Vlad (view profile)

Hi Wessel Lubberhuizen,
Thanks for your response!
In fact, I've carried out some investigations about filter banks.
Now I can say that dct/idct filter bank differs from fft. Sorry for asking about such functionality :)
Actually in case of real-valued filter-banks one should use two different prototype filters and two parrallel phase-shifted dct.
It is told everywhere that real-valued oversampled filter banks has more complex analysis/synthesys than complex-valued.

By the way, I've found the most interesting filter bank implementation in the "Multirate Signal Processing" by Crochiere and Rabiner. It is called Weighted Overlap-Add (WOLA) and it allows to use non-integer oversampling rates.

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08 Nov 2009 Wessel Lubberhuizen

Hi Wu Liangming, Vlad,

The two statements:
x2(i,:) = x2(i,:) * exp(sqrt(-1)*pi*(i-1)/N);
x2(i,2:2:M) = -x2(i,2:2:M);

Are equivalent to:

x2(i,j) = x2(i,j) * exp(sqrt(-1)*pi*((i-1)+j*N)/N)

Indeed, this corresponds to a frequency shift by 0.5 times the subband width.

Vlad, you are right, this can also be done by the fft. In that case, you would first combine x1 and x2 to a single matrix, which is passed though the fft. I'll try to update the code so it incorperates this idea, and also check whether it works with dct instead of fft.

For an approach to reduce the delay of a filter bank, see:

28 Sep 2009 Vlad

Vlad (view profile)

Hi, I'd like to thank you for this job, but I have a question about real-valued signals too.
Is it possible to replace fft/ifft pair with dct/idct?
Due to theory the only difference: fft shifts prototype only to the 0 .. pi/2 range, while dct shifts prototypes to 0..pi/2 range, and its copy to -pi/2 .. 0.
In fact I don't quite understand why you multiply x2 by exp(-1..) before the fft. Again theoretically fft contains the same exp(-1) which shifts prototype filter.

08 Sep 2009 Wu Liangming

I've read your work, and have some questions.
What is the need of doing
x2(i,:) = x2(i,:) * exp(sqrt(-1)*pi*(i-1)/N);
x2(i,2:2:M) = -x2(i,2:2:M);
in the analysis part and it's corresponding in synthesis. It seems to make a frequency shift. I understand the first line will make a frequency shift and I don't know what is the second line doing.
Besides, is it possible to reduce the total latency by any means, even with some complexity gain.

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30 Aug 2009 Wessel Lubberhuizen

Hua, you can make a complex signal real by upsampling and frequency shifting. But why would you want to do that?

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22 Jul 2009 Hua

Hua (view profile)

The output of analysis filter is complex signal. How to make it real?

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15 Feb 2009 Wessel Lubberhuizen

That is correct. A high spectral resolution inherently requires a large temporal delay.

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14 Feb 2009 Maciej Bartkowiak

There is a significant delay introduced by the analysis followed by synthesis. The signal seems to be delayed by N*(L-1)/2 samples

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18 Mar 2008 Wei Zhang

Nice job! Thank you very much!

05 Feb 2008 mahantesh ambole


03 Aug 2007

added comments to the code
fixed linter warnings
various improvements

06 Aug 2007

changed category, description

27 Nov 2008 1.1

removed dependency on Filter Design toolbox

07 Jun 2010 1.2

improved prototype filter design method to achieve -300 dB reconstruction error.

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