Near Perfect Reconstruction Polyphase Filterbank

a near perfect reconstruction polyphase filterbank with two times oversampling

Description

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)

[mahantesh ambole]

good

[Wei Zhang]

Nice job! Thank you very much!

[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

[Wessel Lubberhuizen]

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

[Hua]

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

[Wessel Lubberhuizen]

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

[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.

[Vlad]

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.

[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:
http://iwaenc05.ele.tue.nl/proceedings/papers/S04-05.pdf

[Vlad]

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.

[Rajkumar Ramasamy]

 

[Arash]

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?
thanks

[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.

[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.

[Sanjay Sindhi]

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