This is the estimated result. With the new real and imaginary part matching algorithm, it seems all problems I encountered before have gone.

This is just a part of the generated signal. Compared to the original individual signal, the estimated result seems to be quite accurate to some extent.

So, next step is to move to speech processing. Maybe not too difficult, who knows.

Donnelly, D.;

Computing in Science & Engineering [see also IEEE Computational Science and Engineering]

Volume 8,
Issue 2,
March-April 2006
Page(s):72 – 78

Abstract:

Two assumptions underlie the Fourier transform process: stationarity and linearity. When signals deviate from these conditions, the transform outcomes are suspect. A chirp, which by definition has a frequency that varies with time, doesn’t satisfy these requirements, and its fast Fourier transform (FFT) doesn’t adequately express the changing nature of the signal’s frequency content. In this analysis of a bat chirp, I first examine how the FFT handles a chirp and then how we can use a sequence of windows that individually span only a portion of the total time-domain signal to generate a frequency versus time description of the signal. The trade-off in this kind of windowing is between dynamic response and resolution: we obtain improved dynamics if we use shorter windows, whereas we get better resolution with longer windows. This article and this series concludes with a brief look at the Hilbert-Huang transform, which isn’t constrained by the same assumptions as the FFT. This transform process consists of two independent sets of operations. The first, called empirical mode decomposition, generates a set of intrinsic mode functions (IMFs), from the data. The second step extracts phase information from each IMF and its Hilbert transform. The derivative of the phase with respect to time yields the instantaneous frequency. The net effect of these operations is to transform the time-domain data to frequency versus time data instead of the amplitude versus frequency the FFT obtains.

Full Text:
PDF(464 KB)

IEEE JNL ]]> http://rocksoccer.xtreemhost.com/wordpress/archives/27/feed 0 http://rocksoccer.xtreemhost.com/wordpress/archives/26 http://rocksoccer.xtreemhost.com/wordpress/archives/26#comments Wed, 12 Mar 2008 00:11:43 +0000 Rocksoccer http://rocksoccer.xtreemhost.com/wordpress/archives/26 ScienceDirect – Signal Processing : Underdetermined blind source separation using sparse representations
Abstract

The scope of this work is the separation of N sources from M linear mixtures when the underlying system is underdetermined, that is, when M<N. If the input distribution is sparse the mixing matrix can be estimated either by external optimization or by clustering and, given the mixing matrix, a minimal l1 norm representation of the sources can be obtained by solving a low-dimensional linear programming problem for each of the data points. Yet, when the signals per se do not satisfy this assumption, sparsity can still be achieved by realizing the separation in a sparser transformed domain. The approach is illustrated here for M=2. In this case we estimate both the number of sources and the mixing matrix by the maxima of a potential function along the circle of unit length, and we obtain the minimal l1 norm representation of each data point by a linear combination of the pair of basis vectors that enclose it. Several experiments with music and speech signals show that their time-domain representation is not sparse enough. Yet, excellent results were obtained using their short-time Fourier transform, including the separation of up to six sources from two mixtures.

Full Text: PDF

I think there might be a problem with you pitch shifter when you have two or more frequency components that fall into the same bin. For example: 1) If you have 2 freq. components that fall into separate FFT frequency bins, you are OK. Each freq. component will be shifted by the same ratio. (ex. 10%) 2) If you have 2 freq. components that fall into the same FFT frequency bin, but they appear at different times, you are still OK, each freq. component will proportionally be shifted by the same ratio. 3) However, if you have 2 freq. components that fall into the same FFT frequency bin, and they happen at the same time, BOTH COMPONENTS WILL BE SHIFTED BY THE SAME AMOUNT, which means, different ratios. Is this a known issue?

Indeed, your observation is correct – this is related to the resolution properties of the DFT.
For the case you mention, there are two different scenarios:
1) Consider two closely spaced sinusoids in your input signal that fall into the same bin, for example at 500 and 505 Hz. In this case, they will be shifted both by the same amount (doubling the frequency will move them to 1000 and 1005 Hz instead of 1000 and 1010 Hz) because scaling the bin frequency just heterodynes (frequency shifts) everything within the bin bandwidth. The net effect of this shift might not be prominent enough to be noticed by the listener – depending on the DFT size (and the resulting bin frequency resolution). For small DFT sizes however this might cause the resulting sound to be inharmonic and metallic.
2) Scaling two closely spaced frequencies that are initially NOT located within the same bin might move them closer together for a bin frequency scaling factor < 1.0 which might eventually cause them to fall into the same bin. In that case, the net phase effect should be calculated differently from the way it is done in the code now. I’ve not taken this into account in the present version to keep the code readable.
One might also argue that closely spaced frequencies will cause a low frequency modulation (”beating”) in the signal, which will result in yet another bin getting involved in the process. Therefore, the pitch shifter introduced at the DSP Dimension web site is actually a simplified version of a much more complex process if you really want to do it “right”

Volume 56, Issues 1-2,

1 July 1999,

Pages 79-84

//check hte index

with 1 hz step size

1, full, known f0, different combination
2,

Omega_row and omega_col only give the fundamental frequency of every search. They are different from the
frequency indices. The fundamental frequency of the first signal is represented with
100+omega_row*search_step_size. The fundamental frequency of the second signal is represented with 100+
(omega_row+omega_col+1)*search_step_size. The second index is in fact the difference between the
fundamental frequency of the first signal and that of the second.

The problem is still how to estimate the fundamental frequency. More accurately, is it possible to evaluate the difference of amplitude of two signals in frequency domain?