Extended DFT

Program EDFT produce high-resolution N-point DFT for N greater than the length of data vector X.

EDFT (Extended Discrete Fourier Transform) algorithm produces N-point DFT of sequence X where N is greater than the length of input data. Unlike the Fast Fourier Transform (FFT), where unknown readings outside of X are zero-padded, the EDFT algorithm for calculation of the DFT using only available data and the extended frequency set (therefore, named 'Extended DFT'). EDFT function application is simple and similar to the FFT, besides EDFT have the following additional features:

1. EDFT can extrapolate input sequence X to length N. That is, if applied EDFT to N>length(X), get the results:

F=edft(X,N)=edft(Y)=fft(Y), where Y=ifft(F) and the length(Y)=N.

Y is X plus non-zero forward and backward extrapolation of X to length N and/or interpolation if unknown data inside of X have been replaced by NaN (Not-a-Number).

2. EDFT can increase frequency resolution up to 1/(N*T), where T is sampling period. It is well known, that zero-padding do not increase frequency resolution of DFT, therefore the resolution of FFT algorithm is limited by the length of sequence length(X)*T. Of course, there is no magic, just FFT resolution is equal on all N frequencies, while EDFT is able to increase the resolution on some frequencies and decrease on others. The sum of reciprocals of resolutions along the frequency axis for both algorithms remain equal to N*length(X)*T. Thus, an increase in resolution is dictated by well-known laws of physics, from where, the longer data, the higher the resolution has been achieved.

3. EDFT can estimate amplitudes and phases of sinusoidal components in sequence X. Like as FFT output fft(X,N)/length(X) is proportional to amplitudes of sinusoids in X, also adding a second output argument for EDFT return the amplitude spectrum S of sequence X:

[F,S]=edft(X,N) or [F,S,f]=edft(X,N), where also frequency set used by EDFT is returned in f.

4. Input sequence of EDFT may contain NaN. The proposed algorithm can interpolate and reconstruct of missing readings or even data segments (gaps) inside of sequence X. You just need to replace unknown readings by NaN and run edft(X) or edft(X,N).

5. Is it possible to estimate DFT of nonuniformly (irregularly) sampled input sequence by the proposed algorithm? Yes, it is.

[F,S]=edft(X,N,tk) computes the EDFT of X using the sample points tk.

If N is a vector then EDFT is computed at the query points fn defined in N and N set to be equal length(fn).

If N is a scalar then the query points default to fn=ifftshift(-ceil((N-1)/2):floor((N-1)/2))/N.

If tk is not specified ([]) then tk=(0:length(X)-1).

If the data is nonuniform and/or frequencies fn are not in FFT grid, then IEDFT function, call line: Y=iedft(F,f,tk); can be used as the inverse Fourier transform. Otherwise, Matlab function IFFT can be applied to get uniformly re-sampled version of input sequence X.

6. EDFT can run with a limit to the maximum number of iterations (input argument I) or either in non-iterative (I=1) mode or with additional parameters

[F,S,f]=edft(X,N,tk,I) or [F,S,f,Stopit,A]=edft(X,N,tk,I,W),

where W is the weight vector and consisting of specific weights for each frequency in F. W is proportional to the amplitude spectrum of the signal. So, a`priori knowledge about the form of the input sequence amplitude spectrum S can be used to setup appropriate weight vector W, otherwise the default (equal) weight W=ones(size(F)) will be applied for the first iteration.

'Stopit' is an informative (optional) output parameter. The first row of 'Stopit' showing the number of the performed iterations, the second row indicates breaking of iteration reason and the third row output the algorithm used to compute EDFT.

'A' is the Extended Fourier basis matrix. The EDFT of vector-row is F=X*A or A*X if X is vector-column, and F(:,l)=A(:,:,l)*X(:,l) if input is a matrix. Any NaN in X must be replaced by zero before matrix multiplication.

edft(___) with no output arguments plots N-point Fourier transform F and amplitude spectrum S by frequencies f in the current figure window.

7. Two-dimensional EDFT of array X can be calculated by applying function edft2.m, call line: F=edft2(x,mrows,ncols). The inverse transform to two-dimensional EDFT is Matlab ifft2 function, call line: Y=ifft2(F).

See programs edft.m, iedft.m and edft2.m help for detailed info.

Launch also DEMO program. Demoedft.m allows to verify the proposed algorithm performance over iterations for the simulated test signal and make a comparison with the Matlab function NUFFT.

Download pdf file on site http://arxiv.org/abs/1303.2033 to get more comprehensive insight into suggested algorithm.