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 apply EDFT for N>length(X), get the results:
where 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plus non-zero forward and backward extrapolation of X to length N and/or interpolation if unknown data inside of X have been replaced by NaN (Not-a-Number).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 resolutions along the frequency axis for both algorithms remain equal to N*length(X)*T.
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:
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. EDFT can run with limit to maximum number of iterations (input argument I) or either in non-iterative (I=1) mode
where W is 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 form of the input sequence amplitude spectrum S can be used to setup appropriate weight vector W, otherwise 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 performed iteration, the second row indicate breaking of iteration reason (see EDFT help).
6. Is it possible to estimate DFT of nonuniformly (irregularly) sampled input sequence by proposed algorithm? Yes, it is. As result, the Nonuniform EDFT (NEDFT) program introduced for processing of input sequence X sampled at arbitrary time moments tk. NEDFT call line:
[F,S]=nedft(X,tk,fn) will perform DFT of sequence X(tk) and return outputs F(fn) and S(fn).
If frequencies fn are on different grid then used by FFT and EDFT algorithms, a simple Inverse NEDFT (INEDFT) program should be applied to reconstruct Y(tn), call line:
7. Two-dimensional EDFT of array X can be calculated by applying function edft2.m, call line
See programs edft.m, nedft.m, inedft.m and edft2.m help for detailed info.
Launch also DEMO programs. Demoedft.m and Demonedft.m allows to verify the proposed algorithm's performance over iterations for the simulated test signal.
Download pdf file on site http://arxiv.org/abs/1303.2033 to get more comprehensive insight into suggested algorithm. Run program edft_fig.m to recreate computer simulation results presented in the pdf file.