The connection and difference of Fs,ft,dfs,dtft,dft,fft

For beginners of digital signal processing (DSP), these kinds of transformations are the most headache, they are the theoretical basis of digital signal processing, throughout the processing of the signal.

Learning "Advanced mathematics" and "Signal and system" the two classes of friends, all know that the time domain of any continuous periodic signal can be decomposed into an infinite number of sinusoidal signals, in the frequency domain is expressed as discrete non-periodic signals, that is, the time domain continuous period corresponds to the frequency domain discrete non-periodic characteristics, This is the Fourier series expansion (FS), which is used to analyze continuous periodic signals .

FT is a Fourier transform, which is mainly used for the analysis of continuous non-periodic signals , because the signal is non-periodic, it must contain a variety of frequency signals, so has a time domain continuous non-periodic corresponding frequency domain continuous non-cyclical characteristics.

Both FS and FT are analytical tools for continuous signal spectrum, which are derived from the Fourier series theory. The continuous signals in the time domain have non-periodic characteristics in the frequency domain, but the periodic and aperiodic signals have discrete and continuous points in the frequency domain.

In nature, in addition to the existence of temperature, pressure and other continuous signal in time, there are some discrete signals, discrete signals can be obtained through continuous signal sampling, but also in itself is discrete. For example, a region's annual rainfall or average growth rate, such as signals, the time variable of this type of signal is the year, not at the integer time point of the signal is meaningless. Tools for spectrum analysis of discrete signals include DFS,DTFT and DFT.

DTFT is a discrete-time Fourier transform, which is used for discrete non-periodic sequence analysis , according to continuous Fourier transform requirements for continuous signal must be integrable in time, then for the discrete-time Fourier transform, the discrete sequence above it must satisfy the the condition of the summation of the upper number of the time axis; Because the signal is a non-periodic sequence, it must contain a variety of frequency signals, so the dtft of discrete non-periodic signal after the transformation of the spectrum is continuous, that is, sometimes domain discrete non-periodic correspondence frequency domain continuous period characteristics.


When the discrete signal is a periodic sequence, strictly speaking, the discrete-time Fourier transform is not present, because it does not meet the absolute series of signal sequence and convergence (absolute) and the necessary and sufficient conditions of the Fourier transform, but the use of DFS (discrete Fourier series) This analysis tool can still be Fourier analysis.


We know that the periodic discrete signal is made up of the infinite number of the same period sequence on the time axis, assuming that the period is N, that is, each period sequence has n elements, and such a period sequence has infinitely many, because the infinite multiple period sequence is the same, So it is possible to take only one of the periods to represent the entire sequence, which is drawn out to indicate that the period of the entire sequence characteristic is called the main value period , which is called the sequence of the main values. then using the n corresponding frequency as the base frequency to form the Fourier series expansion of the complex exponential sequence EK (n) =exp (j*2pi*k*n/n), with the main value sequence and complex exponential sequence of correlation (multiply add operation), the main value of each frequency of the spectral component, so that the spectral characteristics of the periodic sequence.


According to DTFT, it is feasible for the finite-length sequence to be used as Z-transform or sequence Fourier transform, or the analysis of the frequency domain and the complex frequency domain of the finite-length sequence has been solved theoretically, but for the digital system, there are some problems in the application of the Z-transform and the sequence Fourier transform. It is important because of the continuity nature of the frequency variable (DTFT transforms the continuous spectrum), which is not convenient for digital operation and storage.

Refer to DFS, you can use a DFS-like analysis method to solve the above problems. The finite-length non-periodic sequence can be assumed to be a main straight period of an infinite long-period sequence, that is, the periodic continuation of the finite-length non-periodic sequence , and the continuation sequence can be processed by DFS, that is, the complex exponential base frequency sequence and the finite long time sequence are correlated. The spectral component of each main value at each frequency is obtained to indicate the spectral information of the "main value period".

Since the DfT borrows from DFS, it assumes that the period of the sequence is infinite, but the interval is defined (the main value interval) in order to conform to the finite length characteristic, which makes the DfT periodic. In addition, theDFT is only a finite discrete frequency representation of a period of time, so it is discrete in frequency, it is equivalent to Dtft transform into a continuous spectrum after the sampling, at this time the sampling frequency is equal to the sequence after the continuation of the period N, that is, the number of main value sequence .

Let's talk about the relationship and the difference between DFS,DTFT,DFT,FFT

DFT and FFT are actually a kind of essence, FFT is a fast algorithm of DFT.


DFS is a discrete Fourier seriers, which carries out series expansions of discrete periodic signals. The DfT is the DFS that takes the principal value, DFS is the periodic extension of the DFT.

Dtft is to discrete time Fourier transformation, which is a sequence of ft, which gets a continuous periodic spectrum, while Dft,fft gets a finite long aperiodic discrete spectrum, not one.


The relationship between DTFT and DFT

We know that the spectrum of a n-point discrete time series Fourier transform (DTFT) is a continuous function that takes (2*PI) as a periodic continuation, by the sampling theorem we know that the time domain is sampled, and the frequency domain period is extended; Similarly, if sampling in the frequency domain, the time domain will also be extended periodically. Discrete Fourier transform (DFT) is based on this theory, in the frequency domain sampling, a period of mining n points (the same number of sequence points), so that the spectrum of the signal is discretized to obtain an important correspondence: an n-point discrete-time signal can be used in the frequency domain of an n-point sequence to uniquely determine, This is what the DFT expression reveals.



As for the DfT of discrete Fourier transform, it is also the analysis and processing of the digital signal transform to the frequency domain, which has great effect on the digital signal processing. Digital signal processing is out of the analog period to process the signal completely dependent on the device, can be directly calculated to signal processing. such as digital filter, just use the coefficient of the system to enter the digital signal for certain calculation, signal out of the system is processed after the data in the time domain expression.


The discrete Fourier transform is not identical to the Fourier transform of the continuous signal in the understanding, mainly is the Fourier transform of the discrete signal involves the periodic continuation, as well as the circular convolution and so on.



FFT, a fast Fourier transform, is a fast algorithm for discrete Fourier transform, which solves the problem that the computational amount of the discrete Fourier transform is very large and impractical, so that the computational amount of the discrete Fourier transform can be reduced by one or several orders of magnitude, so that the discrete Fourier transform is widely used. In addition, the advent of FFT solves a considerable number of computational problems, so that other calculations can also be solved by FFT.




Schematic reference: http://www.cnblogs.com/BitArt/archive/2012/11/24/2786390.html

Below, we use these two properties to illustrate the connection between Dft,dtft,dfs,fft:



Fs,ft,dfs,dtft,dft,fft's contact and differences