Gray-scale image--discrete time Fourier transform of frequency domain filter Fourier transform (DTFT)

Study Dip 22nd Day

Reprint please indicate the source of this article: Http://blog.csdn.net/tonyshengtan, Welcome to reprint, found that the blog is reproduced in some forums, the image can not be normal display, unable to express my views, to this expression is not satisfied ....

The opening crap .originally did not want to write dtft, Reason 1, and the front Fourier transform (FT) derivation process is similar, cause 2, in the image processing DTFT application is not very extensive, but later thought or writes out, reason 1, does not write out I feel the heart is not steadfast, the reason 2,dtft is the DfT close relatives, Do not write the words of the family is incomplete, the next write DFT, in fact, write to this stage, to write something less many, because many are quoted in front of the conclusions and some nature. But write it down, for the sake of your heart. forget who the Chinese previous generation of scientists said, "do scientific research can not fool, you fool it, it will fool you." "I am not a scientific research, but the feeling is to learn the bottom of the heart, not the future where the problem will always doubt their knowledge base has problems." Also, I hope that our contemporary scientific research workers can do a good job in research, all bad teaching and do not study, it is really a bunch of waste. derivation of discrete-time Fourier transform from Fourier series of discrete periodic signalsA certain finite sequence x[n], which in a certain phase n (n1<=n<=n2) is not 0, its external, all 0, using this signal to construct a periodic signal x ' [n],x[n] is X ' [n] a period.

x ' [n] is a periodic signal, so it has a Fourier series, and its Fourier series is periodic:within n x ' [n]=x[n], replace sum content as:
x[n]=0 outside N; so define the function:
so the coefficient ak is proportional to X (E^JW):
where w0=2*pi/n, the upper formula and the first formula together are:
because W0*N=2*PI, so there are:
with the increase of N, the w0 is decreasing, when n approaches infinity, W0 approaches infinity, x ' [N]=x[n], at this time, the upper formula becomes an integral type, the integral variable is w0, because w0=2*pi/n, so the integral interval is 2*PI, there are:
because the period of X (E^JW) E^JW is 2*PI, the integral interval can go to 2*pi intervals of any length, resulting in the following transformations:
This is the discrete-time Fourier transform pair, similarly, the conversion to the frequency domain called the analytic formula, the conversion to the time domain called synthesis formula. Properties

Summaryso far, the four main transformations of the Fourier family have all been deduced below, the next one to write down what the DfT is, then introduce a few common problems, and give the Fourier family tree and the relationship between.

Gray-scale image--discrete time Fourier transform of frequency domain filter Fourier transform (DTFT)