[introduction] think of a problem today,  there are two to be contained here many more boundless bit more discontinuous signal. They are located in 0, 1 between. ? The first signal is from 0 begin to go to 1 advancement,  the half of every ongoing and odd distance, amplitude reduces an in part. ? The 2nd signal is from 0 toward 1 advancement, advance every time the half of odd distance. The rectangular pulse signal that a width appears to be distance length half in the run that takes afore. ? According to Fourier commutation,  these two signal do not satisfy Dirichlet requirement. So what is their Fourier commutation?

Graph the first kind of 1.1.1 discontinuous dot function

Graph the 2nd kind of 1.1.2 discontinuous dot signal

2, signal 1 spectrum

1, spectrum derivation

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Beg the spectrum that takes the first type above all. ? This is its mathematical expression, ? To progression in each, ? ? It expresses a rectangular pulse, ? Height is the negative N of 2 second square, ? Initial drop is 1 subtractive the negative N of 2 second square, ? Terminus is 1 subtract the negative N of 2 is added 1 times square. ? ? Width is added 1 times for the negative N of 2 square. ? ? Draw up the spectrum of this pulse signal. ? Ask an attention, the center of this signal should be located in 1 subtract the of 2 negative N of 3 times is decreased 1 times square. ?

Graph 1.2.1 progression each corresponding Fourier commutation

To the spectrum of former signal, ? ? Need namely progression each spectrum adds up, ? The spectrum that gets signal so. ? ?

The spectrum formula after arranging is below:

Graph the Fourier progression of 1.2.2 signal decomposes formula

Graph the extent chart of the first 2.2 model

2, test and verify is formulary

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This is the signal spectrum formula that final derivation comes out, this also is a progression. ? Through disperse Fourier commutation comes to test and verify below this is formulary.

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This is to pass Python process designing, take losing 10000 between again and again, use data of 100 thousand spectrum to nod, undertake turning over alternating. ? Computational spectrum progression takes 100 step. ? This is the signal weaveform that computation comes out. Can see it and given signal is consistent. ? There is to rush too at 0 o’clock, ? Discontinuous dot has had the others other to rush. ? Accordingly, not only test and verify this formulary effectiveness,  and still can conclude roughly giving this formula should be convergence.

Graph the result of IFFT of the first 1.2.3 signal

3, signal 2 spectrum

1, spectrum derivation

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To the 2nd signal, ? It states the form of boundless progression, ? Among them each signal? Corresponding height is 1, ? The width that is them only and position are different. ? ? What the position of rest of the area that signal place gave out here and its midrib rush is initiative with end position. ? ? The Sinc function of the spectrum correspondence of each pulse. them overlay rises the spectrum that forms whole signal.

Graph the spectrum derivation of 1.3.1 individual pulse

The signal weaveform after derivation is below: Picture picture

The extent chart of the 2nd signal

2, test and verify is formulary

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For test and verify this formulary validity, still pass Python process designing, use disperse Fourier turns over commutation to obtain its corresponding weaveform. ? Take losing the 10000 spectrum in, sampling counts foothold 100 thousand times, ? Undertaking what get signal finally Fourier alternates instead is not undee,  this result is preliminary test and verify formulary validity. ? About this signal error astringent, reentry emulates test and verify all right after. ? Left is primitive signal weaveform,  right is to use weaveform of signal of finite spectrum complex.

Graph the 1.3.2 approximate weaveform that use finite bandwidth to obtain signal

The article had the spectrum that does not have space breakpoint signal to undertake derivation to two, ? They are infinite and progressional form, ? ? Use disperse Fourier commutation to undertake numerical value seeks solution, ? Through emulating undee test and verify the validity of spectrum formula. ? About their spectrum astringent, reentry discusses all right after.

Graph 2.1 signal weaveform and its spectrum

Origin: Zhuo Qing, tsinghuaJoking

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