impulse function as shown in the figure, hisspectrogramis constant 1 and has the property of not decaying. The energy is infinite. We call this spectrumwhite spectrum
Formula method for FT (rectangular signal as an example)
For a rectangular pulse signal with limited energy, his first characteristic is that the spectrum decays as w increases.
Regarding symmetry, we haveJean-Baptiste-Joseph Fourier (French mathematician, 1768-1830)The symmetry of the transformation can be easily concluded, not forgetting that here f (w) is an even function.
(math.) generalized Fourier transform
As shown in the figure, this is an example of the Fourier transform, for the unit step signal there is no way to go directly to the Fourier transform we can first through a other to meet the conditions of Dirichlet's formula to the Fourier transform, and then approximate the Fourier transform of the step signal can be derived. There are many functions that can be used for substitution and approximation, so the solution is often not unique.
When solving with the generalized Fourier transform do not forget that the final limit value is equal to the real limit plus the imaginary limit
Parity virtual reality and conjugate symmetry
parity (odd or even) and reality (idiom); evenly matched
Analyzing a typical signal we can see that if the signal is even symmetric in the time domain, then it is also even symmetric in the frequency domain and is a purely real number signal. If the signal is odd symmetric in the time domain, then the signal is purely imaginary in the frequency domain. Below we discuss the specifics of
Why do we say that the real part is an even function? Because if the real part is an odd function, then the integral value is 0. Similarly, the imaginary part is an odd function.
Let's consider it again in terms of odd and even components
If f (t) is a real function, then he can be decomposed into a superposition of even and odd components. We have the following transformation relation, which applies to the Fourier inverse transform as well
In this way the parity decomposition and the real-imaginary decomposition are unified by the Fourier transform
conjugate symmetry (math.)
Spectral attenuation of signals
Let's summarize the general pattern of signal attenuation
Obviously, we can see that the higher the signal can be to order, the smoother the corresponding signal waveform will be. The corresponding spectrum decays faster. The proportion of high frequency components in the spectrum is also smaller.
Tables of common Fourier transforms
exercise question
1.0
Because the spectrum of a DC signal is a pulsed signal that decays to just a line. Very fast.
2.0
The Fourier transform of an imaginary even function is an imaginary even function, and the Fourier transform of an imaginary odd function is a real odd function.
The solution to this problem uses many properties of the Fourier transform
