Fourier transform, the STFT transform, Gabor transform, wavelet transform, the Haar transform, Hilbert transform

Reference: https://blog.csdn.net/chenaiyanmie/article/details/80246108# Fourier transform

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Fourier transform:

The analysis by integration of the time domain signal to a frequency domain, often complex signal domain to the frequency domain becomes very easy to analyze.

Fourier transform of the conditions: (1) Dirichlet conditions; (2) integrable in the interval;

Limitations of Fourier transform: separates the contact time domain and frequency domain analysis capabilities for the local poor.

S the TFT transformation:

Is windowed Fourier transform, short time Fourier transform.

It overcomes the disadvantages of the lack of capacity of local welfare Fourier transform analysis.

But the window has been determined, it can not be changed, so that the resolution can not be changed according to the change of the frequency signal to be analyzed.

Gabor transformation:

Is the short time Fourier transform STFT window function when taken as a special case of Gaussian function.

STFT Gabor transform and to some extent, solve the problem of local analysis, but for transient signals and non-stationary signals is still difficult to obtain satisfactory results.

Wavelet transform:

Wavelet transform is a frequency-domain transform can be multiresolution analysis.

It overcomes the shortcomings of the first two. Depending on the issue can select different wavelet kernel.

Haar transformation:

Wavelet transformation, mainly for image compression, the image encoding.

Hilbert transform:

We know that the form of the signal is exp (*) exponential, Euler formula can know, a real part and an imaginary part b of this signal is a relationship! Is a is a cos sin, then through a variety of transformed real signal into a complex signal is possible!

Thus, the nature of the Hilbert transform is known that the real part of a complex signal, how to obtain the complex signals (imaginary part is the Hilbert transform of the real part)

Reference: https://www.zhihu.com/question/24783119

Reference: https://zhidao.baidu.com/question/536510541.html

Reference: https://wenku.baidu.com/view/919acf0616fc700abb68fca0.html

Reference: https://wapbaike.baidu.com/item/ Hilbert transform

Hilbert transform, Other:

1. In the field of mathematics and signal processing in a real-valued function of the Hilbert transform (Hilbert transform) - this is denoted H-- signal s (t) and 1 / (πt) convolve to to obtain s' (t).

2. Thus, the results of the Hilbert transform s' (t) may be interpreted as constant input output system (linear time invariant system) linear s (t) is the time, and the impulse response of this system is 1 / (πt) . This is a useful mathematical, used in the description do a real-valued modulated carrier signal of the complex envelope (complex envelope), appears to play an important role in communications theory.

3. Hilbert transform is a well-known mathematician David Hilbert (David Hilbert) named.

Discrete Hilbert transform it should be in order to facilitate the operation of it! Hilbert transformation is used in order to construct the analytical signal , since the analysis by the analytical signal more convenient, but the analytical signal spectrum is 1/2 of the original signal spectrum (positive axle spectrum) personal opinion, it is desirable to adopt

90 ° phase shift: Hilbert conversion will negative frequency components shifted + 90 °, and a positive offset frequency component of -90 °.

 

Hilbert (Hilbert Transform) Hilbert transform a continuous time signal x (t) is equal to the converted output signal h (t) = 1 / πt after the linear system response by having impulse response x * h ( t) [take convolution, the convolution product of the Fourier transform of the Fourier transform]. Since h (t) is the Fourier transform of FIG. 1.

The signal after the Hilbert transform, the amplitude of each frequency component in the frequency domain remains unchanged, but the phase appears 90 ° phase shift. I.e. positive frequency hysteresis π / 2, the former turned negative frequency π / 2, and therefore also known as Hilbert transformer 90 ° phase shifter. It is shown using the block diagram of a direct single sideband signal to produce the phase shift method in a communication system of FIG. FIG x (t) represents the input signal, the phase shift through Hilbert transformer to achieve. Converting the amplitude or phase modulation described by Hubert envelope, it causes the instantaneous frequency and phase analysis is simple, has important theoretical and practical value in a communication system. In communication theory, the Hilbert transform signal analysis tool, digital signal processing, not only for signal conversion, filtering can also be used, it can be made of different types of Hilbert filter.