Signal truncation, energy leakage, and Window Function

It is really hard to do the speckle correlation, and there is no idea at all. I saw the second research. Hey, worry! Recently, we have considered combining the phase correlation with the speckle. As a result, edge effects are encountered when the phase is in the active phase. After reading some papers, you must add a window. I am not very familiar with the window function. I can see this article on the Internet.

Http://yangcui26.blog.163.com/blog/static/37487453200791685722957/

6.4.1 signal truncation and energy leakage effect

Fourier transformation is the main mathematical tool for digital signal processing. It should be noted that Fourier transformation is a study Entire time domain and frequency domain. However, when a computer is used to process engineering test signals, it is impossible to measure and calculate infinite signals. Get its finite time segmentFor analysis. The method is to extract a time segment from the signal, and then use the observed Signal Time segment Periodic ExtensionProcess and obtain Virtual Infinite LengthThen, we can perform Fourier transformation, correlation analysis, and other mathematical processing on the signal.

Figure 6.4-1

The signal after the periodic extension is different from the real signal. The following shows the error caused by this processing from the mathematical point of view. Cosine SignalX (t)The time-domain distribution is infinitely long (-∞, ∞), When the rectangular window function is usedW (t)Returns the truncation signal when it is multiplied.XT (t) = x (t) W (t). Returns the spectrum of the Cosine Signal Based on the Boli leaf transformation relationship.X (ω)Yesω. AtDeltaFunction, while the Rectangular Window FunctionW (t)IsSiNc (ω)Function, truncates the signal according to the convolution theorem in the frequency domain.XT (t)SpectrumXT (ω)
Should be

Returns the spectrum of the truncated signal.XT (ω)And original signal spectrumX (ω)We can see that it is not the original two spectral lines, but the continuous spectrum of two oscillating segments. This indicates that the original signal isTruncationLater, the spectrumDistortion, Originally concentrated inF0The energy at the point is dispersed in two wide frequency bands, which is called a leakage ).

The energy leakage caused by signal truncation is inevitable because of the Window FunctionW (t)Is an infinite function of the frequency band, so even if the original signalX (t)It is a limited bandwidth signal, and it will inevitably become a function of unlimited bandwidth after truncation, that is, the signal energy and distribution in the frequency domain are extended. From the sampling theorem, we can see that no matter how high the sampling frequency is, as long as the signal is truncated, it will inevitably lead to mixing. Therefore, the signal truncation will inevitably lead to some errors, which cannot be ignored in signal analysis.

If the truncation length is increasedTThat is, if the rectangular window is widened, the window spectrumW (ω)Will be compressed to narrow (π/T). In theory, the spectrum is still infinitely wide, but in fact, the frequency component outside the center frequency degrades rapidly, so the leakage error will be reduced. When the window widthTWhen it tends to be infinite, the spectrum windowW (ω)ChangesDelta (ω)Function, andDelta (ω)AndX (ω)The convolution is stillH (ω)This indicates that if the window is infinitely wide, that is, there is no leakage error Without truncation.

Figure 6.4-2

　　Different truncation functions can be used to reduce spectrum energy leakage,The Truncation function is called a window function.Window. The leakage is related to the two sides of the window function spectrum. If the height of the P-sides of the two sides tends to be zero, and the energy is concentrated on the main lobe, the leakage can be closer to the real spectrum. Therefore, different window functions can be used to Intercept Signals in the time domain.

　
6.4.2 common window functions

.. Window functions can be divided into the following main types:

Power window: A power function that uses a time variable, such as a rectangle, triangle, trapezoid, or other time functions.X (t)Higher power;

Trigonometric function window: use trigonometric functions, that is, sine or Cosine functions, to form a composite function, such as Hanning window and Haiming window;
Index window .. : Use exponential time functions, suchE-StFormat, such as Gaussian window.

.. The following describes the properties and features of several common window functions.

(L) Rectangular Window
The rectangular window is a zero-Power window of time variables. The function form is

The corresponding window spectrum is

Rectangular window is used most often. It is a habit of making the signal pass through the rectangular window without adding a window. The advantage of this window is that the main lobe is concentrated, but the disadvantage is that the side lobe is high and there is a negative side lobe (as shown in), which leads to high-frequency interference and leakage, and even a negative spectrum.

Figure 6.4-3

(2) triangular window
.. The triangle window, also known as fejer, is a square shape of the Power window. It is defined

The corresponding window spectrum is

.. Compared with the rectangle window, the width of the main lobe is approximately twice that of the rectangle window, but the side lobe is small and there is no negative side lobe, as shown in.

Figure 6.4-4