The signal after the period extension is different from the real signal. The error caused by this processing will be discussed below from a mathematical point of view. Assuming that the cosine signal x(t) is infinitely long ( -∞, ∞ ) in the time domain, when it is multiplied by the rectangular window function w(t) , the truncated signal xT (t)= x (t)w is obtained (t) . According to the Bolier transform relationship, the spectrum X(ω) of the cosine signal is located at ω . The delta function at , and the spectrum of the rectangular window function w(t) is the sinc(ω) function. According to the frequency domain convolution theorem, the spectrum X T (ω) of the truncated signal x T (t ) should be
Comparing the spectrum X T (ω) of the truncated signal with the spectrum X (ω) of the original signal , it can be seen that it is not the original two spectral lines, but a continuous spectrum of two oscillations. This shows that after the original signal is truncated , its spectrum is distorted , and the energy originally concentrated at f 0 is dispersed into two wider frequency bands. This phenomenon is called spectral energy leakage (Leakage) .
The phenomenon of energy leakage after signal truncation is inevitable, because the window function w(t) is a function with an infinite frequency band, so even if the original signal x(t) is a bandwidth-limited signal, it will inevitably become a function of infinite bandwidth after truncation. , that is, the energy and distribution of the signal in the frequency domain are expanded . It can be known from the sampling theorem that no matter how high the sampling frequency is, as long as the signal is truncated, aliasing will inevitably be caused. Therefore, signal truncation will inevitably lead to some errors, which is a problem that cannot be ignored in signal analysis.
If the truncation length T is increased , that is, the rectangular window is widened, the window spectrum W(ω) will be compressed and narrowed ( π/T decreases). Although theoretically, its spectral range is still infinitely wide, in practice frequency components other than the center frequency attenuate faster, so the leakage error will be reduced . When the window width T tends to infinity, the spectral window W(ω) will become a δ(ω) function, and the convolution of δ(ω) and X(ω) is still H(ω) , which means that if the window Infinite width, i.e. no truncation, there is no leakage error.
Figure 6.4-2
In order to reduce the leakage of spectral energy, different interception functions can be used to truncate the signal. The interception function is called a window function , or simply a window. The leakage is related to the side lobes on both sides of the window function spectrum. If the height of the p side lobes on both sides tends to zero, and the energy is relatively concentrated in the main lobe, it can be closer to the real spectrum . Different window functions are used to truncate the signal.
6.4.2 Commonly used window functions
. . The practical window functions can be divided into the following main types:
Power window: a function that uses a certain power of the time variable, such as a rectangle, a triangle, a trapezoid, or a higher power of other time functions x(t) ;
Trigonometric function window: Apply trigonometric functions, that is, sine or cosine functions, etc., into composite functions, such as Hanning window, Hamming window, etc.;
exponential window . . : Use exponential time function, such as e -st form, such as Gaussian window, etc.
. . The properties and characteristics of several commonly used window functions are described below.
(l) Rectangular window
The rectangular window belongs to the zeroth power window of the time variable, and the function form is
The corresponding window spectrum is
The rectangular window is the most used, and it is customary not to add a window to make the signal pass through the rectangular window. The advantage of this kind of window is that the main lobe is relatively concentrated, and the disadvantage is that the side lobe is relatively high and has negative side lobes (as shown in the figure below), which leads to the introduction of high-frequency interference and leakage in the transformation, and even negative spectral phenomena.
Figure 6.4-3
(2) Triangular window
. . Triangular window, also known as Fejer window, is a square form of power window, which is defined as
The corresponding window spectrum is
. . Compared with the rectangular window, the main lobe width is approximately twice that of the rectangular window, but the side lobes are small and there is no negative side lobe, as shown in the following figure.
Figure 6.4-4
