1. Ideal one-dimensional frequency filtering
1) ideal filter
The purpose of filtering is to suppress interference and improve signal-to-noise ratio. Of course, the most ideal filter is that the effective wave passes completely without distortion and the interference is completely suppressed. Therefore, its frequency response is required to be
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Does this mean that its phase response characteristic is zero? H(ω)=0. Therefore, the ideal filter must be a zero-phase filter and must be non-physically realizable. Of course, this also means that there should be no interference in the effective wave band, otherwise it cannot be filtered out.
The frequency response function diagram of an ideal filter is a rectangle, like a door, so it is also called portal filtering.
2) Ideal low-pass filter
When the spectrum distribution of effective wave and interference is shown in Figure 4-2-3a, an ideal low-pass filter can be designed, and its frequency response is shown in Figure 4-2-4a, and its mathematical model is as follows.
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Figure 4-2-3 Spectrum Example of Effective Wave and Interference Wave
Figure 4-2-4 Frequency and Impulse Response of Ideal Low-pass Filter
The impulse response can be obtained by inverse Fourier transform:
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The impulse response curve is shown in Figure 4-2-4b.
3) Ideal bandpass filter
Generally speaking, there are both high-frequency interference and low-frequency interference (Figure 4-2-3 b), so it is necessary to design a band-pass filter. Its mathematical expression is
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There are two ways to find the impulse response hb(t).
One is the combination of low-pass filters. Because the frequency response of the band-pass filter is equal to the difference between the frequency responses of two low-pass filters with different cutoff frequencies (Figure 4-2-5). According to the linear property of Fourier transform, the impulse response of band-pass filter is also equal to the difference between the impulse responses of two low-pass filters. Therefore, the impulse response of the bandpass filter can be directly written as
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Where f0 is the center frequency of the passband; δ f is half the bandwidth; F0 =(F2+f 1)/2； δf =(F2-f 1)/2
Figure 4-2-5 Frequency Response of Ideal Bandpass Filter
The second method is to directly calculate the inverse Fourier transform of Hb(f), and the same result can be obtained.
4) Ideal high-pass filter
Its frequency response is
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The impulse response can also be calculated by subtraction or directly calculating the inverse Fourier transform, and the following results can be obtained:
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2. Particularity of digital filtering
Digital filtering must be operated on a digital computer, and it faces two special problems. One problem is that digital filtering can only operate on discrete time series, but not on continuous time functions, which is called discreteness. Another problem is that the impulse response can only take finite length instead of infinite length required by theory, that is, finiteness. Because of these particularities, the calculation results can not reach the "ideal" situation of portal filtering, but will be distorted.
1) false door phenomenon
If the impulse response function h(t) is sampled discretely at the sampling interval δ, the frequency characteristics of the sampled impulse response time series H (n δ) include not only the "main gates" corresponding to the frequency characteristics of h(t), but also countless "pseudo-gates" with a period of 1/δ. Let's use a simple frequency convolution theorem to prove this point.
The infinite sequence δ (t) (Figure 4-2-6 b) consisting of countless δ pulses with an interval of δ is called the sampling function, which is multiplied by the impulse response function h(t) (Figure 4-2-6 a) to obtain the sampled impulse response time series H (nδ) (Figure 4-2-6 c). The Fourier transform of δ (t) is the frequency sampling function δ (f) (Figure 4-2-6 e), which consists of countless δ pulses with an interval of 1/δ and an amplitude of1/δ; The Fourier transform of h(t) is H(f) (Figure 4-2-6 d). According to the frequency convolution theorem (the product of time domain corresponds to the convolution of frequency domain), the Fourier transform of H (nδ) should be the convolution of H(f) and δ (f), which is a function with a period of 1/δ (Figure 4-2-6 f). In this way, countless fake doors appeared outside the main entrance.
Because of the existence of the pseudo-gate, the spectrum of some interference waves may appear in the range of the pseudo-gate and be preserved, which can not achieve the expected filtering effect. In addition, when the sampling frequency 1/δ is less than twice the cutoff frequency fc, these periodic filter gates will overlap, and if the frequency response itself is curved, this overlap will also lead to the distortion of the response (Figure 4-2-7).
Figure 4-2-6 Prove the existence of pseudo-gate with frequency convolution theorem
The bidirectional arrows in the figure represent Fourier transform pairs.
Figure 4-2-7 Distortion Caused by Insufficient Sampling Interval
In order to solve the false gate problem, the sampling interval δ can be properly selected, so that the first false gate appears outside the frequency range of the interference wave, which can also prevent the false frequency effect of response distortion.
2) Ghips phenomenon.
The frequency response functions of all ideal filters are discontinuous at the cutoff frequency (fc or f2, f 1), and the inverse Fourier transform (impulse response function) of the discontinuous function must be infinite. In practical calculation, the impulse response function can only be finite, that is, truncated. The frequency response function corresponding to the truncated impulse response is no longer an ideal "gate", but a smooth curve with fluctuation values near this gate. This phenomenon is called Ghips phenomenon.
The frequency convolution theorem can still be used to illustrate this problem. The infinite impulse response h(t) is multiplied by the rectangular function p(t) to obtain the truncated impulse response h'(t) (Figure 4-2-8 a, b, c). The function p(t), also known as the truncated time window, is mathematically defined as:
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Its Fourier transform is
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Fig. 4-2-8 proves the existence of Ghips phenomenon by frequency convolution theorem.
Two-way arrow table Fourier transform pair
Its shape is similar to the impulse response function of an ideal low-pass filter (Figure 4-2-8 e). According to the frequency convolution theorem, the frequency response function H'(f) corresponding to H'(t) should be the result of convolution of H(f) (Figure 4-2-8 d) and P(f). It is a continuous, smooth and fluctuating curve (Figure 4-2-8 f). Mathematically, it can be proved that the fluctuation amplitude is the largest at the discontinuous point, which is about 9% of the original rectangular amplitude. As the distance from the discontinuous point increases, the fluctuation amplitude becomes smaller and smaller.
Because the frequency characteristic curve fluctuates in the passband, the effective wave will be distorted after filtering. In addition, it is also a wave curve outside the passband, which certainly cannot effectively suppress interference.
In order to weaken the influence of Ghips phenomenon, several methods can be adopted. One of them is the edging method. It considers the problem from the perspective of frequency domain, and inserts continuous edges at discontinuous points of rectangular frequency characteristic curve to make the frequency characteristic curve a continuous curve.
For example, the frequency response of low-pass filtering after edge processing is (Figure 4-2-9).
Figure 4-2-9 Frequency Response of Low-pass Filtering after Trimming
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The edge function g(f) requires the following conditions to be met:
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There are many functions that satisfy this condition. Cosine edge function:
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Is one of them. After the edge function is determined, the corresponding h*(t) can be obtained by inverse Fourier transform, and the influence of Ghips phenomenon can be reduced by convolution filtering. The impulse response of low-pass filtering using cosine edge function (4-2- 19) is
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Multiplier method can also be used. This is the way to think about the problem from the perspective of time domain. That is, when truncating h(t), the rectangular time window function is not used, but is replaced by the gradually decaying time window function. A good time window function should have the following characteristics:
A. The time interval should be as long as possible, so that the corresponding spectrum energy is concentrated in its main lobe;
B the shape of the time window should be as smooth as possible without steep inclination.
There are many time window functions that satisfy these conditions. For example, the triangular window widely used in seismic exploration is mathematically expressed as follows
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Its Fourier transform is
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W(f) is the square of sinc function, and it has no negative sidelobe, so it is a good time window.
Although edge method and multiplier method are obtained from different angles, their essence is the same, both of which are to accelerate the attenuation of impulse response function in time domain and reduce the error caused by truncation. The results are similar, which reduces the selectivity of the filter in frequency domain and reduces the steepness of the frequency response curve.