It is obvious that when you ask a layman to explain the term filtering using his common sense. The common and fast reply you would face regarding this term is just to block the unwanted part of something, while allowing its wanted part to pass through a predefined channel.
The same context is applied in geophysical signal processing, as the term Filtering is an important step in signal processing operation that is used to modify the characteristics of the time - varying signals for the purpose either to enhance certain features or to remove a certain part of the signal within a particular frequency range.
Since most signals are complex, it would be easy and comfortable when you analyze and work on enhanced and interesting features rather than work on a whole signal at once. To attempt this, you need to weed out unwanted parts of your signal while remaining with your interested part of the signal based on your predefined frequency range. That is Frequency Filtering in layman terms.
There are many types of filters depend on various circuit designs, operation and functions, for example the Velocity Filter is used to attenuate unwanted noises such as ground roll from straight line seismic events based on apparent velocity, however I have decided to describe to you through this post, the basic physics regarding the operation principle of two (2) types of Frequency Filters because many other filters relied on the same operation. The post is about the Low Pass Filter and High Pass Filter.
Low Pass Filtering is the process by which the frequencies below cut off frequency are passing through it. The primary purpose of using Low Pass Filters is to attenuate (remove) unwanted high frequency components of signals which are above cutoff frequency at which in turn It helps in removal of aliasing effect
High Pass Filtering is the process by which the frequencies above cut off frequency pass through it. The primary purpose of doing high Pass Filtering is to attenuate (remove) unwanted low frequency components of signals which are below the cutoff frequency value, which in turn helps in removal of noise.
It is the time now to describe the basic operation principle of these filters but, before proceeding Let us first consider the basic Resistor Capacitor (RC) circuit design of each filter so as to grasp the idea and an insight easily.
Low Pass Filters consist of resistor (R) that is followed by capacitor (C) as shown in the figure below. As the simple RC series circuit involving alternative voltage, the phase diagram shows that the impedance (z) is equal to square root of the sum of squares of resistor and capacitor components since the input voltage is across both two elements (R and C) such as
z = √(R2 + Xc2).........................(1)
Where Xc - Capacitive reactance, R - Resistance
Input voltage (Vin) = Imz
Put the expression of impedance (z), we will have
Vin = Im × √(R2 + Xc2)........................(2)
Where Im - maximum current
If we consider the figure below, then the output voltage is taken across the capacitor, meaning that the maximum voltage equals the voltage across the capacitor.
Since impedance across the capacitor as Capacitive reactance is given as,
Xc = 1/wC.
Then voltage output can be computed the same as using ohm's law such that,
Voutput = ImXc = Im/wC..........................(3)
By taking the ratio of Vout to Vin, as Filter gain then
Vout/Vin = (1/wC)/√(R2 + (1/wC)2)..........................(4)
The plot of voltage ratios versus w from equation 4 above, show that the low frequency signals pass the filter, this gives it the title Low Pass Filter.
Figure: Schematic representation of Low-pass Filter
High Pass Filters consist of capacitor (C) that is followed by a resistor (R) as shown in the figure below, it's gain can be obtained the same as that of Low pass Filter, except that in Output voltage given by equation 3, will be due to impedance across the Resistor (R) and not capacitor. Then, Vout = ImR
Taking the ratio of voltages Vout to Vin as a Filter gain, while taking Vin from equation 1, then
Vout/Vin = R/√(R2 + (1/wC)2)............................(5)
Then the plot of equation 5, such as Vout/Vin versus w, shows that at low frequencies Vout is small compared to Vin. Then signals with higher frequencies pass the filter while those with low frequencies are blocked. Then the title High pass Filter.
Figure: Schematic representation of High-pass Filter
Figure: An example of seismic trace (t - x) after applying gain and high pass filtering.
Apart from high and low pass frequency filters there is a common trapezoidal shaped filter known as Band-pass Filter which is used to attenuate unwanted part of the signal below and above corner frequencies of the signal. At corner frequency amplitude of the signal decreases to 0.7 relative to the unfiltered part of the signal.
If f1 and f2 are corner frequencies of the signal then, Bandpass Filter attenuate frequencies above f2 and below f1 within the band range f1 to f2, see figure.
Figure: Schematic representation of band-pass Filter.
Filters can be represented in the log-log domain, as straight line segments having the ratio of the output amplitude to the input amplitude, the gain, given in decibel (dB) such as,
dB = 20 log (gain)
This means that the corner frequency is 20 log (0.7) = −3 relative to the flat part of the filter response (below the flat part). The sharpness of the filter is determined by the slope n, often given as the number of poles for the filter. In processing, n is usually between 1 and 8 and typically 4.
Choosing which type of filter to use will depend on your objective, the processing stage and a good approach is to use the filter with good properties.
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