Practical aspects of frequency filtering
Application of a filter in the frequency or time domain (Figures 1.1-24 and 1.1-25) yields basically identical results. In practice, the time-domain approach is favored, since convolution involving a short array, such as a filter operator, is more economical than applying Fourier transforms.
From Figure 1.1-23, the fundamental property of frequency filters can be stated as follows: The broader the bandwidth, the more compressed the filter operator; thus, fewer filter coefficients are required. This property also follows from the fundamental concept that the time span of a time series is inversely proportional to its spectral bandwidth.
In designing a band-pass filter, the goal is to pass a certain bandwidth with little or no modification, and to largely suppress the remaining part of the spectrum as much as practical. At first, it appears that this goal can be met by defining the desired amplitude spectrum for the filter operator as follows:
where f1 and f2 are the cutoff frequencies. This is known as the boxcar amplitude spectrum.
To analyze the properties of such a filter, perform the following sequence of operations:
- Define a boxcar amplitude spectrum and zero-phase spectrum.
- Apply inverse FFT and obtain a filter operator.
- Truncate the operator.
- Apply forward FFT and compute the amplitude spectrum of the truncated operator.
Figure 1.1-26a shows the results of this sequence of operations. The operator is at the top, and the actual and desired (boxcar) amplitude spectra are superimposed at the bottom. Note that the actual spectrum has a ringy character. This is known as the Gibbs phenomenon, and results from representing a boxcar with a finite number of Fourier coefficients. From a practical standpoint, the ringing is undesirable, since some of the frequencies in the passband are amplified, while others are attenuated. Additionally, some of the frequencies in the reject zones on both sides of the boxcar are passed.
How is the Gibbs phenomenon circumvented? Instead of defining the desired passband as a boxcar, assign slopes on both sides as shown in Figure 1.1-26b, and thus define the passband as a trapezoid. Note that the actual and desired amplitude spectra are now closer in agreement and the operator is more compact (it has fewer nonzero coefficients). However, in achieving a more compact operator, the shape of the desired spectrum has been compromised and the passband is broader than intended. The trapezoid slopes must be sufficiently gentle to achieve a satisfactory result as in Figure 1.1-26c, where the actual and desired spectra are approximately equal and the operator is compact. This is most desirable in practice, since it is best to work with operators that are as short as possible. It is recommended that a gentler slope be assigned on the high-frequency side relative to the low-frequency side of the passband. Finally, while defining the passband as a trapezoid, smoothing also must be applied at the corner frequencies (A, B, C, and D, as indicated in Figure 1.1-26c). This must be done because the Fourier transform exists only for continuous functions .
How short can the operator be? Figure 1.1-27 shows a sequence of increasingly longer operators. Solid bars indicate the operator length of the truncated filters. Note that excessive truncation causes a large deviation from the desired amplitude spectrum even though reasonable slopes were provided to the passband. Extension of the operator length brings the desired and actual spectra closer. However, there is a certain length beyond which nearly zero coefficients are added to the operator. The criterion that is used to define the operator length is that the frequency bandwidth is inversely proportional to the effective length of the operator.
Figure 1.1-26 Three zero-phase wavelets (top row) and their respective amplitude spectra (bottom row). (a) The steeply defined slopes of the passband cause ripples in the wavelet and the actual amplitude spectrum. (b) A moderate and (c) gentle slope help eliminate the ripples. Refer to the text for a discussion of corner frequencies A, B, C, and D.
- Bracewell, 1965, Bracewell, R. N., 1965, The Fourier transform and its applications: McGraw-Hill Book Co.
- Analog versus digital signal
- Frequency aliasing
- Phase considerations
- Time-domain operations
- Crosscorrelation and autocorrelation
- Vibroseis correlation
- Frequency filtering
- Bandwidth and vertical resolution
- Time-variant filtering