Magnitude spectrum and phase spectrum

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Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
Series Geophysical References Series
Title Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
Author Enders A. Robinson and Sven Treitel
Chapter 6
ISBN 9781560801481
Store SEG Online Store

What are the amplitude and phase characteristics of a digital filter? The filters we discussed in Chapter 5 operate in the time domain. Therefore, they can be called time-domain digital filters. The numerical examples presented there illustrate digital filtering in the time domain. However, many people are more accustomed to thinking about filtering in the frequency domain. One can study the action of filters profitably either in the time domain or the frequency domain or in a combination of both. The choice of a particular approach depends on the nature of the problem at hand. We shall now proceed to sketch the relation that exists between time-domain and frequency-domain filtering. Before doing so, a brief discussion of simple harmonic motion is in order.

A sinusoidal signal represents simple harmonic motion. Now let this sinusoid be the input to the various digital filters that we considered in Chapter 5. First let us consider the constant filter . We have the diagram shown in Figure 1.

Figure 1.  A constant filter.

The output is


Hence, the output is a rotating vector of length . For example, for , we have the case illustrated in Figure 2.

Figure 2.  Constant filter .

Because both the input vector at time n and the output vector at time n make the same angle with the horizontal axis, we say that the input and output are in phase. Here, we have assumed tacitly that is a positive constant. If, on the other hand, is a real positive constant, say, , we have the situation shown in Figure 3.

Figure 3.  Constant filter .

The output vector is . Because , the output vector can be written as


Thus, we have converted the output vector into the product of the positive constant 0.5 times . The positive constant 0.5 is the length of the output vector and hence is called the magnitude of this output vector. Now the quantity


represents a rotating vector of length l. At time index n = 0, this vector is


which shows that the vector lies on the horizontal axis in the negative direction and thus makes an angle of radians with the positive horizontal axis. The angle is called the phase lead of the vector (a lead represents the amount of advance of a rotating vector). For example, the input vector is advanced by to produce the vector .

Returning now to our example of the filter , we can say that the filter output


can be pictured as a rotating vector of amplitude +0.5 and phase lead radians. If we divide the output by the input , we obtain the filter’s transfer function:


Instead of dealing with angular frequency , one often deals with the cyclic frequency f in Hertz (Hz), where . Thus, the transfer function B(f) of the filter now becomes


The frequency spectrum (or transfer function) can be written in polar form as , which generally is complex. The magnitude of the transfer function is called the filter’s magnitude spectrum (or amplitude spectrum), whereas its phase lead is called the filter’s phase-lead spectrum. Thus, the magnitude spectrum of the filter is 0.5, and its phase-lead spectrum is .

We notice that both the magnitude spectrum and the phase-lead spectrum of this filter are independent of frequency (Figure 4). Because the phase-lead spectrum is constant and equal to , we can say that this filter’s input and output are out of phase by for all f or simply that they are out of phase.

Figure 4.  Magnitude and phase-lead spectra for the constant filter .

In the same way, we can compute the transfer function of any constant filter . The transfer function is


which is just the constant vector .

The next filter we wish to consider is the unit-delay filter Z. The transfer function is now


That is, the transfer function of the unit-delay filter is the vector . The magnitude spectrum of a filter is equal to the magnitude of the filter’s transfer function (i.e., frequency spectrum). Because the vector has unit magnitude, we see that the magnitude spectrum of the unit-delay filter is 1. The phase-lead spectrum of a filter is equal to the phase-lead of the transfer function of the filter. Because the vector makes an angle of with the positive horizontal axis, we see that the phase-lead spectrum of this filter is . Hence, the phase-lead spectrum of this filter is a linear function of frequency f, although its magnitude spectrum is independent of f.

The magnitude and phase-lead spectra are plotted in Figure 5 for , which is the range . Because we have specified that , we have . Summing up, we see that the transfer function of the filter is and that the transfer function of the filter Z is . For the causal FIR filter , we have the transfer function

Figure 5.  Magnitude and phase-lead spectra of a unit-delay filter.


The causal FIR filter has the transfer function


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