4.5 Linear System Filtering in Time and Frequency Domain

There is good reason why the convolution theorem was called "possibly the most important and powerful tool in modern scientific analysis" by Brigham (1974).  We shall soon see that some of the most fundamental mathematical operations in the time or space domain, e.g., differentiation and integration, are simple multiplications in the frequency domain.  In other words, they are filter operations.  Correspondingly, ideal pass band filtering, clearly visualized in the frequency, domain is not nearly so easy to "see" in the time (or space) domain where it is a convolution.

Linear, Time-Invariant System

First let's make clear what is meant by linear, time-invariant filtering which is the only kind that we will address.  With constants a1 and a2, linear means that:

            1. With s(t)–y(t) an input–output pair, the input a1s(t) has an output a1y(t).

            2. The input a1s1(t) + a2s2(t) has the output a1y1(t) + a2y2(t). 

Time invariance means that the system response is independent of time, therefore, if y(t) is the output from input s(t), then the response to s(t-t1) is y(t-t1).  These two properties of linearity and time invariance are the basis for the "amazing" conclusion (e.g., Papoulis, 1962) that such a "linear system" is uniquely determined by a single function, h(t) called the impulse response function (equation 4.1a and Figure 4.4).

Input-Output Relations in Time (or Space) Domain and Frequency Domain

By applying the convolution theorem to the convolution integral we see that a linear system which is completely characterized in the time (or space) domain by its impulse function must also be completely described by its frequency domain counterpart, H(f).  This Fourier transform of the impulse response function is called the transfer function or system response function in the frequency domain.  Sometimes we call the impulse response function the time domain operator and the corresponding transfer function the frequency domain operator.  No matter what terms you use, either of the Fourier transform pair, h(t) or H(f), is the ultimate quantity desired when doing linear filtering.  H(f), of course, is generally complex with real and imaginary parts, or amplitude and phase, whereas h(t) is simply real in applied geophysics. The extremely important relationship between linear filtering in the two Fourier domains is expressed by

(4.5.1)

Here, as before, the double arrows indicate Fourier transform pairs.  The multiplication of the complex spectra S(f) and H(f) means that the resulting amplitude-phase spectrum is the multiplication of the amplitudes of S(f) and H(f) and the addition of their phase spectra, respectively.

Examples of Some Important Practical Filters

Pass Band Filters

Probably the easiest filters to visualize are pure low-pass, high-pass, band-pass, and notch filters expressed in the frequency domain.  Ideally these filters are given by:

 

 

a. Ideal low-pass filter

    (4.5.2)

b. Ideal high-pass filter

 (4.5.3)

c. Ideal band-pass filter

(4.5.4)

d. Ideal notch filter

  (4.5.5)
Figure 4.9 Frequency domain equations and responses for ideal filters: a) low-pass filter, b) high-pass filter, c) band-pass filter, and d) notch filter.

The four filters are plotted in the frequency domain in Figure 4.9.  These are the corresponding system response or transfer functions; their impulse response functions are obtained by inverse Fourier transforms.  Since we recognize the low-pass filter as a rectangle or "boxcar" centered at the origin in the frequency domain, we know that the impulse response function must be sinc function in the time domain, i.e.,

(4.5.6)

Differentation and Integration Filters

There is a fascinating relationship between differentiation in the time (or space) domain and its frequency domain equivalent.  Using the inverse Fourier transform we can write an arbitrary signal, s(t) as

(4.5.7)

Taking the first time derivative yields

(4.5.8a)

where we see by analogy to equation 4.5.7 that

(4.5.8b)

Appealing to the convolution theorem, we see that the simple multiplication of i2pf and S(f) is the frequency domain representation of the first derivative filter in the time domain.  This also could be expressed in the time domain as the convolution of the inverse Fourier transforms of i2pf and S(f) (i.e., yielding d[s(t)]/dt).  The inverse Fourier transform of i2pf is the impulse response function of the time derivative filter; i2pf is the transfer or system response function.  Note that the second time derivative is

(4.5.9a)
(4.5.9b)

The system response function for the second derivative is, therefore, -4p2f2.  Generalizing, we see that the Nth time derivative system response function (transfer function or frequency domain operator) is (i2pf)N.

Once we've established the differentiation operation in frequency domain, we also magically have derived the expression in the frequency domain for the operation of integration.  This is because if we accomplish the derivative operation by multiplying by i2pf, then to undo the derivative (integrate) all we have to do is divide by i2pf.  That is, the frequency domain expression for the indefinite integral s(t)dt is simply 1/(i2pf) x S(f).  The transfer function for integration in the frequency domain is 1/(i2pf).  The double, triple, …, Nth integral transfer functions are expressed in the frequency domain as 1/(i2pf)N.  This is neat stuff, don't you agree?

Continuation of Potential Fields

There are filter operations that work only if the input signal obeys certain physics.  Such a case is a whole class of functions called potential functions.  Potential functions obey one of the most important equations in pure and applied physics, namely Laplace's equation.  This differential equation characterizes forces in static electric, magnetic, and gravity fields, as well as steady state electric current, heat, and fluid flow.  The fundamental relationship applies to the scalar potential functions of these fields.  However, in regions where vector components of the field are in a constant direction, the vector field components also satisfy Laplace's equation.

For example, with this assumption, the vertical component of the acceleration of the Earth's gravity (usually simply called g) and components of the Earth's magnetic field (including the total field component as presented in Figure 1.5) obey Laplace's equation.  Laplace's equation is not valid where sources of gravity or magnetic fields exist but it holds perfectly outside of them such as above the Earth's surface where we make measurements.