The numbers we use in everyday life can be expressed as a one-dimensional graph extending from minus infinity to plus infinity:-
Although some numbers are difficult to represent in some number bases (for example 7 divided by 3 gives 2 and one-third, which is 2.333333 ... in decimal), they become simple in other bases (7/310 = 2.13). Other irrational numbers (like pi), which cannot be fully
represented in any number of digits in any base, still have a position on the range of real numbers we normally use.
Real numbers allow us to solve equations such as x2 = 49. Remember that there are two answers to this, +7 or -7.
The problems start when we try to solve equations like x2 = -49. Our normal range of real numbers does not allow us to solve this and early mathematicians believed that this
equation had no solution. By the middle of the 16th century, however, the Italian mathematician Gerolamo Cardano and his contemporaries were experimenting with solutions to equations that involved the square roots of negative numbers, and, by 1777, the Swiss mathematician Leonhard Euler introduced the symbol i to stand for the square-root of -1.
The imaginary numbers produced by using the symbol i have no physical meaning, but they do allow for the solution of all polynominal equations (the square root of 49 is either 7i or -7i). They also lead to some interesting mathematical solutions like e(pi i) = -1!
Numbers which consist of part real and part imaginary (for example 22.3 + 1.925i) are called complex numbers.
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In the same way that real numbers can be thought of as points on a line, complex numbers can be thought of as points in a plane.
The number a + bi is identified with the point in the plane with x co-ordinate a and y co-ordinate b. The points 1 + 4i and 2 - 2i are plotted here using the convention invented by the Swiss bookkeeper Jean Robert Argand in 1806.
Logically, this type of plot is sometimes referred to as an Argand diagram.
If a complex number in the plane is thought of as a vector joining the origin to that point, then addition of complex numbers corresponds to standard vector addition.
This shows the complex number 3 + 2i obtained by adding the vectors 1 + 4i and 2 - 2i.
The complex number -2 + 3i has the real part -2 and the imaginary part 3. Addition of complex numbers is performed by adding the real and imaginary parts
separately. To add 1 + 4i and 2 - 2i, for example, add the real parts 1 and 2 and then the imaginary parts 4 and -2 to obtain the complex number 3 + 2i. The general rule for addition is:-
(a + bi) + (c + di) = (a + c) + (b + d)i
Multiplication of complex numbers is based on the premise that i × i = -1. This gives the rule:-
(a + bi) × (c + di) = (ac - bd) + (ad + bc)i All of the normal mathematical functions (logs,
trigonometric functions etc.) can be applied to complex numbers (they usually give complex answers). Plots of some part of some of the more esoteric functions of complex numbers give rise to the fractal patterns popular with computer artists!
148 Since points in a plane can be written in
terms of the polar co-ordinates r and p , every complex number z can be written in the form:-
z = r (cos p + i sin p)
Here, r is the modulus, or distance to the origin, and p is the argument of z, or the angle that z makes with the x axis. If z = r (cos p + i sin p) and w = s (cos q + i sin q) are two complex numbers in polar form, then their product in polar form is given by:-
zw = rs (cos (p + q) + i sin (p + q))
We'll go on now and discuss frequencies, and how complex numbers play their part!
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Frequencies
Way back in Chapter 3 we showed how a waveform, consisting of a single
frequency, would appear in the time domain (as a function of time).
We can define such a waveform by three parameters - its frequency (usually in Hertz - the number of cycles per second), its amplitude (the maximum value of the waveform), and its phase - the offset of the central peak from time zero measured as an angle.
Here then is one such single frequency waveform, with some numbers assigned to its three parameters.
The horizontal scale is time (shown in seconds), and the period between successive peaks of the waveform is 40 ms. This corresponds to a frequency of 1/0.04 or 25 Hz.
The amplitude of the waveform is "4", and the waveform begins at a point 1/6th of a wavelength from its maximum value - the phase is 360/6 or 60 degrees (or pi/3 radians).
The actual equation for the type of waveform shown above is:-
a*Cos(2*pi*f*t+p)
where a is the amplitude, f the frequency, t the time and p the phase. pi is either 3.14159...
if we're working in radians, or 180 if we're working in degrees, so the example given (with everything in degrees) is:-
4*Cos(180*25*t+60) = 4*Cos(4500*t+60)
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You can plot functions like this (in the same way that I did) by using a
spreadsheet.
Simply build a column of times incrementing by your chosen sample period, and plug the equation above into the next column (you may need to convert it to radians).
An X/Y plot of these numbers should produce something like the plot shown above!
We could look at the above waveform in the frequency domain, plotting both the amplitude and phase of this waveform as a function of frequency.
This obviously only gives us one point on each graph, but we need two graphs for this representation as opposed to the one in the time domain. Why?
The clue to this is on the previous page. In the frequency domain we need a complex number to specify the waveform. The combination of phase and frequency values in the frequency domain transform into waveforms in the time domain, and allow for complex numbers in both domains, as well as for negative frequencies (a concept that we'll discuss when we get to spatial frequencies!).
Luckily, we don't record either complex or imaginary numbers in our seismic data (which is just as well, as they'd be difficult to display!), and we don't record negative frequencies (if you like, energy from the shot going backwards in time!), so we actually only need about half as many "samples" in the frequency domain as we have in the time domain - more on this later.
Once again, we'll go back to clock watching! Imagine a clock that rotates through 360 degrees in 40 ms.
If the length of the "hand" is four units, and we plot the height of the end of the hand from the centre of the clock, we get the cosine wave shown here.
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Click on the animation to stop it - if you can stop it after 1/6th of a revolution (at 2 o'clock), you'll get the same time function as shown above - a 60 degree phase value!
So, as we might expect from the clock above, we could represent this single frequency as a complex number vector with a radius of "4", and an angle of 60 degrees to the
"Real"-axis in this Argand diagram.
We could express this 25 Hz. component as "2 + 3.464i", and, although it is (slightly) more intuitive to talk about its amplitude and phase, we need to remember that we are dealing with a complex number when we look at combining different frequencies (or different amplitudes and phases of the same frequency) together.
Here's the complete relationship between amplitude (A), phase (p), and the Real (R) and Imaginary (I) components of a complex number expressed in polar co-ordinates, or our single frequency component expressed in Amplitude and Phase converted to its complex equivalent:-
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