You can't do any
serious data
engineering or
spectrum analysis without using imaginary/
complex numbers.
FFTs - in fact all
Fourier/
Laplace/
Z Transforms use
two dimensional re
presentation of numbers.
The whole idea is in some ways related to vectors, in the sense that an
imaginary number can also be
expressed as a length (
amplitude) and an angle (
phase).
Let's define Z to be a
complex number where the real part is called real(Z), the imaginary imag(Z). Amplitude is then named mod(Z) and angle arg(Z) (
modulo and
argument). They can be
calculated using:
mod(Z)
= ( real(Z)
^2
+ imag(Z)
^2 )
^ 1/2 . The length, just like in trigonometry
arg(Z)
= abs(real(Z)
* atan(imag(Z)
/ real(Z)) . The angle, again just like in
trigonometry
Z kan then be written as mod(Z) < arg(Z) (Not really a less than sign in the middle. It means
angle) - eg. 12<
pi/2 is the same as 0+12i (where
i is
the square root of -1)
Quite convenient for some things.
- Multiplying two complex numbers A and B. A * B = mod(A) * mod(B) < arg(A) + arg(B)
- Dividing two complex numbers A and B. A / B = mod(A) / mod(B) < arg(A) - arg(B)
- Taking the power (natural number) of a complex number A^n = mod(A)^n < n * arg(A)
- Taking the squareroot of a complex number: A^1/2 = mod(A)^1/2 < arg(A) / 2 and the same result multiplyed by 1<pi
The rest of the rules are a bit too
complex for
HTML (good excuse, eh?), but I guess you get the point. Good thing to know is that if you take the nth root of a complex number, you get n valid results (just like -1 is in a sense the squareroot of 1, which complex numbers actually take into account: The squareroot of 1<0 is 1<0 and 1< pi which is = -1)
In a
frequency spectrum (of eg. a
sound), all frequencies are represented by an amplitude and a phase. Sounds familiar? It is! Read more in the transformation links!