Re: Wa tis "imaginary signal" in "physical reality" context was [Re: Conversion of data to complex numbers??]

A complex number can represent the projections onto x and y axes of the
end of a line that is fixed at the origin and is rotating with angular
frequency.

The line projects onto the x axis so that x = a.cos(w.t) and onto the y
axis so that y = a.sin(w.t)

Instead of the rotating line, use a bar magnet.
Now take two wire coils at right angles to each other.
The lines of magnetic flux from the rotating bar magnet induce currents
in the coils.
One coil will show current a.cos(w.t) the other will show current
a.sin(w.t)
Together, these two signals let us represent what happens with the
rotating magnet at all times (with only one coil the current seems to
'disappear' at the sine zero-crossing even though the magnet does not
disappear).

Take the magnet as a rotating magnetic field and the coils as RF coils.
Now you have a neat way of representing quadrature RF detection.
The complex arithmetic simply gives an easy way to do the math. The
signal from one of the coils is 'imaginary'.

Another way to think of it is that 'imaginary' has no meaning. 'j'
(square root of -1) only has meaning when it is squared - when it
becomes -1.
Using the rotating line model, multiply a complex number by j and plot
it on the (argand) diagram. It rotates by 90 degrees.
Does that mean anything? Probably not but at least it is close to
meaningful.

If you want to see how I once struggled to find physical meaning for
complex numbers (specifically, z) then you can look at this section of
our (free) on-line DSP tutorial:

http://www.bores.com/courses/intro/iir/5_ztran.htm

In this I was led to invent a concept of 'imaginary frequency' which,
while meaningless in one sense, at least helped me work towards some
physical intuition that was helpful. (You need to follow the reasoning
through several pages, to 'the meaning of z').

Chris
====================
Chris Bore
www.bores.com

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