physical meaning of complex things

[venkatr86]

pls give me an insight into the world of complx things
like Complex numbers to begin with
then complex signals, concept of negative frequency.
i dont get as to how a signal when represented in frequecny domain can have content in the negative half.
then i want to know wat is meant by real and complex signals. wat are analytic signals

[subrahmanyambehara]

hai, please go through the book
modern analog and digital communications by b.p.lathi
u will get good idea

[micky]

Perhaps the easiest way to think of "complex numbers" is as a means of book-keeping two different values at a time for those physical quantities that do have two different aspects to their nature.

For example, a sinewave has an amplitude and it also has a phase (i.e. where in the time axis does one place the t=0 mark). Plain old f(t)=sin(wt) means that you elected for the sinewave to have an amplitude of zero at time t=0. You could just as well have selected your sinewave to be shifted to the left or to the right on the time axis so that, for example, at t=0 it has a value of +1, in which case your function is f(t)=sin(wt+90) (where "90" is degrees of course. Then at t=0, f(t)=sin(90)=1.

1. In the general case you have

f(t)=sin [wt+(phi)]

where phi is the phase offset.

Now that we have agreed that a sine wave has an amplitude and a phase, why not express it is a single "complex" number since complex numbers have two components by birth (never mind the word "imaginary" for now).

Recalling Euler's identity exp(jx) = cos(x)+jsin(x)
where j=sqrt(-1)

immediately suggests a convenient way to using a complex number to denote both amplitude and phase of a sinusoid because the phase "phi" is

phi = tan-1["imaginary part" / "real part"]

where Tan-1 denotes inverse tangent.

In other words, the complex notation can be viewed as a compact notation for handling two aspects of quantities that do have two aspects (such as the amplitude and phase in the case of a sinewave).

Don't let the word "imaginary" bother you. Just call it "the other part" if you like.

You might then say, "so what is the true sine wave expressed in an exponential notation?"
Simple algebra will show you that:

cos(a) = [exp(ja) + exp (-ja)]/2

and

sin(a) = [exp(ja) - exp(-ja)]/2j

You can express the usual Fourier series, for example, in terms of this exponential notation, and it looks much simpler than expanding it in terms of both sines and cosines (to handle the phase).

Michael