What is the relevance of complex signals in communication engineering?

[dexterdev]

Hi PF,
Actually the information signal can be represented as real signal. Then why do we go for complex representations? why complex conjugates etc do appear after all? May be some basics is lacking

-Devanand T

 

[the_emi_guy]

dexterdev,

It is mathematical convenience.

First, think of a complex number as simply an ordered pair of numbers (5,6).
These two numbers are independent of each other and, thus, can be used to describe the state of any system that has two independent degrees of freedom, such as the location of a point on a 2 dimensional plane (Argand diagram).

Next, consider signal processing. We start with a pure sinusoid which, by itself, carries no information. Next we modulate the sinusoid with information. Turns out that there are exactly two independent degrees of freedom when modulating a sinusoid; phase and amplitude. I can modulate the phase without affecting the amplitude, and I can modulate the amplitude without affecting the phase.

Next, note that phase and amplitude are simply the polar coordinate representation of a point on a plane. That same point can be described in rectangular coordinates (x,y).

Finally, I can use a single complex number to describe the state of modulation of my sinusoid, including both of its degrees of freedom. By convention, instead of (x,y), we refer to the rectangular coordinates for the modulation state as (I,Q).

 

[rbj]

in Linear System Theory, exponential functions are what we call "eigenfunctions" of linear, time-invariant systems. because of Euler's formula, sinusoidal functions can be expressed as exponentials (but with imaginary or complex argument). using that representation, every sinusoid can have both its phase and amplitude represented as a single complex value. phase is actually a component of the complex amplitude.

 

[the_emi_guy]

... phase is actually a component of the complex amplitude.

It is more precise to say that amplitude and phase are independent degrees of freedom, both captured in the complex number. After all, if we say that phase is a component of complex amplitude then what shall we call the other component of complex amplitude?

 

[rbj]

It is more precise to say that amplitude and phase are independent degrees of freedom, both captured in the complex number. After all, if we say that phase is a component of complex amplitude then what shall we call the other component of complex amplitude?

it would be "amplitude", i guess. (without the qualifier "complex")

i am saying that the "complex amplitude" of

$$x(t) \ = \ A \ e^{j(\omega t + \phi)} \ = \ A e^{j \phi} \ e^{j \omega t}$$

is

$$A \ e^{j \phi}$$ .

it's a single multiplicative factor and contains both components of information regarding the sinusoid.

 

[Enthalpy]

Lucky you, electrical engineering and signal amplitude is THE use where complex numbers are very concrete. Why do you complaint?

 

[AlephZero]

If you don't like complex numbers for some reason, you can rewrite all the theory using the "phase" (0 degree phase angle) and "quadrature" (90 degrees) components of the signals if you want.

But flip-flopping between "phase-and-quadrature" and "complex" is a good way to make mistakes, since the corresponding formulas differ by a few "randiomly" placed minus signs.

 

[the_emi_guy]

There was a time in history when some folks objected to the idea of negative numbers because they did not map directly to reality. No such thing as a negative number of apples in a basket, or negative distance between cities and so forth.

I do appreciate the OP's desire to want to look under the hood and understand why we are choosing to use complex numbers. Some students will just memorize stuff without thinking about it, and sometimes the big picture doesn't come out in the classroom.

 

[rbj]

There was a time in history when some folks objected to the idea of negative numbers because they did not map directly to reality. No such thing as a negative number of apples in a basket, or negative distance between cities and so forth.

negative quantities do exist in reality. "distance" between cities is the absolute value of relative position. the position is a signed quantity that can be negative. the elevation (above sea level) of the surface of the Dead Sea is a negative 400 meters.

negative quantities are real.

in my opinion (and we've gotten into fights about this terminology here at PF and also at the comp.dsp newsgroup) the term "imaginary" is apropos for numbers that when squared become negative real numbers. those quantities do not really exist in physical reality. they are an abstraction. but negative quantities do really exist in physical reality.

 

[the_emi_guy]

negative quantities do exist in reality. "distance" between cities is the absolute value of relative position. the position is a signed quantity that can be negative. the elevation (above sea level) of the surface of the Dead Sea is a negative 400 meters.

negative quantities are real.

My point was that in former times use of negative numbers was not common. People talked about distances. The dead sea would have been 400 cubits below sea level not -400 cubits above sea level. Educated folk today understand how these mean the same thing.

We have evolved to the point where, today, virtually no one questions the validity and utility of negative numbers, we are introduced to them at a very early age.

But some folk question the existence of imaginary numbers. The OPs objection to imaginary numbers is similar to the historical objection to negative numbers. I'm pointing this out so that the OP can put it in perspective and embrace imaginary numbers.