Complex Exponentials Signal processing

[MikeSv]

Hello everyone.

Iam about to read a course in DSP and I can't get my head why complex exponentials are used as input signals that often?
Is it just to simplify the math?
If not, what exactly is the imaginary part of a complex exponential? Does it "do" anything special compared to a real valued sinusoidal (rotation?)?

Most of the times you just compute Magnitude and Phase with the help of the real and imaginary part, but what is it that is so special about these types of input signals?

Any help would be great!

Thanks in advance,

(Very confused) Mike

 

[NFuller]

Most of the times you just compute Magnitude and Phase with the help of the real and imaginary part, but what is it that is so special about these types of input signals?

Right, and magnitude and phase are definitely of interest when studying signals. I wouldn't say there is anything special about them other than they are simple functions which allow you to probe the behavior of a system without too much effort.

 

[Charles Link]

I can give you an input or two that may be helpful. It is much easier to find the sum of a couple of signals at the same frequency using the complex number algebra rather than using trigonometric identities to compute the result. The same is the case in taking time derivatives of these signals. The complex numbers simplify the calculations over using trigonometric identities.

 

[Bart McCoy]

Complex exponentials (magnitude, <angle ) are just an extremely compact way of representing signals that makes computations easier.

In DSP, we often de-compose signals into different frequency components with a Fourier Transform. Any periodic signal can be broken down into the sum of sinusoids, each with a unique frequency, amplitude, and phase (delay). One such waveform could be 3*cos(314*t - pi/2). The peak amplitude is 3, the radian frequency is 314 rad/sec (100 Hz) and the phase (or delay) is pi/2. The phase is just a constant delay/shift relative to a normal cosine (pi/2 or 90 deg in this case). The complex exponential for this example is 3 angle(pi/2), assuming you always add the negative sign.
You usually accompany it with a frequency vector (or a normalized frequency vector, used in DSP a lot) so that it makes sense.

e.g. The two vectors below form a crude 100 Hz square wave centered around 0, using my crude ASCII mag <angle notation.
f = [ 0 100 300 500 700]
signal = [ 0, 1 <0, 1/3<pi, 1/5<0, 1/7<pi ]

Programs like MATLAB use these very efficiently. You can do all sorts of useful, efficient signal processing with complex exponentials.
A filter simply changes the magnitude & angle of "signal" !

Better than the equivalent:
signal(t) = 0 + 1*cos(2*pi*100*t) + 1/3*cos(2*pi*300*t - pi) + 1/5*cos(2*pi*500*t) + 1/7*cos(2*pi*700*t - pi)