Examples of complex signals?

[ironmathyu5]

I'm new to signal processing and I have been learning DSP techniques and applying them to a project of mine. While doing so I noticed that Fourier transform for example is supposed to decompose a signal with both real and imaginary parts into its sinusoidal components. What I didn't understand is how can a signal have an imaginary component? signals I'm dealing with are usually vibration data and real world sensors. But I couldn't think of any signal with an imaginary part.

[Digital Larry]

In the case of my guitar playing, the imaginary part is where people scream to hear more of it!

I found this:  https://www.quora.com/What-is-the-physical-significance-of-imaginary-signals

What came to mind is that real world signals don't have imaginary parts because they don't extend infinitely in time forwards and backwards, but I'm not sure that's the real reason.

DL

[ElectricDruid]

I'd say that Real and Imaginary parts of a signal are simply a way of dealing with the phase. You either provide the signal phase as a separate piece of information, so in the form y = sin(x+p) where p gives you the phase, or you express the same thing as a mixture of sin and cosine ("real" and "imaginary") parts, in the form y = a*sin(x) + b*cos(x). The net result is the same.
(I think its that trig identity: sin(a+b) = sin(a)*cos(b) +sin(b)*cos(a)..but don't quote me on that)

I'd also argue that what's being done isn't really "complex numbers" in any sense much. It's just that you can express the signal in various ways, and one way uses two dimensions, and since mathematicians already had a notation for "2D numbers" they used that, so we finish up with "Imaginary" parts in our signals. There's nothing imaginary about any of it, really.

[Rob Strand]

The question you should ask is what does it mean to have a complex signal.    Once you get that you will know there's no need to ask for an example.

When you use an Oscilloscope you only see real signal cos(wt) or sin(wt), or in general A cos(wt + phi), the imaginary part does not exist.    To make a stronger point imagine a mechanical system like a speaker cone.  The position of the speaker is real, we can measure it.   We have all the information to understand and analyse the system with real signals.

When you deal with complex signals,  A exp(j(wt + phi)), is it a mathematical abstraction.    It works because linear systems by definition must follow the rule of superposition.
The key points are:
- The complex signal contains the same information as the real signal.  It's just a different representation of the same thing.
- The whole purpose of the mathematical abstraction is to make computations easier.   

You can see why it holds together from these notes,
https://www.cs.ccu.edu.tw/~wtchu/courses/2012s_DSP/Lectures/Lecture%203%20Complex%20Exponential%20Signals.pdf   

If you have ever solved equations for electrical or mechanical systems you will know that it can be tedious.  You are continually messing with both the cos part and the sin part and using trig identities like cos(A+B).    The exponential form makes life a lot easier,  that's why we use Laplace transforms and complex numbers to solve AC circuits, not differential equations.

There's even a real number equivalent of the Fourier transform.  It is very slow to compute even for a computer.   The invention of the FFT was a big step forward but it needs complex signals to see the patterns and make it fast.

The FFT produces complex results.   That's where you get stuck.   There's one step missing, the one which converts the complex signal back to the underlying real signal it represents.

You can summarize all the above processes as,
- Real signal
- Convert to complex signal equivalent
- Do processing or calculations
- Convert complex signal back to real result.

The complex numbers are just a stepping stone to make the calculations easier.

[elseif]

I'm new to signal processing and I have been learning DSP techniques and applying them to a project of mine. While doing so I noticed that Fourier transform for example is supposed to decompose a signal with both real and imaginary parts into its sinusoidal components. What I didn't understand is how can a signal have an imaginary component? signals I'm dealing with are usually vibration data and real world sensors. But I couldn't think of any signal with an imaginary part.

One real-world example of a signal that can be represented by a complex component is single-sideband modulation.  In the days of yore, it was created by analog mixers and filters.  The complex representation leads to the Hartley and Weaver methods of generation which lend nicely to a DSP implementation. The output signal is the sum of real and imaginary parts.