How to convert wave from real to complex and vice versa?

Question

I have wave expressed by array of real numbers (double in C++). But I want to express it as a complex. I tried to create complex variable and assign to its real the array of my wave and to its imaginary just array of zeros. OK it's complex now. But I feel there is something wrong. It's not authentic complex. It's just surrogate. For example I can't manipulate in phase (only reverse phase). When I multiply it by for example $ i^{0.4} $ it's just changing amplitude little bit but phase is without change.

In the other hand if I have authentic complex wave how to express it as a real numbers but including information from imaginary values?

For any help great thanks in advance.

Answer 1

Rather than saying the questions you are asking are non-sensical, I will say they reveal a lack of understanding of complex numbers. There is lots of material to be had with a few simple searches. I would also recommend that you read my blog article The Exponential Nature of the Complex Unit Circle which gives an explanation of Euler's equation which is absolutely essential to understand in order to comprehend the meaning of DFT bin values.

For your first question: Technically, a real number is a complex number with a zero imaginary component. So setting the real part equal to your values and the imaginary part to zero is the correct method. When you multiplied it by $ i^{0.4} $ it should have rotated each value a tenth of a cycle. Thus the amplitude stays the same, but the phase shifts by $\pi/5$ radians.

For your second question: A complex number consists of two values. A real number consists of one. Therefore converting a complex number to a real number is a non-sensical proposition. However, you can calculate the magnitude, which is a real number. You can also ask what the real part is, which is also a real number. The imaginary part is a real number multiplied by $i$.

A real tone is a sinusoidal, flat as a pancake. A complex tone is a spiral, like a slinky. The definitions I use in my blog articles are:

$$ S_n = A \cos( \alpha n + \phi ) $$

For a real tone, and:

$$ S_n = A e^{i (\alpha n + \phi)} $$

For a complex one.

Notice that a real valued tone is just the average of two complex tones:

$$ \frac{ A e^{i (\alpha n + \phi)} + A e^{-i (\alpha n + \phi)} }{2} = A \frac{ e^{i (\alpha n + \phi)} + e^{-i (\alpha n + \phi)} }{2} = A \cos( \alpha n + \phi )$$

Because

$$ \cos( \theta ) = \frac{ e^{i\theta} + e^{-i\theta} }{2} $$

Hope this helps.

Ced

Answer 2

I decided to answer my own question cause I need more space. To all of you who says:

... converting a complex number to a real number is a non-sensical proposition.

I prepared some funny graph which is very popular in all Fourier concern videos on youtube. Probably all of you met it before:

As you probably know that rounding and wraping like snail lines are moving on plane of real an imaginary values. I made it little bit different then everybody, I mean here horizontal values are imaginary, and real are vertical values. So the graph on the right is just wave real values and time, but converted in some way from complex values? Am I wrong? If I have imaginary numbers all zeros, then that rounding snail would be just vertical line going up and down, but right graph representation of wave real values would be the same as it is now.

But when I turn the knob - which represent the $ complexValue * i^{knobValue} $ My real representation of wave (right graph) also change according to the knob value, and according to imaginary values: So the right graph is dependent of real and imag values. But still shows only real values. That's why I think the right graph is in some way complex numbers converted to real numbers but including in some way information from imaginary.

Where I am wrong?