simultaneous inverse fast fourier transform of two real functions

Question

I'm trying to calculate the inverse Fourier Transform of two real functions with a single IFFT. The best and most straightforward explanation I've found so far is here, where it says:

Use the fact that the FFT is linear and form the sum of the first transform plus i times the second. You have two vectors, x1 and x2, with discrete Fourier Transforms X1 and X2 respectively. Then

x1 = Re[ IDFT[ X1 + i X2 ] ]

and

x2 = Im[ IDFT[ X1 + i X2 ] ].

The problem is that I don't get where the 'i' parameter comes from. Any hint on this would be much appreciated.

Thanks in advance.

EDIT:

After doing some experiments I finally made it work, but now I'm more confused than before as it didn't work as I expected and had to use some imagination to figure out the correct formulas.

I just made up a new complex array where:

Re[n] = X1Re[n] - X2Im[n]
Im[n] = X2Re[n] + X1Im[n]

After doing an IFFT on it x1 = Re and x2 = Im, so wouldn't it be correct to express it like this?

x1 = Re[ IDFT[ X1 - i X2 ] ]
x2 = Im[ IDFT[ X2 + i X1 ] ].

Answer 1

Are you wondering what the 'i' represents? In this case, I believe 'i' is referring to sqrt(-1), the imaginary unit vector.

Then:

Re[ IDFT[ X1 + i X2 ] ]

will be the 'real' part of that transform (anything without an 'i') and

Im[ IDFT[ X1 + i X2 ] ]

will be the 'imaginary' part of that transform (anything multiplied by an 'i').

It is possible I've misunderstood your question and this answer is much too simplistic; if it is, no insult was intended to your intelligence, I just misunderstood you.

Answer 2

If you want to ignore the mathematics of complex variables, multiplying by i is just notation for how you swap and scale a pair of vectors to produce another pair of vectors. And the complex vectors X1 and X2 can each be considered to be just pairs of real-valued vectors (with a "complex" relationship under the transforms of interest). The swap and scale makes the two component vectors more easily separable, after some arithmetic and transforms, into the real valued vector of interest.