MATLAB: Applying Inverse Fast Fourier Transform to an array of real numbers

Question

I'm trying to understand the difference between applying the Discrete Cosine Transform (DCT) and the Inverse Discrete Fourier Transform (IDFT) to the log Mel-filterbank energies as explained in the answer here. So, I tried the following pretty simple example in MATLAB:

x1 = [1. 2. 3.];
X1 = fft(x1);
x2 = ifft(abs(X1));
x3 = ifft(x1);

The classes of the results are:

Why do x2 and x3 have two different classes?

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Update:

Here is the file 'myVoice.wav' in the following test:

[signal, fs] = audioread('myVoice.wav');
segmentLength = 240;
x = signal(20490 : 20490 + segmentLength - 1);
x = x .* hamming(segmentLength);
res = ifft(abs(fft(x)) .^ 2);

Answer 1

Since $\vert X_1 \vert$ is a real even function, its FFT (IFFT) is a real function. This is a basic property of Fourier transforms.

The power spectrum can be real but not even, in which case the ifft will give complex coefficients. In the example you linked, they start with a non-even real signal and then take the FFT. This will give complex values coefficients for the FFT. The answer then says that

The problem is that we have only our log-energies, no phase information, so after applying the ifft we get complex-valued coefficients

So, after taking the FFT, the phase is discarded, and this loss of information makes it impossible to get back to the original real signal. (The Fourier transform is bijective). Once the phase is discarded you have a real but non-even function, the IFFT of which will yield complex coefficients.