Constellation Mapping In OFDM

Question

If i am not mistaken, the two following hold :

a. In OFDM Conceptualization, the constellation mapping at Tx (or de-mapping at Rx) is done in Frequency domain

b. However the basic theory (Including the math) of constellation mapping, example PSK/QAM, is always in Time Domain. I have not found any explanation in the frequency domain.

So :

1. Is there a way to convert between the two domains ? (To at least understand the underlying math and the ready-easy-see connection between the two domains ?)

2. Or is there something else going on ? Am i missing something ?

Basically : If i am writhing a MATLAB/Python code...do i do the Modulation (Demodulation) in Time (As per Constellation Mapping concept) Domain or the Frequency (As per OFDM Concept Formulation and usage ) Domain

My concerns arise due to the following :

a. https://en.wikipedia.org/wiki/File:OFDM_transmitter_ideal.png (Constellation Mapping/Modulaton is clearly in Frequency domain.)

b. https://electronicscoach.com/phase-shift-keying.html (PSK concepts are explained in time domain)

So the input bit stream, which could be random data, or say audio-video data is in time domain, it is first modulated by say QPSK. Second, the modulated data is converted to frequency domain ( by dft) , then the mapping to each subcarrier...and then the entire mapped data at transmitter ( in frequency domain) is converted back to time domain using idft, for transmission through the air? Correct? And then finally the same is done in reverse order at receiver ..correct ?

Answer 1

The mapping is in the frequency domain. Each bin of the DFT represents a separate carrier frequency, and what you are doing by mapping in the frequency domain is setting the magnitude and phase for each of those bins, over that current frame in time that the DFT you are creating applies to. The time domain aspect is that for each "symbol duration" that you would be familiar with from traditional QPSK/QAM, the DFT frame changes, and thus each subcarrier gets a new magnitude and phase in each time slot according to the QAM or QPSK constellation.

The process is summarized in the block diagram below:

Consider a very simple case of a 4 sub-carrier OFDM waveform with a QPSK modulation, without getting into adding the cyclic prefix. The DFT would be 4 bins and the modulation on each bin would appear as follows:

So a given frame would be four samples in the frequency domain taking on one of four values: $X_k \in (\frac{1}{\sqrt{2}}(1+j, 1-j, -1+j, -1-j))$ depending on the data. The 4 samples in frequency would become four samples in time through an inverse FFT, resulting in the same 4 samples in time that we would expect to have if we created 4 QPSK waveforms on four closely spaced frequencies and summed them together...which is what gets transmitted over the air as a time domain signal. OFDM is basically a massively parallel implementation of QAM modulations spaced at frequencies where (with proper timing and synchronization) each carrier is orthogonal to the other carriers.

It may be easier to conceptualize if you created the QPSK modulations in time, and modulated them onto each of those separate carrier frequencies which correspond exactly to a bin spacing in a DFT, and then over a time interval consistent with one symbol period (no data changing) you take the DFT of that waveform. You will end up with what we show above, the data has mapped to each bin as the carrier for that data with the appropriate magnitude and phase that was set for that carrier.

Please see this related post that elaborates this point in greater detail.