# OFDM IFFT/FFT processing

After the serial-to-parallel to break input bitstream into groupings we do the constellation mapping into values of phase/amplitude (or I, Q). For 64 sub-carriers this gives us 64 pairs of amplitude(n)/phase(n).

an = amplitude(n) * cos(phase(n))

bn = amplitude(n) * sin(phase(n))

sub-carrier(n) = an * cos (n * delf * t) + bn * sin(n * delf * t)

Why do we need the IFFT. We could just compute the 64 sin/cosine time functions, sum them and then up-convert to passband. We lose the complex notation (an + j bn) where we keep the an and bn terms separate -- and do the up-conversion separately for both the an and bn sums.

At the receiver, we could just do a real sample to get 2*64 = 128 samples of the summed sub-carriers. These real samples could be input to an FFT to get back the an and bn terms. This is in contrast to separating out the an and bn terms at the receiver This would simplify the hardware.

• When computing 64 sin/cosine time functions, will the var be always 1? .. Assumed it's always 1, that is similar to the real transforms matrix such as DCT, DST based OFDM, so that's the benefits of using sin/cosine time function?? Could you explain these points? Jul 15 '21 at 1:44
• What do you man by "var"? I researched "DCT, DST based OFDM" and see mention of only real values. Do you have a good reference on the "DCT, DST based OFDM"? My original question related to potentially avoiding the separate treatment of the IFFT/FFT real and imaginary paths at the cost of 2X the sample at the receiver. Jul 16 '21 at 11:53
• var = variance. var is supposed to be 1 in both time and frequency domains. About you question, yes I mean when you use that mentioned method, will the time and frequency domains variance be 1? Regarding the references, just check on google, OFDM based on DCT ........ etc. Jul 18 '21 at 0:36
• Still not sure about the var -- var of exactly what? I looked at the DCT -- seems it results in real data but 2X number of sub-carriers. I am suggesting to use classic Fourier series w/o the j so the single sub-carrier sin/cos terms will carrier the encoded amp/phase data. at the tx: s(t) = sum[an * cos (freq(n) * t) + bn * sin(freq(n) * t)] and the an/bn carry the amp/phase data. Transmit as real and receive as real with 2X sampling to recover the an/bn terms. In the eq above, the freq(n) does not have to be equally-spaced and n need not be power of 2. Jul 20 '21 at 13:04
• var is the variance of the signal. That variance of the signal must be 1 in both time and frequency domains. Jul 21 '21 at 2:02

For your sub-carrier formula, you forgot the $$j$$ in front of the $$b_n$$; without that, you lose all your data, because your real and imaginary parts get summed. Can't do that! The whole idea of complex equivalent baseband is that the real anand imaginary parts are independent; this does not only apply to OFDM, but to any baseband technique; for example, QPSK wouldn't be possible if you just added your real and imaginary parts.