# Orthogonal Frequency Division Multiplexing (OFDM)

I am not a professional but I'm trying to figure out how OFDM works. It is clear to me that OFDM works with closely spaced subcarriers with orthogonal frequency. Each subcarrier gets modulated with a conventional digital modulation scheme, let say for the sake of this example QAM-4.

Lets say that I have a stream of data 0101 1100 1111 0100 broken up in four parts to be send with OFDM.

What confuses me is this. I know that QAM requires two carriers and uses symbols with two bits. Does this mean that fow every broken part of data stream, OFDM will actually use two subcarriers?

QAM does not require any subcarriers at all, it modulates the carrier signal at the carrier frequency.Both the in-phase (I) and the quadrature (Q) components in QAM are modulated at the carrier frequency, the only difference is that the carrier waves used for modulation are $90^0$ out of phase i.e the I-component is modulated with $\cos(2\pi f_ct)$ whilst the Q-component is modulated with $\sin(2\pi f_ct)$, where $f_c$ is the carrier frequency.So the use of QAM does not change the number of subcarriers used in a OFDM scheme.And for clarification in a M-QAM scheme, each symbol will represent $\sqrt{M}$ bits, a symbol does not necessarily have to represent 2 bits.

In OFDM, we have $n$ different subcarriers so instead of transmitting at the carrier frequency $f_c$ we will transmit at a set of frequencies $\{f_i\} \text{ where } i = 1,..,n$ and where all frequencies in $\{f_i\}$ are very close to $f_c$.So if for example we use 4 subcarriers and we have your bitstream of $0101110011110100$, we would first map the bitstream to parallel symbol streams $S_{\# 3} S_{\# 2} S_{\# 1} S_{\# 0}$ and transmit symbol $S_{\# 0}$ at $f_0$, $S_{\# 1}$ at $f_1$, ... e.t.c Because we have 4 subcarriers all 4 symbols will be transmitted in parallel.

• If I understood correctly, every symbol is transmited at different subcarrier frequency? What if the data stream was broken down in three parts? My guess is that since the amplitude and phase of the subcarrier is changed, we can transmit any symbol with any subcarrier? – Quirik Aug 10 '15 at 14:40
• E.g. last bits of data stream are mapped to symbols 00, 01, 11, 11 and then ransmitted at f0, f1, f2, and f3 subcarrier frequencies? – Quirik Aug 10 '15 at 14:48
• No, any symbol can be transmitted at any of the subcarrier frequencies.So even if you have 00 00 11 11 you will still transmit them at $f_0$, $f_1$, $f_2$ & $f_3$.The actual symbol to be transmitted has no connection to which frequency it will be transmitted on. – KillaKem Aug 13 '15 at 9:02
• QAM will use one subcarrier frequency e.g. f0, and its I and Q component, which is 90 degrees out of phase to transmit symbol S0. Is this correct? – Quirik Dec 11 '15 at 15:35
• QAM uses one carrier at $f_c$, you can't call it a subcarrier because there is only one frequency used in QAM modulation. – KillaKem Dec 12 '15 at 10:42

The reason why OFDM is confusing is mainly because it is never fully presented. There are always gaps and holes in the details. For old-school OFDM, you see details of multiple orthogonal carriers (eg. a fundamental sinusoid and harmonics of it are all orthogonal), where each orthogonal carrier amplitude may be altered (eg. either make the amplitude equal to zero or make it a finite real constant value - which can be changed with time - this is using each carrier as a binary data-carrying component).

And then, you see new-school method of 'OFDM', which involves a sequence of QAM vectors (or QAM complex numbers) - a sequence of these, where each complex number in that sequence is a QAM symbol. This sequence can merely be imagined to be a make-believe 'frequency domain sequence'.

This is the main thing. We merely imagine that our data begins with a made-up 'frequency domain' sequence. It merely exists on 'paper' to begin with. This sequence of complex numbers goes into a IFFT processing unit. So basically an IFFT is applied to that sequence of complex numbers. Cyclic prefixing is then applied to the IFFT sequence (to deal with multipath effects later during wireless transmission). After cyclic-prefixing is applied, the resulting sequence is longer than the original IFFT sequence (because appending a cyclic prefix obviously leads to a longer sequence). At this point, it is then necessary to think of a way of sending this new longer sequence (such as wirelessly) - keeping in mind that each 'value' of this longer sequence is a complex number.

Each complex number has a real part and imaginary part. So this is where we can clock out these complex numbers - one number at a time. The rate of clocking out these stored complex values needs to be known and precise, and this rate needs to be known at the receiving side too. The real part of each number can modulate a single sinusoid broadcast carrier. The imaginary part of each number can modulate a 90-degrees phase-shifted version of that same single sinusoid carrier. This is quadrature modulation. These quadrature waveforms can then be added (summed) and then transmitted (eg. wirelessly). Now, sources do say that this transmitted waveform is supposed to be 'OFDM'. But it is really just quadrature modulation.

And - at the receiving side, quadrature demodulation is then carried out, in order to acquire the two (in-phase and quadrature) waveforms, and then some procedures then need to be carried out, such as finding pairs of repeated sequence patterns seen in the incoming waveforms --- this is for the purpose of synchronisation, and the beginning of procedures for channel estimation, followed by data recovery.

There are always a bunch of details left out in discussions about OFDM. I have only mentioned some details. But hopefully it helps people to get closer toward understanding OFDM and also differences between old-school multi-carrier OFDM and the new-school (IFFT) method.

One extra note is - the classical (old-school) method and the IFFT method are quite different. They can be compared - but they are two different kettles of fish.