# Spectral location of individual OFDM symbols

The generic IFFT OFDM transmitter takes $N$ serial input data symbols, parallelizes them, maps them to their signal constellation points, and processes all $N$ resultant complex values using the IFFT to produce $N$ subcarriers (assuming no virtual carriers).

The IFFT takes $N$ input values and produces $N$ output values, such that changing a single input value will change all output values. Therefore each input symbol contributes to each of the $N$ subcarriers; each symbol is "shredded" across all subcarriers.

However, several sources state that in OFDM "each modulation symbol is mainly confined to a relatively narrow bandwidth" (Communications Engineering Desk Reference, Elsevier, 2009, 1st ed, p.287).

Are these two different OFDM implementations, or is the IFFT implemented in a different way to achieve this?

• Each bin in the FFT generates a time waveform that is N samples long. The IFFT is a superposition of all these time waveforms. The "shredding" as you call it is across time, not frequency. – John Dec 5 '14 at 14:04

The IFFT transforms a discrete signal to the time domain. Therefore, the input signal of the IFFT should be interpreted as a frequency domain signal.

For OFDM this means that the subcarriers are defined before the IFFT. The IFFT ouput is the time domain representation that will eventually be transmitted over the channel. Before it can be transmitted it must first be converted to an analog signal. Analyzing this signal in time domain, e.g. by viewing it on an oscilloscope, won't tell you anything about the subcarrier bandwidth.

To see the bandwidth of one OFDM subcarrier you have to analyze the analog signal in the frequency domain, e.g. by using a spectrum analyzer. If you modulate all subcarriers the spectrum will look nearly rectangular, with a flat top, because the subcarriers are close together and overlap. If you modulate only one subcarrier (all inputs of the IFFT are set to zero except one), the spectrum will be much narrower, its bandwidth approximately reduced by a factor that is equal to the number of subcarriers. And it will have a sinc-like shape.

• If you set all IFFT inputs to zero, except one, you still get non-zero output values at all $N$ outputs (OK there may be some input values that may result in certain outputs being zero). I am assuming the actual mathematical IFFT equation is implemented in the transmitter (isn't this typical in OFDM transmitters?) – aslan Dec 5 '14 at 15:20
• @JDVlok What you say is true and OFDM TXs use an ordinary IFFT. But you seem to be mixing up time and frequency domain. If the input to the IFFT is all zero except for one bin than the IFFT output is a discrete-time, complex sinusoidal. In general, more than one sample of a discrete-time complex sinusoidal is unequal to zero. The term "narrowband" refers to frequency domain but what you are analyzing is the time domain. – Deve Dec 5 '14 at 15:28
• The IFFT input takes $N$ independent symbols concurrently, and then "converts" them to the time domain (weighting with $e^{j2\pi fmn/N}$ gives the frequency view). The time domain signal is then split up into real and imaginary components which are modulated onto $cos$ and $-sin$ (after repeating each sample sufficiently such that the carrier waves are formed). Taking the FFT of the modulated signal should display the OFDM subcarriers. However, whether you view the IFFT input as a frequency input or not each IFFT input doesn't seem to map directly to each subcarrier. What am I missing? – aslan Dec 5 '14 at 16:46
• @JDVlok The modulated, radio frequency signal is analog. How can you take the FFT of that signal? Of course, you can model this signal in a simulation and do a spectral analysis. If you do that right you should see one narrowband subcarrier if one single subcarrier is non-zero. If you do not see this, then sth. is wrong with your simulation model or the spectral analysis. – Deve Dec 5 '14 at 16:54

Your 2nd paragraph ends incorrectly. Each IFFT input is "shredded" across the all values in the time domain, but only effects one subcarrier in the frequency domain. Just as adding a single pure sinusoid will effect every value in the time domain (except at its zero crossings), but take up only one frequency (essentially no bandwidth) in the frequency domain.

• I agree, changing a single IFFT input value affects only one subcarrier, although all time domain values (IFFT output) are affected. – aslan Dec 6 '14 at 15:16