# Visualising OFDM subcarriers in the frequency domain

Im struggling to understand why in some images the frequency domain shows clear bars with amplitude and phase and with 0 frequency variation (it is clear that the frequency of the waves does not change in the image on the left).

Yet in others it shows it sloping or curving upwards:

Why isn't this a bar when the frequency of the signal clearly doesn't vary?

Same thing in OFDM. The constellation points should map to a single 312.5KHz aligned frequency and modify their amplitude and phase, yet when you look at the frequency domain on the OFDM carriers, they're sinc shaped waves instead of being bars. How is this possible?

## 1 Answer

Note that this illustration is just that, an illustration: The spectrum on the right can't be explained by the time-domain signal excerpt on the left alone. What's shown on the right is the spectrum that's typical for a tone with some phase noise, no matter where that comes from. But since frequency is the time derivative of phase: no, if you get something like the right spectrum, you don't only have a tone of a single, constant, frequency.

Why isn't this a bar when the frequency of the signal clearly doesn't vary?

Because only periodic signals have line spectra. Your OFDM signal isn't periodic (for longer than an OFDM symbol), so it can't have a line spectrum.

Since any information transmission requires change of some parameters of a wave to actually contain any information, no information-carrying signal can have (with sufficiently long observation) have a line spectrum. End of story!

You'll notice that when reading up on pulse shaping, for modulations¹ where there's no DC offset², the pulse shaping defines the spectral shape.

In OFDM, the pulse shape is a rectangle the length of the OFDM signal. Hence, the sinc-shaped spectrum. (and the trick about OFDM is that the zeros of the individual sincs happen to fall exactly on the maxima of the other sincs, so that they don't interfere, hence the "O" in OFDM.)

¹ no matter whether we're considering a single-carrier system or just one subcarrier of an OFDM system

² i.e. all practical digital modulations aside from a few power detection-based specialities.

• So it's something to do with all the harmonics also appearing on the frequency domain graph and the curve in the frequency domain is just a curve connecting the frequencies together. But the main point here is that the subcarrier is a separate entity regardless of the constellation map changes that has been fashioned to have a sinc shaped frequency domain to allow for ICI removal due to their orthogonality. This can only mean I assume that a single subcarrier looks like a rectangle function in the time domain..? – Lewis Kelsey Nov 11 '18 at 5:01
• No, it wouldn't be a rectangle function as that averages out at 0 frequency, rather it would average out at a wave (sinusoid) with a constant frequency in the time domain. – Lewis Kelsey Nov 11 '18 at 5:04
• And the harmonics of this wave overlap in the frequency domain by virtue of the carrier spacing and the fact the harmonics create a sinc shape – Lewis Kelsey Nov 11 '18 at 5:10
• an OFDM subcarrier has rectangle pulse shape (ask any text book). I don't understand "as that averages out at 0 frequency": Whatever that means, but the Fourier transform of a rectangle ("boxcar windows") is a sinc, definitely. I don't know what you mean with harmonics – only peridic signals have harmonics. A rectangle isn't periodic. – Marcus Müller Nov 11 '18 at 9:18
• So a subcarrier wave consists of lots of waves being sent at frequencies in the 312.5KHz range (sampled from the sinc), and amplitudes, and this looks like a rectangle function in the time domain? – Lewis Kelsey Nov 11 '18 at 9:31