OFDM Subcarrier Spacing, Guard Intervals, and Channel Width

A radio channel is never flat. Multipath reflections arrive at the receiver with different delays, and the sum of those delayed copies is a filter with peaks and notches spaced according to the delay spread of the environment. An indoor office with 50 nanoseconds of delay spread has a coherence bandwidth of a few megahertz, so its notches fall inside a 20 MHz channel and destroy any signal whose bandwidth covers them. A single carrier modulated at tens of megabits per second occupies enough bandwidth to straddle several notches, and an equaliser wide enough to undo the resulting intersymbol interference becomes expensive and fragile.

Orthogonal frequency-division multiplexing avoids the equaliser by making each subcarrier narrow enough that the channel looks flat across it. The price is a different set of problems, each governed by a constant that the standard fixes.

Orthogonality as a definition, not an approximation

Two sinusoids are orthogonal over an interval if the integral of their product is zero. For subcarriers spaced by exactly $1/T$ over a symbol of length $T$, that holds: the integral of $\cos(2\pi f_k t)\cos(2\pi f_m t)$ over the symbol is zero whenever $f_k \neq f_m$ and both are integer multiples of $1/T$. Orthogonality means the subcarriers do not interfere even though their spectra overlap; a receiver can separate them by correlating with the right frequency, with no filtering needed.

The practical consequence is that the whole set of subcarriers is generated by a single inverse discrete Fourier transform and recovered by a forward one. A 64-point transform produces 64 samples, which are transmitted as one symbol. The transform is what makes OFDM cheap: an implementation does one FFT per symbol regardless of how many subcarriers it carries.

For the 5 GHz OFDM PHY the transform is 64 points over a 20 MHz channel, giving subcarrier spacing of 20 MHz / 64 = 312.5 kHz. Of the 64 subcarriers, 52 are used: 48 carry data, 4 are pilots used for phase and frequency tracking. The centre subcarrier is left null because a direct-conversion receiver has a DC offset there, and the outermost subcarriers are guard bands that keep the spectrum inside the channel mask. The symbol is 3.2 microseconds long, the reciprocal of the spacing.

The cyclic prefix buys delay tolerance at a fixed cost

A symbol transmitted through a multipath channel arrives as several delayed copies. The last samples of one symbol overlap the first samples of the next, which is intersymbol interference. OFDM’s defence is to copy the final portion of each symbol and prepend it, extending the symbol by a guard interval while keeping the transform period unchanged.

The prefix works because of a property of the discrete Fourier transform: a circular shift of the input is a phase rotation of the output. If the channel delay is shorter than the prefix, the received samples over the transform window are a circular convolution of the symbol with the channel, and the receiver can undo it with a single complex multiply per subcarrier, a one-tap equaliser. If the delay exceeds the prefix, the convolution stops being circular and the orthogonality breaks; the subcarriers interfere with each other, an effect called inter-carrier interference, and no per-subcarrier equaliser can repair it.

The guard interval is therefore a hard limit on how much delay spread the system tolerates. The legacy 5 GHz PHY uses a 0.8 microsecond prefix, one quarter of the 3.2 microsecond symbol, for a 4 microsecond symbol period. At the speed of light a microsecond of excess path is about 300 metres, so 0.8 microseconds tolerates reflections whose extra path is a few hundred metres, generous indoors and marginal in a large metal structure. The high-throughput amendment later added an optional short guard interval of 0.4 microseconds for environments with less spread.

The cost is proportional and unavoidable. A 0.8 microsecond prefix on a 4 microsecond symbol spends 20 percent of airtime transmitting a copy of data already sent. Shortening the prefix to 0.4 microseconds lowers that overhead to about 11 percent and raises throughput by roughly the same proportion, at the price of tolerance for delay spread beyond 0.4 microseconds. That is the entire trade: prefix length versus tolerance for delay spread, with throughput as the currency.

Channel width and the transform size

Widening the channel does not change the subcarrier spacing in the legacy design. A 40 MHz channel is served by a 128-point transform at the same 312.5 kHz spacing, so it carries twice as many subcarriers and the symbol duration is unchanged. The 80 and 160 MHz channels of the later 5 GHz amendments use 256- and 512-point transforms at the same spacing, and the prefix stays at 0.8 microseconds in the mandatory mode.

Holding spacing constant keeps the prefix fraction and the delay tolerance constant while the bandwidth grows. It also means that wider channels are more likely to contain a notch, because the channel’s coherence bandwidth does not grow with the channel width. A 160 MHz channel has a good chance of containing a deep fade somewhere, and since the coding and interleaving spread each coded bit across subcarriers, the loss is absorbed rather than fatal, but the effective SNR is lower than a narrow channel’s.

Wider channels also contend with more neighbours. In the 5 GHz band the channels are adjacent and overlapping, so a station using 160 MHz must defer to any 20 MHz transmission anywhere inside its span. This is why wide channels help a lone station and hurt a dense deployment, and why access points commonly reduce their channel width when the channel is busy. The mechanism by which a wide-channel station learns that part of its span is occupied is clear channel assessment on the primary channel, by energy detection and by preamble detection, and the interaction with overlapping cells is covered in Spatial reuse, OBSS, and BSS coloring.

The 802.11ax reversal: narrower subcarriers

The high-efficiency amendment changed the constant. It defines subcarrier spacing of 78.125 kHz, one quarter of the legacy value, which makes each symbol four times longer: 12.8 microseconds instead of 3.2. The transform for a 20 MHz channel is 256 points rather than 64.

The motivation is efficiency rather than raw peak rate. A symbol four times longer carries the same number of bits per subcarrier at the same modulation, so the data rate per subcarrier per unit time is unchanged; what changes is that the fixed overheads of prefix, preamble, and interframe spacing become a smaller fraction of the transmission. The prefix becomes 0.8, 1.6, or 3.2 microseconds against a 12.8 microsecond symbol, so the mandatory prefix costs 6 percent of airtime rather than 20. Outdoor delay spreads of several microseconds, which break the legacy prefix, fit inside the longest option.

Longer symbols have a second effect that matters for a dense network: they tolerate timing errors and frequency offsets better, because a given absolute error is a smaller fraction of the symbol. A cheap crystal with a few parts-per-million error produces a phase drift that is more damaging over a short symbol than a long one.

The cost is latency granularity. A transmission cannot be shorter than one symbol, and a symbol is now 12.8 microseconds plus prefix. Small frames pay a larger minimum. The amendment compensates by making the smallest resource unit 26 subcarriers, about 2 MHz, rather than a whole 20 MHz channel, so a small frame occupies a narrow slice of the band for a longer time instead of the whole band briefly. That is the mechanism behind multi-user transmission, described in OFDMA scheduling and multi-user transmission.

Pilots, nulls, and what the subcarrier map hides

The subcarrier map is not simply “all subcarriers carry data”. Four pilots in the legacy 20 MHz channel are known symbols inserted at fixed positions; the receiver compares them to what it expects and derives a phase and amplitude correction that it applies to the data subcarriers. Without the pilots, residual carrier frequency offset between transmitter and receiver would rotate the constellation over the symbol and no amount of equalisation would fix it.

The pilots cost capacity: 4 of 52 subcarriers, about 8 percent, in the legacy channel. Later amendments scale the pilot count with the width and change their pattern between symbols so that a receiver can track a drifting channel. In the high-efficiency amendment pilots within a resource unit are fewer and their density adapts to the RU size, which is one of the reasons small allocations carry proportionally more overhead.

The null at DC and the guard subcarriers are also part of the map. A receiver that assumed a data subcarrier at DC would decode noise; a transmitter that filled the guard band would violate the spectral mask and interfere with the adjacent channel. Both are fixed by the standard and appear in every implementation’s tone plan, which is why the number of usable subcarriers in a 20 MHz channel is 242 rather than 256 in the later amendment and 52 rather than 64 in the legacy one.

What the physical layer leaves to the MAC

OFDM converts a hostile channel into a set of parallel, approximately flat subchannels. It does not decide how many bits to put on each one. That decision, the modulation and coding scheme, depends on the signal-to-noise ratio at the receiver, which the transmitter does not know directly and must infer from feedback or from the pattern of failures. The mechanism that makes that inference is described in Rate adaptation and MCS selection, and it is where the physical layer’s constants turn into the throughput a user observes.