## Understanding OFDM - Part 2

By CWNP On 10/06/2009 - 15 CommentsIn Part 1 of this “blogtorial”, Rick provides a high level comparison of the way OFDM is used in IEEE 802.11a/g. In Part 2, he gives a more detailed description of OFDM subcarriers and the four modulation types used in Wi-Fi.

##### Understanding OFDM

###### Part 2 – Subcarriers Unstrung

By Rick Murphy

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**Review of Part 1**

- OFDM is used in current and soon-to-be released wireless communications technologies
- FDM can be compared to the single notes played sequentially by a lead guitarist
- OFDM can be compared to the multi-note chords produced by a rhythm guitarist
- Like a chord produced by a guitar, an OFDM symbol contains multiple information components
- 802.11a/g uses a form of OFDM which creates 64 subcarriers
- Only 48 of these subcarriers are used to represent bits from the input user data stream

**Overview**

OFDM is simple in theory. A fast, serial data stream (your baseband user data for instance) is divided by means of a “serial-to-parallel” conversion, into multiple, slower data streams. In addition, the allocated RF channel, which in 802.11a/g is 20 MHz wide, is divided into many smaller sub-channels known as “subcarriers” (also called “bins” or “tones”) through a mathematical function known as an Inverse Fast Fourier Transform (IFFT). The IFFT converts an input signal from the time domain by mapping its baseband frequencies onto their representative phases and amplitudes in preparation for modulation to the passband.

**Subcarriers**

In 802.11a/g, the subcarriers created by IFFT are each 312.5 KHz wide. If you divide the channel bandwidth (20 MHz) by the subcarrier bandwidth (312.5 KHz) you will see that 802.11a/g comprises 64 subcarriers. This is referred to as 64-point FFT/IFFT.

Each of these subcarriers is capable of being individually modulated to carry part of the current input data stream. But, not all of the subcarriers are used in that way. 802.11a/g assigns the first six subcarriers located at the lower end of the channel to act as guard subcarriers. This is to protect against Inter-channel Interference (ICI) with the adjacent lower channel. Likewise, the last five subcarriers, located at the upper end of the channel are used to guard against ICI with the adjacent upper channel. These subcarriers are inactive or null. These 11 subcarriers (5 + 6) make up the “guard band” between any two adjacent channels. The center subcarrier, called the Direct Conversion subcarrier (DC), is also inactive. The reservation of these 12 non-populated subcarriers (11 guards and 1 DC), leaves a remainder of 52 populated subcarriers. Of these 52 populated subcarriers, 802.11a/g reserves four to be used as “pilot subcarriers”, which carry only timing and frequency information to help the receiver synchronize with the transmitted signal. Although, the pilots are used to carry information, it’s not information taken from the input data stream. A final tally shows that of the 64 subcarriers created by IFFT in an 802.11a/g 20 MHz channel; only 48 of them are available to be modulated with input data.

**Some Simple Calculations**

In 802.11a/g and even the new 802.11n, a new modulation symbol can be transmitted every 4 microseconds. Based on that, it’s possible to signal 250,000 symbol transitions every second (one second/4 microseconds = 250,000) on each of the subcarriers.

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**Symbols per Channel Including Nulls (Gross)**

Remember that there are a total of 64 subcarriers in 802.11. If it’s possible to transmit 250,000 symbols per second on each of the subcarriers, and if we were to use all of the 64 subcarriers for input data, then the full channel capacity would equal 16,000,000 symbols per second.

**Symbols per Channel Excluding Nulls and Pilots (Net)**

But since only 48 of the possible 64 subcarriers are used to carry user data in 802.11 that means that the actual channel capacity is only equal to 12,000,000 data symbols per second.

** “Symbols per second” = “baud rate”**

Another way of saying, “the number of symbols per second” is to say “baud rate”, which is a well known telecommunications term. The baud rate is defined as, “the number of distinct symbol changes made to the transmission medium per second” (Wikipedia). So, it is just as accurate to reword the previous lines to say that each 802.11 OFDM subcarrier has a baud rate of 250,000 baud (250 kilobaud or kBd). Likewise, the theoretical 802.11 OFDM channel capacity is 16,000,000 baud (16 Megabaud or MBd), but the maximum data transmission rate is limited to 12,000,000 baud (12 Mbd). Just for perspective, when I wrote “Guitar Tuner” on a Commodore VIC-20 back in 1981, 300 baud modems were “state-of-the-art”. We’ve made some progress since then.

**Modulation Coding Schemes (MCS)**

If you are already familiar with the dynamic rate shift levels used in 802.11a/g, and know that they can automatically adjust, according to environmental conditions, between 54, 48, 36, 24, 18, 12, 9, and 6 Megabits per second (Mbps), you may wonder, like me, how the data rates available in 802.11a/g correlate with the baud rate. The thing to keep in mind here is that the baud rate and bit rate are NOT necessarily the same thing. Sometimes they’re equal and sometimes they’re not. The baud rate, as we have seen, is the number of possible symbol changes per second. The bit rate is determined by the type of modulation that is being used to produce the symbol. Some types of modulation methods can represent more than one bit of information with each symbol. This is done through the creative use of phase and amplitude modifications of the transmitted symbol. The receiver has the job of matching the modulated symbol to a reference pattern in order to recognize the bit sequence that was intended by the transmitter.

**Modulation Levels**

Depending on the quality of the channel, 802.11a/g/n may select one of four types of phase modulation, namely Binary Phase Shift Keying (BPSK), Quadrature Phase Shift Keying (QPSK), 16-level Quadrature Amplitude Modulation (16 QAM), and 64-level Quadrature Amplitude Modulation (64 QAM). BPSK is the simplest and has the greatest propagation capabilities, but only represents a single information bit with each symbol (baud). BPSK is normally selected under poor channel conditions such as if there is a lot of interference in the area or if the transmitter and receiver are a significant distance apart. When BPSK is in use, it’s correct to say that the baud rate and bit rate are equal. With any of the other modulation types used in 802.11a/g/n, the baud rate and bit rate are not equal, since those are able to signal more than one bit per symbol. For example, QPSK, which is used under good channel conditions, represents two information bits per baud (symbol). When channel conditions are very good, 16 QAM modulation may be selected and used to represent four information bits per baud. The highest bit-rate modulation used in 802.11a/g/n, at six bits per baud, is 64 QAM. But it can only be selected when the quality of the channel is excellent. This usually means that before 64 QAM would be selected for transmitting, the sender and receiver would need to be very close to each other, with very little interference in the area.

**I and Q Constellations**

In order to picture these modulation methods, they are usually graphed on a chart using Cartesian (“x” and “y”) coordinates. The “x” axis is called the In-phase (I) axis and the “y” axis is called the Quadrature (Q) axis. For that reason these charts are typically referred to as “I and Q” charts and this family of digital, modulation techniques is frequently called “I and Q” modulation. The charts look something like a view of the night sky, with the dots that represent the ideal phase and amplitude positions looking like stars or satellites in orbit. For that reason, an image showing the locations where I and Q mapped symbols are expected to be received is called an “I and Q constellation”.

**Maximum Channel Bit Rate**

Now that we have determined the maximum baud rate of the channel as being 12 Mbd, we can calculate the maximum bit rates for the channel based on the type of modulation is use. Using BPSK (1 bit per symbol) there would be a maximum of 12 Mbps. With QPSK (2 bits per symbol) there would be a maximum of 24 Mbps. With 4 bit per symbol, 16-QAM, the maximum rises to 48 Mbps. Finally, using 64-QAM, which represents six bits of information with each symbol, the 12 megabaud channel is capable of 72 Mbps. So that’s 72, 48, 24, and 12 Mbps max channel bit rates. “What about the 54, 48, 36, 24, 18, 12, 9, and 6 Mbps data rates that we all know are the ones used by 802.11a/g”, you may ask? Well, a funny thing happens on the way to the air interface. Some of those bits per second are stolen! But in order to do that story justice we’ll discuss it in part three of this commentary.

*To be continued…*

**Summary of Part 2**

- 802.11 FFT/IFFT divides the 20 MHz channels into 64 subcarriers each 312.5 kHz wide
- The center subcarrier is not used to carry information
- Eleven subcarriers are used to provide guard bands to prevent ICI but do not carry information
- Four subcarriers are reserved for use as Pilot subcarriers, but do not carry input data

48 subcarriers are left to transmit input data - 802.11a/g symbol rate per subcarrier is 250 kBd
- 802.11a/g symbol rate per channel is 12 MBd
- 802.11a/g bit rate per channel is 72 Mbps (64 QAM), 48 Mbps (16 QAM), 24 Mbps (QPSK), and 12 Mbps (BPSK)

*Watch for “Understanding OFDM - Part 3 – “So You Say You Want a Convolution”*

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