rs and 12 virtual subcarriers that perform a guard band function. The 52 'useful' subcarriers are equally spaced around the centre frequency of each channel. The four pilot signal carriers (-21, -7, 7 and 21) are used to model and track frequency offsets. The modulation options and associated gross data rates are shown in Table 2.13 - the coding rate options are also shown.2 A 54 Mbps OFDM carrier using 64 QAM has a symbol rate of 4 microseconds. The symbol length in single carrier 64 QAM would be 50 nanoseconds. The guard interval is 0.8 microseconds; the data burst is 3.2 microseconds long. The job of the IDFT (inverse discrete Fourier transform) is to do the maths to calculate the sine and cosine waveform values needed to create the composite time-domain sine and cosine waveforms that will represent these 64 frequency subcarriers each of which has a specific phase and amplitude. The back end of the waveform is copied to the front in order to provide a guard period and the waveform is clocked out at 20 MHz to form the 3.2-microsecond modulated burst with the 0.8-microsecond guard period. In the receiver, a DFT is applied to the composite time-domain waveform and (all being well), the waveform across the burst is resolved into 64 frequency subcarriers each with the phase and amplitude information needed to decide which 4 bits were originally modulated onto the signal. The process exploits a number of convenient properties of sinusoidal waveforms. 2.13.5 Properties of a Sinusoid A cosine /sine function has a period of 360 degrees. The cosine curve is the same shape as the sine curve but displaced by 90 degrees. Providing the system is linear, a sinusoidal input results in a sinusoidal output. The average of the values described by any true sinusoidal waveform is zero if averaged over an even number of cycles. Multiplying cos(x) by sin(x) produces a third waveform that is smaller in amplitude and at a higher frequency, but that will sum to zero when averaged out. The cosine (real) computation only measures the cos