## WiMax Spectrum Efficiency

The capacity of each sub carrier depends on the modulation order, which can be BPSK (1 bit per sub carrier), QPSK (2 bits per sub carrier), 16QAM (4 bits per sub carrier), or 64QAM (6 bits per sub carrier) in the case of the OFDM PHY. In general more power is required for using higher order modulation in order to achieve the same range performance.

In the OFDM PHY there are 256 sub carriers spanning the sampling spectrum which is defined as:

Eq. 1) Fs = FLOOR(n · BW / 8000) · 8000 ,

Where n is the sampling factor, a constant dependent on the channel size, and BW is the channel size in units of Hz. The number of sub carriers corresponds to the size of the FFT/IFFT used to receive and transmit the OFDM symbols. To reduce the complexity of the digital processing algorithms it is desirable to use FFT sizes that are powers of 2.

For channels in the 3.5 GHz band the licensed channels are multiples of 1.75 MHz and n = 8/7. For a channel width of 3.5 MHz the sampling spectrum is 4.0 MHz. The 256 sub carriers are equally distributed across the sampling spectrum implying a spacing of:

Eq. 2) Δf = Fs/256 .

For example Δf = Fs/256 = 15,625 Hz for a 3.5 MHz channel.

Notice that changing the channel width changes both the sub carrier spacing and the symbol time. This implies a range of practical channel sizes for fixed applications but quickly becomes unworkable for mobile applications where the design approach of scaling the FFT size to the channel width is used with the OFDMA PHY.

In order to provide increased inter-channel interference margin and ease the radio filtering constraints, not all of the 256 sub carriers are energized.

There are 28 lower and 27 upper “guard” sub carriers plus the DC sub carrier that are never energized. Of the 256 total sub carriers therefore, only 200 are used which leaves a total occupied spectrum of Δf · 200 = 3.125 MHz for a 3.5 MHz channel.

This example implies a raw, occupied bandwidth efficiency of 89% (3.125/3.5 = 89%), but the number varies for other channel bandwidths and sampling factors. This is the first example we have encountered of what can be considered to be channel overhead that decreases the channel capacity, in this case it is required by design to improve the channel quality when adjacent spectrum is occupied.

Not all of the 200 occupied sub carriers are used to carry data traffic. There are eight pilot sub carriers that are dedicated for channel estimation purposes, leaving 192 data sub carriers for user and management traffic. In order to calculate the raw channel capacity it is useful to understand how many bits each data sub carrier can carry.

The raw sub carrier capacity, before taking out the overhead added by redundant error correction bits, is given by the modulation order: 6 bits/sub carrier for 64QAM, 4 bits/sub carrier for 16 QAM, and so on. For example, a channel able to support 64QAM modulation could send six bits for each data carrier per symbol. But how long is a symbol?

The orthogonality of the sub carriers is achieved by maintaining an inverse relationship between the sub carrier spacing and the symbol time. So the useful symbol time is just the inverse of the sub carrier spacing:

Eq. 3) Tb = 1/Δf.

For example, a 3.5 MHz channel has a useful symbol time of 1/15625 = 64 us. However for multi-path channels, we must make allowances for variable delay spread and time synchronization errors. In OFDM, this is accomplished by repeating a fraction of the last portion of the useful symbol time and appending it to the beginning of the symbol for a resulting symbol time of:

Eq. 4) Ts = Tb + G · Tb,

Where G is a fraction:

Eq. 5) G = 1/2m, m = {2,3,4,5}.

The repeated symbol fraction is called the “cyclic prefix”. Larger cyclic prefix implies increased overhead (decreased capacity since the cyclic prefix carries no new information) but larger immunity to ISI from multi-path and synchronization errors.

For a 3.5 MHz channel the useful symbol time is 64 us and the minimum total symbol time is Ts = 64 us + 64/32 us = 66us. The raw channel capacity per symbol is:

Eq. 6) Craw = 192 · k / Ts,

Where k is the bits per symbol for the modulation being used.

Assuming 64QAM modulation (6 bits per symbol):

192 data sub carriers x 6 bits/sub carrier / 66 us = 17.45 Mbps.

Notice that the modulation rates are designed so that an FEC coded block just fits in one symbol time when all 192 sub carriers are used.

For instance for 64QAM, 144 Bytes = 1152 bits / 6 bits/symbol = 192 sub carriers.

The useful channel capacity per symbol is:

Eq. 7) C = Craw x OCR,

Where OCR is the overall coding rate given in the table. For example, for a 3.5 MHz channel the useful channel capacity per symbol assuming the highest rate modulation and coding is:

C = 17.45 Mbps x 3/4 = 13.1 Mbps.2

It is useful to summarize the discussion of the channel capacity is terms of the spectral efficiency. Spectral efficiency is expressed in units of bits per second per Hz and is obtained by dividing the channel capacity by the channel width:

Eq. 8) E = C / BW.

We can see that our 3.5 MHz channel has a spectral efficiency (so far) up to 13.1 Mbps / 3.5 MHz = 3.74 b/s/Hz. The spectral efficiency is a useful figure of merit to keep in mind because it lets you quickly calculate the capacity for other channel sizes that WiMAX supports.

2 By now at least some readers must be wondering what happened to the often-hyped 75 Mbps channel capacity for WiMAX? Taking the very largest channel size, 20 MHz, highest coding rate, and minimum cyclic prefix, the raw channel size using equation 6 is: Craw = 192 x 6 b/sub carrier / 11.3 us = 102.0 Mbps.

The useful channel size from equation 7 is: C = Craw x ¾ = 76.5 Mbps. Of course we have said nothing about the (short) range of such a hypothetical channel, and we should be aware that this is before taking out other PHY and MAC layer overhead that, as we will see, is significant. To be blunt, talking about 75 Mbps WiMAX channels for MAN applications is about as meaningful as quoting the top end speed marked on the speedometer of a family minivan.

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