openofdm/docs/source/eq.rst

Sub-carrier Equalization and Pilot Correction ============================================

• Module: equalizer.v
• Input: I (16), Q (16)
• Output: I (16), Q (16)

This is the first module in frequency domain. There are two main tasks: sub-carrier gain equalization and correcting residue phase offset using the pilot sub-carriers.

Sub-carrier Structure

The basic channel width in 802.11a/g/n is 20 MHz, which is further divided into 64 sub-carriers (0.3125 MHz each).

Sub-carriers in 802.11 OFDM

fig_subcarrier shows the sub-carrier structure of the 20 MHz band. 52 out of 64 sub-carriers are utilized, and 4 out of the 52 (-7, -21, 7, 21) sub-carriers are used as pilot sub-carrier and the remaining 48 sub-carriers carries data. As we will see later, the pilot sub-carriers can be used to correct the residue frequency offset.

Each sub-carrier carries I/Q modulated information, corresponding to the output of 64 point FFT from sync_long.v module.

FFT of the Perfect and Two Actual LTS

To plot fig_lts_fft:

lts1 = samples[11+160:][32:32+64]
lts2 = samples[11+160:][32+64:32+128]
fig, ax = plt.subplots(nrows=3, ncols=1, sharex=True);
ax[0].plot([c.real for c in np.fft.fft(lts)], '-bo');
ax[1].plot([c.real for c in np.fft.fft(lts1)], '-ro');
ax[2].plot([c.real for c in np.fft.ff t(lts2)], '-ro');
plt.show()

fig_lts_fft shows the FFT of the perfect LTS and the two actual LTSs in the samples. We can see that each sub-carrier exhibits different magnitude gain. In fact, they also have different phase drift. The combined effect of magnitude gain and phase drift (known as channel gain) can clearly be seen in the I/Q plane shown in fig_lts_fft_iq.

FFT in I/Q Plane of The Actual LTS

To map the FFT point to constellation points, we need to compensate for the channel gain. This can be achieved by normalize the data OFDM symbols using the LTS. In particular, the mean of the two LTS is used as channel gain (H):

$$H[i] = \frac{1}{2}(LTS_1[i] + LTS_2[i])\times L[i], i \in [-26,\ldots, -1, 1, \ldots, 26]$$

where L[i] is the sign of the LTS sequence:

$$\begin{aligned} L_{-26,26} = \{ &1, 1, –1, –1, 1, 1, –1, 1, –1, 1, 1, 1, 1, 1, 1, –1, –1, 1,\\ &1, –1, 1, –1, 1, 1, 1, 1, 0, 1, –1, –1, 1, 1, –1, 1, –1, 1,\\ &–1, –1, –1, –1, –1, 1, 1, –1, –1, 1, –1, 1, –1, 1, 1, 1, 1\} \end{aligned}$$

And the FFT output at sub-carrier i is normalized as:

$$S'[i] = \frac{S[i]}{H[i]}$$