From 4e3c57c5f33bc84edcc7774649b5069c72c40f75 Mon Sep 17 00:00:00 2001 From: Jinghao Shi Date: Wed, 12 Apr 2017 15:49:17 -0400 Subject: [PATCH] working on eq --- docs/source/eq.rst | 91 +++++++++++++++++++++ docs/source/files/lts.txt | 128 +++++++++++++++--------------- docs/source/freq_offset.rst | 23 +++++- docs/source/images/lts.png | Bin 106321 -> 84706 bytes docs/source/images/lts_16.png | Bin 0 -> 104653 bytes docs/source/images/lts_fft.png | Bin 0 -> 74924 bytes docs/source/images/lts_fft_iq.png | Bin 0 -> 27740 bytes docs/source/images/match_size.png | Bin 72465 -> 67414 bytes docs/source/images/quadrant.png | Bin 0 -> 33333 bytes docs/source/images/subcarrier.png | Bin 0 -> 11938 bytes docs/source/index.rst | 1 + docs/source/sync_long.rst | 91 ++++++++++++++------- docs/source/verilog.rst | 51 +++++++++++- 13 files changed, 288 insertions(+), 97 deletions(-) create mode 100644 docs/source/eq.rst create mode 100644 docs/source/images/lts_16.png create mode 100644 docs/source/images/lts_fft.png create mode 100644 docs/source/images/lts_fft_iq.png create mode 100644 docs/source/images/quadrant.png create mode 100644 docs/source/images/subcarrier.png diff --git a/docs/source/eq.rst b/docs/source/eq.rst new file mode 100644 index 0000000..ab09438 --- /dev/null +++ b/docs/source/eq.rst @@ -0,0 +1,91 @@ +Sub-carrier Equalization and Pilot Correction +============================================ + +- **Module**: :file:`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). + +.. _fig_subcarrier: +.. figure:: /images/subcarrier.png + :align: center + + Sub-carriers in 802.11 OFDM + +:numref:`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 :file:`sync_long.v` module. + + +.. _fig_lts_fft: +.. figure:: /images/lts_fft.png + :align: center + :scale: 80% + + FFT of the Perfect and Two Actual LTS + +To plot :numref:`fig_lts_fft`: + +.. code-block:: python + + 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() + +:numref:`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 :numref:`fig_lts_fft_iq`. + + +.. _fig_lts_fft_iq: +.. figure:: /images/lts_fft_iq.png + :align: center + :scale: 80% + + 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 (:math:`H`): + +.. math:: + + H[i] = \frac{1}{2}(LTS_1[i] + LTS_2[i])\times L[i], i \in + [-26,\ldots, -1, 1, \ldots, 26] + +where :math:`L[i]` is the sign of the LTS sequence: + +.. math:: + + 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\} + +And the FFT output at sub-carrier :math:`i` is normalized as: + +.. math:: + + S'[i] = \frac{S[i]}{H[i]} + diff --git a/docs/source/files/lts.txt b/docs/source/files/lts.txt index d2441b8..e99a602 100644 --- a/docs/source/files/lts.txt +++ b/docs/source/files/lts.txt @@ -1,67 +1,3 @@ --1.559999999999999998e-01 -0.000000000000000000e+00 -1.200000000000000025e-02 --9.800000000000000377e-02 -9.199999999999999845e-02 --1.059999999999999970e-01 --9.199999999999999845e-02 --1.150000000000000050e-01 --3.000000000000000062e-03 --5.399999999999999939e-02 -7.499999999999999722e-02 -7.399999999999999634e-02 --1.270000000000000018e-01 -2.100000000000000130e-02 --1.219999999999999973e-01 -1.700000000000000122e-02 --3.500000000000000333e-02 -1.509999999999999953e-01 --5.600000000000000117e-02 -2.199999999999999872e-02 --5.999999999999999778e-02 --8.100000000000000255e-02 -7.000000000000000666e-02 --1.400000000000000029e-02 -8.200000000000000344e-02 --9.199999999999999845e-02 --1.310000000000000053e-01 --6.500000000000000222e-02 --5.700000000000000205e-02 --3.899999999999999994e-02 -3.699999999999999817e-02 --9.800000000000000377e-02 -6.199999999999999956e-02 -6.199999999999999956e-02 -1.189999999999999947e-01 -4.000000000000000083e-03 --2.199999999999999872e-02 --1.610000000000000042e-01 -5.899999999999999689e-02 -1.499999999999999944e-02 -2.400000000000000050e-02 -5.899999999999999689e-02 --1.370000000000000107e-01 -4.700000000000000011e-02 -1.000000000000000021e-03 -1.150000000000000050e-01 -5.299999999999999850e-02 --4.000000000000000083e-03 -9.800000000000000377e-02 -2.599999999999999881e-02 --3.799999999999999906e-02 -1.059999999999999970e-01 --1.150000000000000050e-01 -5.500000000000000028e-02 -5.999999999999999778e-02 -8.799999999999999489e-02 -2.100000000000000130e-02 --2.800000000000000058e-02 -9.700000000000000289e-02 --8.300000000000000433e-02 -4.000000000000000083e-02 -1.110000000000000014e-01 --5.000000000000000104e-03 -1.199999999999999956e-01 1.559999999999999998e-01 0.000000000000000000e+00 -5.000000000000000104e-03 @@ -126,3 +62,67 @@ 1.059999999999999970e-01 1.200000000000000025e-02 9.800000000000000377e-02 +-1.559999999999999998e-01 +0.000000000000000000e+00 +1.200000000000000025e-02 +-9.800000000000000377e-02 +9.199999999999999845e-02 +-1.059999999999999970e-01 +-9.199999999999999845e-02 +-1.150000000000000050e-01 +-3.000000000000000062e-03 +-5.399999999999999939e-02 +7.499999999999999722e-02 +7.399999999999999634e-02 +-1.270000000000000018e-01 +2.100000000000000130e-02 +-1.219999999999999973e-01 +1.700000000000000122e-02 +-3.500000000000000333e-02 +1.509999999999999953e-01 +-5.600000000000000117e-02 +2.199999999999999872e-02 +-5.999999999999999778e-02 +-8.100000000000000255e-02 +7.000000000000000666e-02 +-1.400000000000000029e-02 +8.200000000000000344e-02 +-9.199999999999999845e-02 +-1.310000000000000053e-01 +-6.500000000000000222e-02 +-5.700000000000000205e-02 +-3.899999999999999994e-02 +3.699999999999999817e-02 +-9.800000000000000377e-02 +6.199999999999999956e-02 +6.199999999999999956e-02 +1.189999999999999947e-01 +4.000000000000000083e-03 +-2.199999999999999872e-02 +-1.610000000000000042e-01 +5.899999999999999689e-02 +1.499999999999999944e-02 +2.400000000000000050e-02 +5.899999999999999689e-02 +-1.370000000000000107e-01 +4.700000000000000011e-02 +1.000000000000000021e-03 +1.150000000000000050e-01 +5.299999999999999850e-02 +-4.000000000000000083e-03 +9.800000000000000377e-02 +2.599999999999999881e-02 +-3.799999999999999906e-02 +1.059999999999999970e-01 +-1.150000000000000050e-01 +5.500000000000000028e-02 +5.999999999999999778e-02 +8.799999999999999489e-02 +2.100000000000000130e-02 +-2.800000000000000058e-02 +9.700000000000000289e-02 +-8.300000000000000433e-02 +4.000000000000000083e-02 +1.110000000000000014e-01 +-5.000000000000000104e-03 +1.199999999999999956e-01 diff --git a/docs/source/freq_offset.rst b/docs/source/freq_offset.rst index e17b48d..8312095 100644 --- a/docs/source/freq_offset.rst +++ b/docs/source/freq_offset.rst @@ -20,22 +20,26 @@ visually how each correction step helps in the final constellation plane. .. _fig_cons: .. figure:: /images/cons.png :align: center + :scale: 80% Constellation Points Without Any Correction .. figure:: /images/cons_w_coarse.png :align: center + :scale: 80% Constellation Points With Only Coarse Correction .. figure:: /images/cons_w_coarse_fine.png :align: center + :scale: 80% Constellation Points With both Coarse and Fine Correction .. _fig_cons_full: .. figure:: /images/cons_w_coarse_fine_pilot.png :align: center + :scale: 80% Constellation Points With Coarse, Fine and Pilot Correction @@ -49,7 +53,7 @@ The coarse CFO can be estimated using the short preamble as follows: .. math:: - \alpha_{ST} = \frac{1}{16}\angle(\sum_{i=0}^{N}\overline{S[i]}S[i+16]) + \alpha_{ST} = \frac{1}{16}\angle(\sum_{i=0}^{N-1}\overline{S[i]}S[i+16]) where :math:`\angle(\cdot)` is the phase of complex number and :math:`N \le 144 (160 - 16)` is the subset of short preambles utilized. The intuition is that the @@ -69,5 +73,22 @@ set :math:`N=64`. The ``prod_avg`` in :numref:`fig_sync_short` is fed into a ``moving_avg`` module with window size set to 64. +Fine CFO Correction +------------------- + +A finer estimation of the CFO can be obtained with the help of long training +sequence inside the long preamble. + +The long preamble contains two identify training sequence (64 samples each at 20 +MSPS), the phase offset can be calculated as: + +.. math:: + + \alpha_{LT} = \frac{1}{64}\angle(\sum_{i=0}^{63}\overline{S[i]}S[i+64]) + +This step is omitted in |project| due to the limited resolution of phase +estimation and rotation in the look up table. .. [1] Sourour, Essam, Hussein El-Ghoroury, and Dale McNeill. "Frequency Offset Estimation and Correction in the IEEE 802.11 a WLAN." Vehicular Technology Conference, 2004. VTC2004-Fall. 2004 IEEE 60th. Vol. 7. 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**Module**: :file:`sync_long.v` +- **Input**: ``I (16), Q (16), phase_offset (32), short_gi (1)`` +- **Output**: ``long_preamble_detected (1), fft_re (16), fft_im (16)`` + After detecting the packet, the next step is to determine precisely where each OFDM symbol starts. In 802.11, each OFDM symbol is 4 |us| long. At 20 MSPS sampling rate, this means each OFDM symbol contains 80 samples. The task is to @@ -14,20 +18,20 @@ achieved using the long preamble following the short preamble. 802.11 OFDM Packet Structure (Fig 18-4 in 802.11-2012 Std) As shown in :numref:`fig_training`, the long preamble duration is 8 |us| (160 -samples), and contains two identical long training sequence (LTS), 64 samples each. -The LTS is known and we can use `matched filter -`_ to find it. +samples), and contains two identical long training sequence (LTS), 64 samples +each. The LTS is known and we can use `cross correlation +`_ to find it. -The match *score* at sample :math:`i` can be calculated as follows. +The cross validation *score* at sample :math:`i` can be calculated as follows. .. math:: - :label: eq_matched + :label: eq_cross_corr - Y[i] = \sum_{k=0}^{63}(S[i+k]\overline{H[63-k]}) + Y[i] = \sum_{k=0}^{63}(S[i+k]\overline{H[k]}) where :math:`H` is the 64 sample known LTS in time domain, and can be found in -Table L-6 in :download:`802.11-2012 std ` (index 64 to -127). A numpy readable file of the LTS (64 samples) can be found :download:`here +Table L-6 in :download:`802.11-2012 std ` (index 96 to +159). A numpy readable file of the LTS (64 samples) can be found :download:`here `, and can be read like this: .. code-block:: python @@ -39,7 +43,7 @@ Table L-6 in :download:`802.11-2012 std ` (index 64 to .. figure:: /images/lts.png :align: center - Long Preamble and Matched Filter Result + Long Preamble and Cross Correlation Result To plot :numref:`fig_lts`, load the data file (see :ref:`sec_sample`), then: @@ -53,30 +57,30 @@ To plot :numref:`fig_lts`, load the data file (see :ref:`sec_sample`), then: fig, ax = plt.subplots(nrows=2, ncols=1, sharex=True) ax[0].plot([c.real for c in samples][:500]) # lts is from the above code snippet - ax[1].plot([abs(c) for c in np.convolve(samples, lts, mode='same')][:500], '-ro') + ax[1].plot([abs(c) for c in np.correlate(samples, lts, mode='valid')][:500], '-ro') plt.show() -:numref:`fig_lts` shows the long preamble samples and also the result of matched -filter. We can clearly see two spikes corresponding the two LTS in long +:numref:`fig_lts` shows the long preamble samples and also the result of cross +correlation. We can clearly see two spikes corresponding the two LTS in long preamble. And the spike width is only 1 sample which shows exactly the beginning of each sequence. Suppose the sample index if the first spike is :math:`N`, then -the 160 sample long preamble starts at sample :math:`N-33`. +the 160 sample long preamble starts at sample :math:`N-32`. This all seems nice and dandy, but as it comes to Verilog implementation, we -have to make a few compromises. +have to make a compromise. -First, from :eq:`eq_matched` we can see for each sample, we need to perform 64 +From :eq:`eq_cross_corr` we can see for each sample, we need to perform 64 complex number multiplications, which would consume a lot FPGA resources. -Therefore, we need to reduce the matched filter size. The idea is to only use -a portion instead of all the LTS samples. +Therefore, we need to reduce the size of cross validation. The idea is to only +use a portion instead of all the LTS samples. .. _fig_match_size: .. figure:: /images/match_size.png :align: center - Matched Filter with Various Size (8, 16, 32, 64) + Cross Correlation with Various Size (8, 16, 32, 64) :numref:`fig_match_size` can be plotted as: @@ -86,17 +90,44 @@ a portion instead of all the LTS samples. fig, ax = plt.subplots(nrows=5, ncols=1, sharex=True) ax[0].plot([c.real for c in lp]) - ax[1].plot([abs(c) for c in np.convolve(lp, lts[:8], mode='same')], '-ro') - ax[2].plot([abs(c) for c in np.convolve(lp, lts[:16], mode='same')], '-ro') - ax[3].plot([abs(c) for c in np.convolve(lp, lts[:32], mode='same')], '-ro'); - ax[4].plot([abs(c) for c in np.convolve(lp, lts, mode='same')], '-ro') + ax[1].plot([abs(c) for c in np.correlate(lp, lts[:8], mode='valid')], '-ro') + ax[2].plot([abs(c) for c in np.correlate(lp, lts[:16], mode='valid')], '-ro') + ax[3].plot([abs(c) for c in np.correlate(lp, lts[:32], mode='valid')], '-ro'); + ax[4].plot([abs(c) for c in np.correlate(lp, lts, mode='valid')], '-ro') plt.show() -:numref:`fig_match_size` shows the long preamble (160 samples) as well as -matched filter with different size. It can be seen that using the first 16 -samples of LTS is good enough to exhibit two narrow spikes. Therefore, |project| -use matched filter of size 16 for symbol alignment. And the first sample of the -long preamble starts at :math:`N_{16}-57`, where :math:`N_{16}` is the index of -the first spike when the filter size is 16 (for completeness, it is -:math:`N_{32}-49` when filter size is -32). +:numref:`fig_match_size` shows the long preamble (160 samples) as well as cross +validation with different size. It can be seen that using the first 16 samples +of LTS is good enough to exhibit two narrow spikes. Therefore, |project| use +cross correlation of first 16 samples of LTS for symbol alignment. To confirm, +:numref:`fig_lts_16` shows the cross correlation of the first 16 samples of LTS +on the actual packet. The two spikes are not as obvious as the ones in +:numref:`fig_lts`, but are still clearly visible. + +.. _fig_lts_16: +.. figure:: /images/lts_16.png + :align: center + + Cross Validation using the First 16 Samples of LTS + +To find the two spikes, we keep a record of the max correlation sample for the +first 64 samples (since the first spike is supposed to be at the 32th sample). +Similarly, we also keep a record of the max correlation sample for the second 64 +samples. For further eliminate false positives, we also check if the two spike +sample indexes are :math:`64 \pm 1` apart. + + +FFT +--- + +Now we have located the start of each OFDM symbol, the next task is to perform +FFT on the last 64 data samples inside each symbol. For this we utilize the +`XFFT core +`_ +generated by Xilinx ISE. Depend on if `short guard interval (SGI) +`_ is used, the first 16 or 8 +samples of each OFDM symbol need to be skipped. + +But before performing FFT, we need to first apply the frequency offset +correction (see :ref:`freq_offset`). This is achieved via the ``rotate`` module +(see :ref:`rotate`). diff --git a/docs/source/verilog.rst b/docs/source/verilog.rst index 2ac2282..69e9dd8 100644 --- a/docs/source/verilog.rst +++ b/docs/source/verilog.rst @@ -9,7 +9,9 @@ quantization and look up table. In |project|, these approximations are used. Magnitude Estimation -------------------- -**Module**: :file:`complex_to_mag.v` +- **Module**: :file:`complex_to_mag.v` +- **Input**: ``i (32), q (32)`` +- **Output**: ``mag (32)`` In the ``sync_short`` module, we need to calculate the magnitude of the ``prod_avg``, whose real and imagine part are both 32-bits. To avoid 32-bit @@ -36,10 +38,15 @@ second cycle, ``max`` and ``min`` are determined. In the final cycle, the magnitude is calculated. +.. _sec_phase: + Phase Estimation ---------------- -**Module**:: :file:`phase.v` +- **Module**: :file:`phase.v` +- **Input**: ``i (32), q (32)`` +- **Output**: ``phase (32)`` +- **Note**: The returned phase is scaled up by 512 (i.e., :math:`int(\theta *512)`) When correcting the frequency offset, we need to estimate the phase of a complex number. The *right* way of doing this is probably using the `CORDIC @@ -102,3 +109,43 @@ Refer to `this guide `_ on how to create a look up table in Xilinx ISE. The generated module is stored in :file:`verilog/coregen/atan_lut.v`. + + + +.. _rotate: + +Rotation +-------- + +- **Module**: :file:`/verilog/rotate.v` +- **Input**: ``i (16), q (16), phase (32)`` +- **Output**: ``out_i (16), out_q (16)`` +- **Note**: The input phase is assumed to be scaled up by 512. + +To rotate a complex number :math:`C=I+jQ` by :math:`\theta` degree, we can +multiply it by :math:`e^{j\theta}`, as shown in :eq:`eq_rot`. + +.. math:: + :label: eq_rot + + C' = (I+jQ)\times(\cos(\theta)+j\sin(\theta)) + +Again, this can be done using the CORDIC algorithm. But similar to +:ref:`sec_phase`, we use the look up table. + + +.. _fig_quadrant: +.. figure:: /images/quadrant.png + :align: center + :scale: 60% + + Quadrant in I/Q Plane + +As shown in :numref:`fig_quadrant`, we split the I/Q plane into 8 quadrants, +:math:`\pi/4` each. To avoid storing nearly duplicate entries in the table, we +first map the phase to be rotated (:math:`[-\pi, \pi]`) into the :math:`[0, +\pi/4]` range. Next, since the incoming phase is scaled up by 512, each quadrant +is further split into :math:`402=int(\pi/4*512)` sectors. And the +:math:`\cos(\theta)` and :math:`\sin(\theta)` values (scaled up by 2048) are +stored in the look up table. The table is generated by the +:file:`scripts/gen_rot_lut.py`.