doc

docs/source/decode.rst

@@ -0,0 +1,104 @@
+Decoding
+========
+
+Now we have corrected the residual CFO and also have corrected the channel gain,
+the next step is to map the FFT output to actual data bits. This is the reverse
+process of encoding a packet.
+
+1. demodulation: complex number to bits
+#. deinterleaving: shuffle the bits inside each OFDM symbol
+#. Convolution decoding: remove redundancy and correct potential bit errors
+#. Descramble.
+
+Step 1 and 3 depend on the modulation and coding scheme, which can be obtained
+from the SIGNAL field. The SIGNAL field is encoded in the first OFDM symbol
+after the long preamble and is always BPSK modulated regardless of the actual
+modulation. Recall that in
+802.11a/g, one OFDM symbol contains 48 data sub-carriers, which corresponds to
+48 data bits in BPSK scheme. The SIGNAL field is also convolutional encoded at
+1/2 rate so there are 24 actual data bits in the SIGNAL field.
+
+Next, we first go through the decoding process and then explain the format of
+both legacy (802.11a/g) and the HT (802.11n) SIGNAL format.
+
+Demodulation
+------------
+
+- **Module**: :file:`demodulate.v`
+- **Input**: ``rate (7), cons_i (16), cons_q (16)``
+- **Output**: ``bits (6)``
+
+This step maps the complex number in the FFT plane into bits. :numref:`fig_mod`
+shows the constellation encoding schemes for BPSK, QPSK, 16-QAM and 64-QAM.
+also supported in |project|.
+
+.. _fig_mod:
+.. figure:: /images/mod.png
+    :align: center
+    :scale: 80%
+
+    BPSK, QPSK, 16-QAM and 64-QAM Constellation Bit Encoding
+
+Inside each OFDM symbol, each sub-carrier is mapped into 1, 2, 4 or 6 bits
+depending on the modulation.
+
+Deinterleaving
+--------------
+
+Inside each OFDM symbol, the encoded bits are interleaved to map adjacent bits
+into non-adjacent sub-carriers and also alternatively into less or more
+significant bits in the constellation bits.
+
+To understand how the block interleaver works, first we need to define a few
+parameters. Here we only consider 802.11a/g and 802.11n single spatial stream
+mode.
+
+.. table:: Modulation Dependent Parameters (802.11a/g)
+    :align: center
+
+    +------------+-------------+----------+------------------+------------------+------------------+
+    | Modulation | Coding Rate | Bit-Rate | :math:`N_{BPSC}` | :math:`N_{CBPS}` | :math:`N_{DBPS}` |
+    +------------+-------------+----------+------------------+------------------+------------------+
+    | BPSK       | 1/2         | 6        | 1                | 48               | 24               |
+    +------------+-------------+----------+------------------+------------------+------------------+
+    | BPSK       | 3/4         | 9        | 1                | 48               | 36               |
+    +------------+-------------+----------+------------------+------------------+------------------+
+    | QPSK       | 1/2         | 12       | 2                | 96               | 48               |
+    +------------+-------------+----------+------------------+------------------+------------------+
+    | QPSK       | 3/4         | 18       | 2                | 96               | 72               |
+    +------------+-------------+----------+------------------+------------------+------------------+
+    | 16-QAM     | 1/2         | 24       | 4                | 192              | 96               |
+    +------------+-------------+----------+------------------+------------------+------------------+
+    | 16-QAM     | 3/4         | 36       | 4                | 192              | 144              |
+    +------------+-------------+----------+------------------+------------------+------------------+
+    | 64-QAM     | 2/3         | 48       | 6                | 288              | 192              |
+    +------------+-------------+----------+------------------+------------------+------------------+
+    | 64-QAM     | 3/4         | 54       | 6                | 288              | 216              |
+    +------------+-------------+----------+------------------+------------------+------------------+
+
+where:
+
+- :math:`N_{BPSC}`: number of bits per sub-carrier
+- :math:`N_{CBPS}`: number of coded bits per OFDM symbol
+- :math:`N_{DBPS}`: number of data bits per OFDM symbol
+
+Let :math:`s=max(N_{BPSC}/2,
+
+The interleaving process involves two permutations. Accordingly, the
+de-interleaving process also contains two permutation to reverse the
+interleaving.
+
+The first permutation of de-interleaving is:
+
+.. math::
+
+    i = s\times\lfloor\frac{j}{s}\rfloor + (j+\lfloor16\times\frac{j}{N_{CBPS}}\rfloor)\%s, j\in[0, N_{CBPS}-1]
+
+where :math:`s=max(N_{BPSC}/2, 1)`
+
+The second permutation is defined as:
+
+.. math::
+
+    k = 16\times i - (N_{CBPS}-1)\times\lfloor\frac{16\times i}{N_{CBPS}}\rfloor
+

docs/source/eq.rst

@@ -31,6 +31,9 @@ Each sub-carrier carries I/Q modulated information, corresponding to the output
 of 64 point FFT from :file:`sync_long.v` module. 
 
 
+Sub-Carrier Equalization
+------------------------
+
 .. _fig_lts_fft:
 .. figure:: /images/lts_fft.png
     :align: center
@@ -72,20 +75,141 @@ 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]
+    [-26, 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, -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, 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\}
+    &-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]}
+    Y[i] = \frac{X[i]}{H[i]}, i \in [-26, 26]
 
+where :math:`X[i]` is the FFT output at sub-carrier :math:`i`.
+
+
+.. _fig_raw_fft:
+.. figure:: /images/raw_fft.png
+    :align: center
+    :scale: 80%
+
+    FFT Without Normalization
+
+.. _fig_norm_fft:
+.. figure:: /images/norm_fft.png
+    :align: center
+    :scale: 80%
+
+    FFT With Normalization
+
+:numref:`fig_raw_fft` and :numref:`fig_norm_fft` shows the FFT before and after
+normalization using channel gain.
+
+
+Residual Frequency Offset Correction
+------------------------------------
+
+We can see from :numref:`fig_norm_fft` that the FFT output is tilted slightly.
+This is caused by residual frequency offset that was not compensated during the
+coarse CFO correction step.
+
+This residual CFO can be corrected either by :ref:`sec_fine_cfo`, or/and by the
+pilot sub-carriers. Ideally we want to do both, but since the fine CFO is
+usually beyond the resolution of the phase look up table, we skip it in the
+:file:`sync_long.v` module and only rely on the pilot sub-carriers.
+
+Regardless of the data sub-carrier modulation, the four pilot sub-carriers (-21,
+-7, 7, 21) always contains BPSK modulated pseudo-random binary sequence.
+
+
+The polarity of the pilot sub-carriers varies symbol to symbol. For 802.11a/g,
+the pilot pattern is:
+
+.. math::
+
+    p_{0,\ldots,126} = \{
+    &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, 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, 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,-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, 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 pilot sub-carriers at OFDM symbol :math:`n` (starting at 0 from the first
+symbol after the long preamble) is then:
+
+.. math::
+
+    P^{(n)}_{-21, -7, 7, 21} = \{p_{n\%127}, p_{n\%127}, p_{n\%127}, -p_{n\%127}\}
+
+
+For 802.11n at 20MHz bandwidth with single spatial stream, the n'th pilot
+sub-carriers are:
+
+.. math::
+    P^{(n)}_{-21, -7, 7, 21} = \{\Psi_{n\%4}, \Psi_{(n+1)\%4}, \Psi_{(n+2)\%4},
+    \Psi_{(n+3)\%4}\}
+
+And:
+
+.. math::
+    \Psi_{0, 1, 2, 3} = \{1, 1, 1, -1\}
+
+
+In other words, the pilot sub-carries of the first few symbols are:
+
+
+.. math::
+
+    P^{(0)}_{-21, -7, 7, 21} = \{1, 1, 1, -1\}\\
+    P^{(1)}_{-21, -7, 7, 21} = \{1, 1, -1, 1\}\\
+    P^{(2)}_{-21, -7, 7, 21} = \{1, -1, 1, 1\}\\
+    P^{(3)}_{-21, -7, 7, 21} = \{-1, 1, 1, 1\}\\
+    P^{(4)}_{-21, -7, 7, 21} = \{1, 1, 1, -1\}\\
+    \cdots
+
+For other configurations (e.g., spatial stream, bandwidth), the pilot
+sub-carrier pattern can be found in Section 20.3.11.10 in
+:download:`802.11-2012 std <./files/802.11-2012.pdf>`.
+
+
+The residual phase offset at symbol :math:`n` can then be estimated as:
+
+.. math::
+
+    \theta_n = \angle(\sum_{i\in\{-21, -7, 7, 21\}}\overline{X^{(n)}[i]}\times P^{(n)}[i]\times H[i])
+
+
+Combine this phase offset and the previous channel gain correction together, the
+adjustment to symbol :math:`n` is:
+
+.. math::
+
+    Y^{(n)}[i] = \frac{X^{(n)}[i]}{H[i]}e^{j\theta_n}
+
+
+.. _fig_pilot_fft:
+.. figure:: /images/pilot_fft.png
+    :align: center
+    :scale: 80%
+
+    Residual CFO Correction Using Pilot Sub-Carriers
+
+:numref:`fig_pilot_fft` shows the effect of correcting the residual CFO using
+pilot sub-carriers. Each sub-carrier can then be mapped to constellation points
+easily.
+
+In |project|, the above tasks are implemented by the :file:`equalizer.v` module.
+It first stores the first LTS, and then calculates the mean of the two LTS and
+store it as channel gain.
+
+For each incoming OFDM symbol, it first obtains the polarity of the pilot
+sub-carriers in current symbol, then calculates the residual CFO using the pilot
+sub-carriers and also performs the channel gain correction.

docs/source/freq_offset.rst

@@ -44,7 +44,7 @@ visually how each correction step helps in the final constellation plane.
     Constellation Points With Coarse, Fine and Pilot Correction
 
 :numref:`fig_cons` to :numref:`fig_cons_full` shows the constellation points of
-a 64-QAM modulated 802.11a packet.
+a 16-QAM modulated 802.11a packet.
 
 Coarse CFO Correction
 ---------------------
@@ -73,6 +73,8 @@ 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.
 
 
+.. _sec_fine_cfo:
+
 Fine CFO Correction
 -------------------
 

docs/source/index.rst

@@ -3,8 +3,8 @@
    You can adapt this file completely to your liking, but it should at least
    contain the root `toctree` directive.
 
-Welcome to |project|'s documentation!
+|project|: Verilog Implementation of 802.11 OFDM Decoder
-=====================================
+========================================================
 
 .. toctree::
    :maxdepth: 2
@@ -15,6 +15,7 @@ Welcome to |project|'s documentation!
    freq_offset
    sync_long
    eq
+   decode
    setting
    verilog