# Physical Layer¶

## 4FSK generation¶

M17 standard uses 4FSK modulation running at 4800 symbols/s (9600 bits/s) with a deviation index h=0.33 for transmission in 9 kHz channel bandwidth. Channel spacing is 12.5 kHz. The symbol stream is converted to a series of impulses which pass through a root-raised-cosine (alpha=0.5) shaping filter before frequency modulation at the transmitter and again after frequency demodulation at the receiver.

The bit-to-symbol mapping is shown in the table below.

Table 1 Dibit symbol mapping to 4FSK deviation
Information bits Symbol 4FSK deviation
Bit 1 Bit 0
0 1 +3 +2.4 kHz
0 0 +1 +0.8 kHz
1 0 -1 -0.8 kHz
1 1 -3 -2.4 kHz

The most significant bits are sent first, meaning that the byte 0xB4 (= 0b10’11’01’00) in type 4 bits (see Bit types) would be sent as the symbols -1 -3 +3 +1.

## Preamble¶

Every transmission starts with a preamble, which shall consist of at least 40 ms of alternating -3, +3… symbols. This is equivalent to 40 milliseconds of a 2400 Hz tone

## Bit types¶

The bits at different stages of the error correction coding are referred to with bit types, given in Table 2.

 Type 1 Data link layer bits Type 2 Bits after appropriate encoding Type 3 Bits after puncturing (only for convolutionally coded data, for other ECC schemes type 3 bits are the same as type 2 bits) Type 4 Decorrelated and interleaved (re-ordered) type 3 bits

Type 4 bits are used for transmission over the RF. Incoming type 4 bits shall be decoded to type 1 bits, which are then used to extract all the frame fields.

## Error correction coding schemes and bit type conversion¶

Two distinct ECC/FEC schemes are used for different parts of the transmission.

### Subsequent frames¶

A 48-bit (type 1) chunk of the LSF is partitioned into 4 12-bit parts and encoded using Golay (24, 12) code. This produces 96 encoded bits of type 2. These bits are used in the Link Information Channel (LICH).

16-bit FN and 128 bits of payload (144 bits total) are convolutionally encoded in a manner analogous to that of the link setup frame. A total of 148 bits is being encoded resulting in 296 type 2 bits. These bits are punctured to generate 272 type 3 bits.

96 type 2 bits of LICH are concatenated with 272 type 3 bits and re-ordered to form type 4 bits for transmission. This, along with 16-bit sync in the beginning of frame, gives a total of 384 bits

The LICH chunks allow for late listening and indepedent decoding to check destination address. The goal is to require less complexity to decode just the LICH and check if the full message should be decoded.

### Golay (24,12)¶

The Golay (24,12) encoder uses the polynomial 0xC75 to generate the 11 check bits. The check bits and an overall parity bit are appended to the 12 bit data, resulting in a 24 bit encoded chunk.

\begin{align} G =& x^{11} + x^{10} + x^6 + x^5 + x^4 + x^2 + 1 \end{align}

The output of the Golay encoder is shown in the table below.

 Field Data Check bits Parity Position 23..12 11..1 0 Length 12 11 1

Four of these 24-bit blocks are used to reconstruct the LSF.

### Convolutional encoder¶

 [ECC] Moreira, Jorge C.; Farrell, Patrick G. “Essentials of Error‐Control Coding” Wiley 2006, ISBN: 9780470029206

The convolutional code shall encode the input bit sequence after appending 4 tail bits at the end of the sequence. Rate of the coder is R=½ with constraint length K=5 [NXDN]. The encoder diagram and generating polynomials are shown below

\begin{align} G_1(D) =& 1 + D^3 + D^4 \\ G_2(D) =& 1+ D + D^2 + D^4 \end{align}

The output from the encoder must be read alternately.

 [NXDN] NXDN Technical Specifications, Part 1: Air Interface; Sub-part A: Common Air Interface

### Code puncturing¶

Removing some of the bits from the convolutional coder’s output is called code puncturing. The nominal coding rate of the encoder used in M17 is ½. This means the encoder outputs two bits for every bit of the input data stream. To get other (higher) coding rates, a puncturing scheme has to be used.

Two different puncturing schemes are used in M17 stream mode:

1. $$P_1$$ leaving 46 from 61 encoded bits
2. $$P_2$$ leaving 11 from 12 encoded bits

Scheme $$P_1$$ is used for the initial LICH link setup info, taking 488 bits of encoded data and selecting 368 bits. The $$gcd(368, 488)$$ is 8 which, when used to divide, leaves 46 and 61. A full puncture pattern requires the output be divisible by the number of encoding polynomials. For this case the full puncture matrix should have 122 entries with 92 of them being 1.

Scheme $$P_2$$ is for frames (excluding LICH chunks, which are coded differently). This takes 296 encoded bits and selects 272 of them. Every 12th bit is being punctured out, leaving 272 bits. The full matrix shall have 12 entries with 11 being 1.

The matrix $$P_1$$ can be represented more concisely by duplicating a smaller matrix with a flattening.

\begin{align} S_{} = & \begin{bmatrix} a & \vec{r_1} & c \\ b & \vec{r_2} & X \end{bmatrix} \\ S_{full} = & \begin{bmatrix} a & \vec{r_1} & c & b & \vec{r_2} \\ b & \vec{r_2} & a & \vec{r_1} & c \end{bmatrix} \end{align}

The puncturing schemes are defined by their partial puncturing matrices:

.. only:: latex \setcounter{MaxMatrixCols}{32} \begin{align} P_1 = & \begin{bmatrix} 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & X \end{bmatrix} \\ P_2 = & \begin{bmatrix} 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 0 \end{bmatrix} \end{align}

The complete linearized representations are:

Listing 1 linearized puncture patterns
P1 = [1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0,
1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0,
1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1,
0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1,
0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1,
1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1]

P2 = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0]


### Interleaving¶

For interleaving a Quadratic Permutation Polynomial (QPP) is used. The polynomial $$\pi(x)=(45x+92x^2)\mod 368$$ is used for a 368 bit interleaving pattern [QPP]. See appendix Table 24 for pattern.

 [QPP] Trifina, Lucian, Daniela Tarniceriu, and Valeriu Munteanu. “Improved QPP Interleavers for LTE Standard.” ISSCS 2011 - International Symposium on Signals, Circuits and Systems (2011): n. pag. Crossref. Web. https://arxiv.org/abs/1103.3794

### Data decorrelator¶

To avoid transmitting long sequences of constant symbols (e.g. 010101…), a simple algorithm is used. All 46 bytes of type 4 bits shall be XORed with a pseudorandom, predefined stream. The same algorithm has to be used for incoming bits at the receiver to get the original data stream. See Table 23 for sequence.