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## Analytical Expressions Used in `bercoding` Function and Bit Error Rate Analysis App

This section covers the main analytical expressions used in the `bercoding` function and the Bit Error Rate Analysis app.

### Common Notation

This table describes the additional notations used in analytical expressions in this section.

DescriptionNotation
Energy-per-information bit-to-noise power-spectral-density ratio

`${\gamma }_{b}=\frac{{E}_{b}}{{N}_{0}}$`

Message length

$K$

Code length

$N$

Code rate

`${R}_{c}=\frac{K}{N}$`

### Block Coding

This section describes the specific notation for block coding expressions, where ${d}_{\mathrm{min}}$ is the minimum distance of the code.

Soft Decision

For BPSK, QPSK, OQPSK, 2-PAM, 4-QAM, and precoded MSK, equation 8.1-52 in [1]) applies,

`${P}_{b}\le \frac{1}{2}\left({2}^{K}-1\right)Q\left(\sqrt{2{\gamma }_{b}{R}_{c}{d}_{\mathrm{min}}}\right)$`

For DE-BPSK, DE-QPSK, DE-OQPSK, and DE-MSK,

`${P}_{b}\le \frac{1}{2}\left({2}^{K}-1\right)\left[2Q\left(\sqrt{2{\gamma }_{b}{R}_{c}{d}_{\mathrm{min}}}\right)\left[1-Q\left(\sqrt{2{\gamma }_{b}{R}_{c}{d}_{\mathrm{min}}}\right)\right]\right]$`

For BFSK coherent detection, equations 8.1-50 and 8.1-58 in [1] apply,

`${P}_{b}\le \frac{1}{2}\left({2}^{K}-1\right)Q\left(\sqrt{{\gamma }_{b}{R}_{c}{d}_{\mathrm{min}}}\right)$`

For BFSK noncoherent square-law detection, equations 8.1-65 and 8.1-64 in [1] apply,

`${P}_{b}\le \frac{1}{2}\frac{{2}^{K}-1}{{2}^{2{d}_{\mathrm{min}}-1}}\mathrm{exp}\left(-\frac{1}{2}{\gamma }_{b}{R}_{c}{d}_{\mathrm{min}}\right)\sum _{i=0}^{{d}_{\mathrm{min}}-1}{\left(\frac{1}{2}{\gamma }_{b}{R}_{c}{d}_{\mathrm{min}}\right)}^{i}\frac{1}{i!}\sum _{r=0}^{{d}_{\mathrm{min}}-1-i}\left(\begin{array}{c}2{d}_{\mathrm{min}}-1\\ r\end{array}\right)$`

For DPSK,

`${P}_{b}\le \frac{1}{2}\frac{{2}^{K}-1}{{2}^{2{d}_{\mathrm{min}}-1}}\mathrm{exp}\left(-{\gamma }_{b}{R}_{c}{d}_{\mathrm{min}}\right)\sum _{i=0}^{{d}_{\mathrm{min}}-1}{\left({\gamma }_{b}{R}_{c}{d}_{\mathrm{min}}\right)}^{i}\frac{1}{i!}\sum _{r=0}^{{d}_{\mathrm{min}}-1-i}\left(\begin{array}{c}2{d}_{\mathrm{min}}-1\\ r\end{array}\right)$`

Hard Decision

For general linear block code, equations 4.3 and 4.4 in [9], and 12.136 in [6] apply,

`$\begin{array}{l}{P}_{b}\le \frac{1}{N}\sum _{m=t+1}^{N}\left(m+t\right)\left(\begin{array}{c}N\\ m\end{array}\right){p}^{m}{\left(1-p\right)}^{N-m}\\ t=⌊\frac{1}{2}\left({d}_{\mathrm{min}}-1\right)⌋\end{array}$`

For Hamming code, equations 4.11 and 4.12 in [9] and 6.72 and 6.73 in [7] apply

`${P}_{b}\approx \frac{1}{N}\sum _{m=2}^{N}m\left(\begin{array}{c}N\\ m\end{array}\right){p}^{m}{\left(1-p\right)}^{N-m}=p-p{\left(1-p\right)}^{N-1}$`

For rate (24,12) extended Golay code, equations 4.17 in [9] and 12.139 in [6] apply:

`${P}_{b}\le \frac{1}{24}\sum _{m=4}^{24}{\beta }_{m}\left(\begin{array}{c}24\\ m\end{array}\right){p}^{m}{\left(1-p\right)}^{24-m}$`

where ${\beta }_{m}$ is the average number of channel symbol errors that remain in corrected N-tuple format when the channel caused m symbol errors (see table 4.2 in [9]).

For Reed-Solomon code with $N=Q-1={2}^{q}-1$,

`${P}_{b}\approx \frac{{2}^{q-1}}{{2}^{q}-1}\frac{1}{N}\sum _{m=t+1}^{N}m\left(\begin{array}{c}N\\ m\end{array}\right){\left({P}_{s}\right)}^{m}{\left(1-{P}_{s}\right)}^{N-m}$`

For FSK, equations 4.25 and 4.27 in [9], 8.1-115 and 8.1-116 in [1], 8.7 and 8.8 in [7], and 12.142 and 12.143 in [6] apply,

`${P}_{b}\approx \frac{1}{q}\frac{1}{N}\sum _{m=t+1}^{N}m\left(\begin{array}{c}N\\ m\end{array}\right){\left({P}_{s}\right)}^{m}{\left(1-{P}_{s}\right)}^{N-m}$`

otherwise, if ${\mathrm{log}}_{2}Q/{\mathrm{log}}_{2}M=q/k=h$, where h is an integer (equation 1 in [10]) applies,

`${P}_{s}=1-{\left(1-s\right)}^{h}$`

where s is the SER in an uncoded AWGN channel.

For example, for BPSK, $M=2$ and ${P}_{s}=1-{\left(1-s\right)}^{q}$, otherwise ${P}_{s}$ is given by table 1 and equation 2 in [10].

### Convolutional Coding

This section describes the specific notation for convolutional coding expressions, where ${d}_{free}$ is the free distance of the code, and ${a}_{d}$ is the number of paths of distance d from the all-zero path that merges with the all-zero path for the first time.

Soft Decision

From equations 8.2-26, 8.2-24, and 8.2-25 in [1] and 13.28 and 13.27 in [6] apply,

`${P}_{b}<\sum _{d={d}_{free}}^{\infty }{a}_{d}f\left(d\right){P}_{2}\left(d\right)$`

The transfer function is given by

`$\begin{array}{l}T\left(D,N\right)=\sum _{d={d}_{free}}^{\infty }{a}_{d}{D}^{d}{N}^{f\left(d\right)}\\ {\frac{dT\left(D,N\right)}{dN}|}_{N=1}=\sum _{d={d}_{free}}^{\infty }{a}_{d}f\left(d\right){D}^{d}\end{array}$`

where $f\left(d\right)$ is the exponent of N as a function of d.

This equation gives the results for BPSK, QPSK, OQPSK, 2-PAM, 4-QAM, precoded MSK, DE-BPSK, DE-QPSK, DE-OQPSK, DE-MSK, DPSK, and BFSK:

`${P}_{2}\left(d\right)={{P}_{b}|}_{\frac{{E}_{b}}{{N}_{0}}={\gamma }_{b}{R}_{c}d}$`

where ${P}_{b}$ is the BER in the corresponding uncoded AWGN channel. For example, for BPSK (equation 8.2-20 in [1]),

`${P}_{2}\left(d\right)=Q\left(\sqrt{2{\gamma }_{b}{R}_{c}d}\right)$`

Hard Decision

From equations 8.2-33, 8.2-28, and 8.2-29 in [1] and 13.28, 13.24, and 13.25 in [6] apply,

`${P}_{b}<\sum _{d={d}_{free}}^{\infty }{a}_{d}f\left(d\right){P}_{2}\left(d\right)$`

When d is odd,

`${P}_{2}\left(d\right)=\sum _{k=\left(d+1\right)/2}^{d}\left(\begin{array}{c}d\\ k\end{array}\right){p}^{k}{\left(1-p\right)}^{d-k}$`

and when d is even,

`${P}_{2}\left(d\right)=\sum _{k=d/2+1}^{d}\left(\begin{array}{c}d\\ k\end{array}\right){p}^{k}{\left(1-p\right)}^{d-k}+\frac{1}{2}\left(\begin{array}{c}d\\ d/2\end{array}\right){p}^{d/2}{\left(1-p\right)}^{d/2}$`

where p is the bit error rate (BER) in an uncoded AWGN channel.