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.