Analytical Expressions Used in berawgn Function and Bit Error Rate Analysis App

These sections cover the main analytical expressions used in the berawgn function and Bit Error Rate Analysis app.

M-PSK

From equation 8.22 in [2],

$$P_s=\frac{1}{\pi }{\displaystyle \underset{0}{\overset{(M-1)\pi /M}{\int }}\exp \left(-\frac{k{E}_b}{N_0}\frac{{\sin }^2\left[\pi /M\right]}{{\sin }^2\theta }\right)d\theta }$$

This expression is similar, but not strictly equal, to the exact BER (from [4] and equation 8.29 from [2]):

$$P_b=\frac{1}{k}\left({\displaystyle \sum _{i=1}^{M/2}(w_i^{\text{'}})P_i}\right)$$

where $$w_i^{\text{'}}=w_i+w_{M-i}$$, $$w_{M/2}^{\text{'}}=w_{M/2}$$, $$w_i$$ is the Hamming weight of bits assigned to symbol i,

$$\begin{array}{c}{P}_i=\frac{1}{2\pi }{\displaystyle \underset{0}{\overset{\pi (1-(2i-1)/M)}{\int }}\exp \left(-\frac{k{E}_b}{N_0}\frac{{\sin }^2\left[(2i-1)\pi /M\right]}{{\sin }^2\theta }\right)d\theta }\\ -\frac{1}{2\pi }{\displaystyle \underset{0}{\overset{\pi (1-(2i+1)/M)}{\int }}\exp \left(-\frac{k{E}_b}{N_0}\frac{{\sin }^2\left[(2i+1)\pi /M\right]}{{\sin }^2\theta }\right)d\theta }\end{array}$$

For M-PSK with M = 2, specifically BPSK, this equation 5.2-57 from [1] applies:

$$P_s=P_b=Q\left(\sqrt{\frac{2E_b}{N_0}}\right)$$

For M-PSK with M = 4, specifically QPSK, these equations 5.2-59 and 5.2-62 from [1] apply:

$$\begin{array}{c}{P}_s=2Q\left(\sqrt{\frac{2E_b}{N_0}}\right)\left[1-\frac{1}{2}Q\left(\sqrt{\frac{2E_b}{N_0}}\right)\right]\\ P_b=Q\left(\sqrt{\frac{2E_b}{N_0}}\right)\end{array}$$

DE-M-PSK

For DE-M-PSK with M = 2, specifically DE-BPSK, this equation 8.36 from [2] applies:

$$P_s=P_b=2Q\left(\sqrt{\frac{2E_b}{N_0}}\right)-2Q^2\left(\sqrt{\frac{2E_b}{N_0}}\right)$$

For DE-M-PSK with M = 4, specifically DE-QPSK, this equation 8.38 from [2] applies:

$$P_s=4Q\left(\sqrt{\frac{2E_b}{N_0}}\right)-8Q^2\left(\sqrt{\frac{2E_b}{N_0}}\right)+8Q^3\left(\sqrt{\frac{2E_b}{N_0}}\right)-4Q^4\left(\sqrt{\frac{2E_b}{N_0}}\right)$$

From equation 5 in [3],

$$P_b=2Q\left(\sqrt{\frac{2E_b}{N_0}}\right)\left[1-Q\left(\sqrt{\frac{2E_b}{N_0}}\right)\right]$$

OQPSK

For OQPSK, use the same BER and SER computations as for QPSK in [2].

DE-OQPSK

For DE-OQPSK, use the same BER and SER computations as for DE-QPSK in [3].

M-DPSK

For M-DPSK, this equation 8.84 from [2] applies:

$$P_s=\frac{\sin (\pi /M)}{2\pi }{\displaystyle \underset{-\pi /2}{\overset{\pi /2}{\int }}\frac{\exp \left(-(k{E}_b/N_0)(1-\cos (\pi /M)\cos \theta )\right)}{1-\cos (\pi /M)\cos \theta }d\theta }$$

This expression is similar, but not strictly equal, to the exact BER (from [4]):

$$P_b=\frac{1}{k}\left({\displaystyle \sum _{i=1}^{M/2}(w_i^{\text{'}})A_i}\right)$$

where $$w_i^{\text{'}}=w_i+w_{M-i}$$, $$w_{M/2}^{\text{'}}=w_{M/2}$$, $$w_i$$ is the Hamming weight of bits assigned to symbol i,

$$\begin{array}{l}{A}_i=F\left(\left(2i+1\right)\frac{\pi }{M}\right)-F\left(\left(2i-1\right)\frac{\pi }{M}\right)\\ F(\psi )=-\frac{\sin \psi }{4\pi }{\displaystyle \underset{-\pi /2}{\overset{\pi /2}{\int }}\frac{\exp \left(-k{E}_b/N_0(1-\cos \psi \cos t)\right)}{1-\cos \psi \cos t}dt}\end{array}$$

For M-DPSK with M = 2, this equation 8.85 from [2] applies:

$$P_b=\frac{1}{2}\exp \left(-\frac{E_b}{N_0}\right)$$

M-PAM

From equations 8.3 and 8.7 in [2] and equation 5.2-46 in [1],

$$P_s=2\left(\frac{M-1}{M}\right)Q\left(\sqrt{\frac{6}{M^2-1}\frac{k{E}_b}{N_0}}\right)$$

From [5],

$$\begin{array}{c}{P}_b=\frac{2}{M{\log }_{2}M}\times \\ {\displaystyle \sum _{k=1}^{{\log }_{2}M}{\displaystyle \sum _{i=0}^{(1-2^{-k})M-1}\left\{{(-1)}^{\lfloor \frac{i{2}^{k-1}}{M}\rfloor }\left(2^{k-1}-\lfloor \frac{i{2}^{k-1}}{M}+\frac{1}{2}\rfloor \right)Q\left((2i+1)\sqrt{\frac{6{\log }_{2}M}{M^2-1}\frac{E_b}{N_0}}\right)\right\}}}\end{array}$$

M-QAM

For square M-QAM, $$k={\log }_{2}M$$ is even, so equation 8.10 from [2] and equations 5.2-78 and 5.2-79 from [1] apply:

$$P_s=4\frac{\sqrt{M}-1}{\sqrt{M}}Q\left(\sqrt{\frac{3}{M-1}\frac{k{E}_b}{N_0}}\right)-4{\left(\frac{\sqrt{M}-1}{\sqrt{M}}\right)}^2{Q}^2\left(\sqrt{\frac{3}{M-1}\frac{k{E}_b}{N_0}}\right)$$

From [5],

$$\begin{array}{c}{P}_b=\frac{2}{\sqrt{M}{\log }_2\sqrt{M}}\\ \times {\displaystyle \sum _{k=1}^{{\log }_2\sqrt{M}}{\displaystyle \sum _{i=0}^{(1-2^{-k})\sqrt{M}-1}\left\{{(-1)}^{\lfloor \frac{i{2}^{k-1}}{\sqrt{M}}\rfloor }\left(2^{k-1}-\lfloor \frac{i{2}^{k-1}}{\sqrt{M}}+\frac{1}{2}\rfloor \right)Q\left((2i+1)\sqrt{\frac{6{\log }_{2}M}{2(M-1)}\frac{E_b}{N_0}}\right)\right\}}}\end{array}$$

For rectangular (non-square) M-QAM, $$k={\log }_{2}M$$ is odd, $$M=I\times J$$, $$I=2^{\frac{k-1}{2}}$$, and $$J=2^{\frac{k+1}{2}}$$. So that,

$$\begin{array}{c}{P}_s=\frac{4IJ-2I-2J}{M}\\ \times Q\left(\sqrt{\frac{6{\log }_2(IJ)}{(I^2+J^2-2)}\frac{E_b}{N_0}}\right)-\frac{4}{M}(1+IJ-I-J)Q^2\left(\sqrt{\frac{6{\log }_2(IJ)}{(I^2+J^2-2)}\frac{E_b}{N_0}}\right)\end{array}$$

From [5],

$$P_b=\frac{1}{{\log }_2(IJ)}\left({\displaystyle \sum _{k=1}^{{\log }_{2}I}{P}_I(k)}+{\displaystyle \sum _{l=1}^{{\log }_{2}J}{P}_J(l)}\right)$$

where

$$P_I(k)=\frac{2}{I}{\displaystyle \sum _{i=0}^{(1-2^{-k})I-1}\left\{{(-1)}^{\lfloor \frac{i{2}^{k-1}}{I}\rfloor }\left(2^{k-1}-\lfloor \frac{i{2}^{k-1}}{I}+\frac{1}{2}\rfloor \right)Q\left((2i+1)\sqrt{\frac{6{\log }_2(IJ)}{I^2+J^2-2}\frac{E_b}{N_0}}\right)\right\}}$$

and

$$P_J(k)=\frac{2}{J}{\displaystyle \sum _{j=0}^{(1-2^{-l})J-1}\left\{{(-1)}^{\lfloor \frac{j{2}^{l-1}}{J}\rfloor }\left(2^{l-1}-\lfloor \frac{j{2}^{l-1}}{J}+\frac{1}{2}\rfloor \right)Q\left((2j+1)\sqrt{\frac{6{\log }_2(IJ)}{I^2+J^2-2}\frac{E_b}{N_0}}\right)\right\}}$$

Orthogonal M-FSK with Coherent Detection

From equation 8.40 in [2] and equation 5.2-21 in [1],

$$\begin{array}{l}{P}_s=1-{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\left[Q\left(-q-\sqrt{\frac{2k{E}_b}{N_0}}\right)\right]}^{M-1}\frac{1}{\sqrt{2\pi }}\exp \left(-\frac{q^2}{2}\right)dq}\\ P_b=\frac{2^{k-1}}{2^k-1}{P}_s\end{array}$$

Nonorthogonal 2-FSK with Coherent Detection

For $$M=2$$, equation 5.2-21 in [1] and equation 8.44 in [2] apply:

$$P_s=P_b=Q\left(\sqrt{\frac{E_b(1-\mathrm{Re}\left[\rho \right])}{N_0}}\right)$$

$$\rho$$ is the complex correlation coefficient, such that:

$$\rho =\frac{1}{2E_b}{\displaystyle \underset{0}{\overset{T_b}{\int }}{\tilde{s}}_1(t){\tilde{s}}_2^{*}(t)dt}$$

where $${\tilde{s}}_1(t)$$ and $${\tilde{s}}_2(t)$$ are complex lowpass signals, and

$$E_b=\frac{1}{2}{\displaystyle \underset{0}{\overset{T_b}{\int }}{\left|{\tilde{s}}_1(t)\right|}^{2}dt}=\frac{1}{2}{\displaystyle \underset{0}{\overset{T_b}{\int }}{\left|{\tilde{s}}_2(t)\right|}^{2}dt}$$

For example, with

$${\tilde{s}}_1(t)=\sqrt{\frac{2E_b}{T_b}}{e}^{j2\pi f_{1}t},{\tilde{s}}_2(t)=\sqrt{\frac{2E_b}{T_b}}{e}^{j2\pi f_{2}t}$$

then

$$\begin{array}{c}\rho =\frac{1}{2E_b}{\displaystyle \underset{0}{\overset{T_b}{\int }}\sqrt{\frac{2E_b}{T_b}}{e}^{j2\pi f_{1}t}\sqrt{\frac{2E_b}{T_b}}{e}^{-j2\pi f_{2}t}dt}=\frac{1}{T_b}{\displaystyle \underset{0}{\overset{T_b}{\int }}{e}^{j2\pi (f_1-f_2)t}dt}\\ =\frac{\sin (\pi \Delta f{T}_b)}{\pi \Delta f{T}_b}{e}^{j\pi \Delta ft}\end{array}$$

where $$\Delta f=f_1-f_2$$.

From equation 8.44 in [2],

$$\begin{array}{l}\mathrm{Re}\left[\rho \right]=\mathrm{Re}\left[\frac{\sin (\pi \Delta f{T}_b)}{\pi \Delta f{T}_b}{e}^{j\pi \Delta ft}\right]=\frac{\sin (\pi \Delta f{T}_b)}{\pi \Delta f{T}_b}\cos (\pi \Delta f{T}_b)=\frac{\sin (2\pi \Delta f{T}_b)}{2\pi \Delta f{T}_b}\\ \Rightarrow P_b=Q\left(\sqrt{\frac{E_b(1-\sin (2\pi \Delta f{T}_b)/(2\pi \Delta f{T}_b))}{N_0}}\right)\end{array}$$

where $$h=\Delta f{T}_b$$.

Orthogonal M-FSK with Noncoherent Detection

From equation 5.4-46 in [1] and equation 8.66 in [2],

$$\begin{array}{l}{P}_s={\displaystyle \sum _{m=1}^{M-1}{(-1)}^{m+1}\left(\begin{array}{c}M-1\\ m\end{array}\right)\frac{1}{m+1}\exp \left[-\frac{m}{m+1}\frac{k{E}_b}{N_0}\right]}\\ P_b=\frac{1}{2}\frac{M}{M-1}{P}_s\end{array}$$

Nonorthogonal 2-FSK with Noncoherent Detection

For $$M=2$$, this equation 5.4-53 from [1] and this equation 8.69 from [2] apply:

$$P_s=P_b=Q(\sqrt{a},\sqrt{b})-\frac{1}{2}\exp \left(-\frac{a+b}{2}\right)I_0(\sqrt{ab})$$

where

$$a=\frac{E_b}{2N_0}(1-\sqrt{1-{\left|\rho \right|}^2}),b=\frac{E_b}{2N_0}(1+\sqrt{1-{\left|\rho \right|}^2})$$

Precoded MSK with Coherent Detection

Use the same BER and SER computations as for BPSK.

Differentially Encoded MSK with Coherent Detection

Use the same BER and SER computations as for DE-BPSK.

MSK with Noncoherent Detection (Optimum Block-by-Block)

The upper bound on error rate from equations 10.166 and 10.164 in [6]) is

$$\begin{array}{c}{P}_s=P_b\\ \le \frac{1}{2}\left[1-Q\left(\sqrt{b_1},\sqrt{a_1}\right)+Q\left(\sqrt{a_1},\sqrt{b_1}\right)\right]+\frac{1}{4}\left[1-Q\left(\sqrt{b_4},\sqrt{a_4}\right)+Q\left(\sqrt{a_4},\sqrt{b_4}\right)\right]+\frac{1}{2}{e}^{-\frac{E_b}{N_0}}\end{array}$$

where

$$\begin{array}{cc}{a}_1=\frac{E_b}{N_0}\left(1-\sqrt{\frac{3-4/{\pi }^2}{4}}\right),& b_1=\frac{E_b}{N_0}\left(1+\sqrt{\frac{3-4/{\pi }^2}{4}}\right)\\ a_4=\frac{E_b}{N_0}\left(1-\sqrt{1-4/{\pi }^2}\right),& b_4=\frac{E_b}{N_0}\left(1+\sqrt{1-4/{\pi }^2}\right)\end{array}$$

CPFSK Coherent Detection (Optimum Block-by-Block)

The lower bound on error rate (from equation 5.3-17 in [1]) is

$$P_s>K_{{\delta }_{\min }}Q\left(\sqrt{\frac{E_b}{N_0}{\delta }_{\min }^2}\right)$$

The upper bound on error rate is

$${\delta }_{\min }^2>\underset{1\le i\le M-1}{\min }\left\{2i\left(1-\text{sinc}(2ih)\right)\right\}$$

where h is the modulation index, and $$K_{{\delta }_{\min }}$$ is the number of paths with the minimum distance.

$$P_b\cong \frac{P_s}{k}$$