Horst Schubert

Template:Probability fundamentals Pairwise error probability is the error probability that for a transmitted signal ($X$) its corresponding but distorted version ($\hat{X}$) will be received. This type of probability is called ″pair-wise error probability″ because the probability exists with a pair of signal vectors in a signal constellation. It's mainly used in communication systems.

Expansion of the definition

In general, the received signal is a distorted version of the transmitted signal. Thus, we introduce the symbol error probability, which is the probability $P(e)$ that the demodulator will make a wrong estimation $(\hat{X})$ of the transmitted symbol $(X)$ based on the received symbol, which is defined as follows:

$P(e)\triangleq \frac{1}{M}\sum _x\mathbb{\mathbb{P}}(X\ne \hat{X}|X)$

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The pairwise error probability $P(X\to \hat{X})$ is defined as the probability that, when $X$ is transmitted, $\hat{X}$ is received.

$P(e|X)$ can be expressed as the probability that at least one $\hat{X}\ne X$ is closer than $X$ to $Y$.

Using the upper bound to the probability of a union of events, it can be written:

$P(e|X)\le \sum _{\hat{X}\ne X}P(X\to \hat{X})$

Finally:

$P(e)=\frac{1}{M}\sum _{X\in S}P(e|X)\le \frac{1}{M}\sum _{X\in S}\sum _{\hat{X}\ne X}P(X\to \hat{X})$

Closed form computation

For the simple case of the additive white Gaussian noise (AWGN) channel:

$Y=X+Z,Z_i\sim \mathcal{𝒩}(0,\frac{N_0}{2}{I}_n)$

The PEP can be computed in closed form as follows:

$\begin{array}{rl}P(X\to \hat{X})& =\mathbb{\mathbb{P}}(||Y-\hat{X}|{|}^2<\left|\right|Y-Z\left|{|}^2\right|X)\\ & =\mathbb{\mathbb{P}}(||(X+Z)-\hat{X}|{|}^2<\left|\right|(X+Z)-X|{|}^2)\\ & =\mathbb{\mathbb{P}}(||(X-\hat{X})+Z|{|}^2<\left|\right|Z|{|}^2)\\ & =\mathbb{\mathbb{P}}(||X-\hat{X}|{|}^2+\left|\right|Z|{|}^2+2(Z,X-\hat{X})<|\left|Z\right|{|}^2)\\ & =\mathbb{\mathbb{P}}(||X-\hat{X}|{|}^2<2(Z,X-\hat{X}))\\ & =\mathbb{\mathbb{P}}(||X-\hat{X}|{|}^2/2<(Z,X-\hat{X}))\end{array}$

$(Z,X-\hat{X})$ is a Gaussian random variable with mean 0 and variance $N_0\left|\right|X-\hat{X}|{|}^2/2$.

For a zero mean for a zero mean Gaussian random variable:

$P(X>x)=Q(x)=\frac{1}{\sqrt{2\mathrm{\Pi }}}\underset{x}{\overset{+\mathrm{\infty }}{\int }}{e}^{-}\frac{X^2}{2}$

Hence,

$\begin{array}{rl}P(X\to \hat{X})& =Q(\frac{\left|\right|X-\hat{X}|{|}^2}{2}.\sqrt{\frac{2}{N_0\left|\right|X-\hat{X}|{|}^2}})\\ & =Q\left(\frac{\left|\right|X-\hat{X}\left|\right|}{\sqrt{2N_0}}\right)\end{array}$

See also

• Signal Processing
• Telecommunication
• Electrical engineering
• Random variable

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Further reading

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534