Pairwise error probability

Key takeaways

Pairwise error probability belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Pairwise error probability to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Pairwise error probability from memory before moving on to harder problems.

Reference excerpt

Pairwise error probability is the error probability that for a transmitted signal ( X {\displaystyle X} ) its corresponding but distorted version ( X ^ {\displaystyle {\widehat {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 ) {\displaystyle P(e)} that the demodulator will make a wrong estimation ( X ^ ) {\displaystyle ({\widehat {X}})} of the transmitted symbol ( X ) {\displaystyle (X)} based on the received symbol, which is defined as follows:

P ( e ) ≜ 1 M ∑ x P ( X ≠ X ^ | X ) {\displaystyle P(e)\triangleq {\frac {1}{M}}\sum _{x}\mathbb {P} (X\neq {\widehat {X}}|X)}

where M is the size of signal constellation. The pairwise error probability P ( X → X ^ ) {\displaystyle P(X\to {\widehat {X}})} is defined as the probability that, when X {\displaystyle X} is transmitted, X ^ {\displaystyle {\widehat {X}}} is received.

P ( e | X ) {\displaystyle P(e|X)} can be expressed as the probability that at least one X ^ ≠ X {\displaystyle {\widehat {X}}\neq X} is closer than X {\displaystyle X} to Y {\displaystyle Y} . Using the upper bound to the probability of a union of events, it can be written:

P ( e | X ) ≤ ∑ X ^ ≠ X P ( X → X ^ ) {\displaystyle P(e|X)\leq \sum _{{\widehat {X}}\neq X}P(X\to {\widehat {X}})}

Finally:

P ( e ) = 1 M ∑ X ∈ S P ( e | X ) ≤ 1 M ∑ X ∈ S ∑ X ^ ≠ X P ( X → X ^ ) {\displaystyle P(e)={\tfrac {1}{M}}\sum _{X\in S}P(e|X)\leq {\tfrac {1}{M}}\sum _{X\in S}\sum _{{\widehat {X}}\neq X}P(X\to {\widehat {X}})}

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

Y = X + Z , Z i ∼ N ( 0 , N 0 2 I n ) {\displaystyle Y=X+Z,Z_{i}\sim {\mathcal {N}}(0,{\tfrac {N_{0}}{2}}I_{n})\,\!}

The PEP can be computed in closed form as follows:

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Pairwise error probability

Start with the simplest possible case. Write down what Pairwise error probability claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Pairwise error probability before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Pairwise error probability ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Pairwise error probability

In research: Pairwise error probability appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Pairwise error probability in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Pairwise error probability is common in secondary-school and first-year university syllabi. It links to neighbouring topics Probability theory, Signal processing, so understanding it makes those chapters shorter.
In everyday life: Look for Pairwise error probability outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Pairwise error probability in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Pairwise error probability means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Pairwise error probability out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Pairwise error probability in simple terms?

Pairwise error probability is the error probability that for a transmitted signal ( X {\displaystyle X} ) its corresponding but distorted version ( X ^ {\displaystyle {\widehat {X}}} ) will be received. This type of probability is called ″pair-wise error probability″ because the probability exists…

Why does Pairwise error probability matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Pairwise error probability?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Pairwise error probability.