Recipe

Compute a block's per-symbol distortion

Use when: you have a source block $x^n$ and its reconstruction $\hat x^n$ , and need the total or per-symbol distortion under Hamming or squared-error distortion

Goal: the per-symbol distortion $d(x^n,\hat x^n)=\frac1n\sum d(x_i,\hat x_i)$ , using the measure that matches the alphabet

Taught in Distortion measures (Hamming, squared error)

Quick recipe

Decide which distortion measure fits the alphabet
Compute the single-symbol distortion for each position
Sum the per-position costs and divide by the block length $n$
Interpret the result in the units the chosen measure produces

1
Check whether the source's values are discrete, where wrong is just wrong with no notion of "how far," or continuous, where the size of a miss varies.

Example

A bit-string reconstruction calls for Hamming distortion; a sensor reading calls for squared-error distortion.
- discrete alphabet — use Hamming distortion, $d(x,\hat x)=0$ if $x=\hat x$ and $1$ otherwise.
- continuous alphabet — use squared-error distortion, $d(x,\hat x)=(x-\hat x)^2$ .
Why: neither measure is universally correct; the choice is part of the problem's own modeling decision, made before any number gets computed.
2
Apply the formula chosen in Step 1 to each pair $x_i,\hat x_i$ in the block.

Example

For $x=1001101$ and $\hat x=1101100$ : positions 2 and 5 disagree (each costing $1$ under Hamming), the rest agree (costing $0$ ).

Why: the block distortion is built entirely out of these individual per-position costs.
3
Average the costs from Step 2 over the whole block.

Example

Two disagreements out of $7$ positions: $\dfrac{2}{7}\approx0.286$ .

Why: dividing by $n$ turns a raw total into a per-symbol rate, which is what makes distortion comparable across blocks of different lengths.
4
Match the interpretation to whichever measure was chosen back in Step 1.

Example

For a sensor reading $x=4.5$ reproduced as $\hat x=4.8$ : $d(4.5,4.8)=(4.5-4.8)^2=0.09$ , in squared units, not the same units as the original reading.
- Hamming distortion was used — read the result as a fraction of positions wrong, a probability once averaged over the source's own randomness.
- squared-error distortion was used — read the result in squared units of the original quantity, not the original units themselves.
Why: the two measures aren't comparable to each other or interchangeable in meaning, even when the raw numbers happen to look similar in size.

Common mistakes

Applying squared-error distortion to a discrete alphabet where "how far wrong" has no meaning, such as a flipped bit.
Forgetting to divide by $n$ , reporting a raw total instead of the per-symbol rate the theory is built on.
Reporting a squared-error result as if it were in the original units, instead of squared units.
Mixing the two measures within the same block instead of picking one and applying it consistently to every symbol.