The purpose of block coding is to increase the Euclidean distance between the transmitted signals, thus improving so the performances (P(e)) for a given transmitted power at the price of a bandwidth expansion.
To compare codes, we can employ the Hamming distances since the Euclidean ones depend on Hamming distances.
There are two alternatives:
Assuming that we are employing an antipodal signaling scheme.
The code word ci=(ci1,ci2,…,cin) is mapped into the sequence:
with ψ (t) a signal of duration T and energy E.
The hard demodulator makes binary decions on the components of the received vector r and then it delivers the received word to the decoder that estimates the code word by calculating the distance between the received word and all M possible code words.
Therefore it finds the code word that is closest to it in the Hamming distance sense.
In hard-decion decoding three basic steps can be recognized:
Decoding is realized on the base of standard array that identifies the 2k decision regions.
The construction of the standard array allows us to divide the observation space, constituted by 2n vectors, into 2k decision regions.
A standard array is generated in the following way:
Each row of the standard array is called a coset and the first element of each coset is called the coset leader.
Standard Array Properties
Theorem 1. All elements of the standard array are distinct.
Proof: Let us assume two elements of the array are equal.
This can happen in two cases.
1. The elements belong to the same row (coset ). In this case we have:
which is impossible because the elements belong to different columns.
2. The two elements belong to two different cosets:
belong to the same coset which is impossible since by assumption k ≠ 1
Theorem 2: if y1 and y2 belong to the same coset
From this theorem we conclude that each coset can be uniquely identified by the product . We can define the syndrome as
with s (called syndrome) a (n-k) – vector that depends only on the coset leader.
Each coset can be identified uniquely by a syndrome s
Since dijH ≥ dminH and since Q(·) is decreasing function:
Using the Union Bound we obtain:
Example: Compare the performances of an uncoded data transmission system with the performances of a code system using a (7,4) Hamming code with R=10 Kbit/sec, the received power is P=10-6W, N0/2 =10-11W/Hz and the modulation scheme is binary PSK.
Hamming code → dHmin=3
Using this code the error probability decreases by a factor of 12.
In this case mod.+AWGN+ demod. hard can be modeled by a BSC:
because the code is linear and therefore they depend only on the weight distribution
The correct decision probability can be calculated as the probability that a correctable error occurs ( see standard array).
It can be also estimated with the union baud.
Hard-decision decoding scheme can obtain the same performances as the optimum one (soft-decision decoding scheme) paying in Eb/N0 <2 dB for common applications.
If we employ an 8-level quantizer, the performance difference with soft decision reduces to Eb/N0 < 0.1 dB.
Consider the example of the prvious example and we want to employ a hard decision decoding:
Pe=P (a non – correctable error configurationoccurs)=
= P (error configuration occurs with weight >1)=
In this case the error probability decreases by a factor 2 (12 in the soft-decision case).
13. Cyclic Codes