dc

• f c denotes center frequency
• Negative Frequencies contain no Additional Info
Characteristics:
• Complex valued signal
• No information loss, truely equivalent
Let us consider DN = {(xi , yi) : i = 1, .., N} iid realizations of the joint observation-class
phenomenon (X(u), Y (u)) with true probability measure P(X,Y) defined on (X ×Y, σ(FX ×
FY )). In addition, let us consider a family of measurable representation functions D, where
any f(·) ∈ D is defined in X and takes values in Xf . Let us assume that any representation
function f(·) induces an empirical distribution Pˆ Xf ,Y on (Xf ×Y, σ(Ff ×FY )), based on the
training data and an implicit learning approach, where the empirical Bayes classification rule
is given by: gˆf (x) = arg maxy∈Y Pˆ Xf ,Y (x, y).
Turbo codes
UNIT 6
In information theory, turbo codes (originally in French Turbocodes) are a class of highperformance
forward error correction (FEC) codes developed in 1993, which were the first
practical codes to closely approach the channel capacity, a theoretical maximum for the code
rate at which reliable communication is still possible given a specific noise level. Turbo
codes are finding use in 3G mobile communications and (deep
space) satellite communications as well as other applications where designers seek to achieve
reliable information transfer over bandwidth- or latency-constrained communication links in
the presence of data-corrupting noise. Turbo codes are nowadays competing with LDPC
codes, which provide similar performance.
Prior to turbo codes, the best constructions were serial concatenated codes based on an
outer Reed-Solomon error correction code combined with an inner Viterbi-decoded short
constraint length convolutional code, also known as RSV codes.
In 1993, turbo codes were introduced by Berrou, Glavieux, and Thitimajshima (from
Télécom Bretagne, former ENST Bretagne, France) in their paper: "Near Shannon Limit
Error-correcting Coding and Decoding: Turbo-codes" published in the Proceedings of IEEE
International Communications Conference. In a later paper, Berrou gave credit to the
"intuition" of "G. Battail, J. Hagenauer and P. Hoeher, who, in the late 80s, highlighted the
interest of probabilistic processing.". He adds "R. Gallager and M. Tanner had already
imagined coding and decoding techniques whose general principles are closely related,"
although the necessary calculations were impractical at that time.
The first class of turbo code was the parallel concatenated convolutional code (PCCC). Since
the introduction of the original parallel turbo codes in 1993, many other classes of turbo code
have been discovered, including serial versions and repeat-accumulate codes. Iterative Turbo
decoding methods have also been applied to more conventional FEC systems, including
Reed-Solomon corrected convolutional codes