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Extra resources for Channel Coding Theory: Course Held at the Department for Automation and Information, July 1970

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F with a domain V such that (~)='ll} P {f (3. 2} == 1-A (0<4<1). ) Then, ( 3. 3) . 2. , > h(A) = (3. 4} 1 X Proof. 5} Since for denote the RV taking the value f ( ~ ) :fo 11 and 0 if H(Td~) X = value (name! I~) = H('r\lx,~)+ H(xiTf) . :E and f ( 1 1 T· ... S A tog/ N -1). (3. 6) Further, on account of (3. 2) and (3. £ H( x) = - 1=> { - P{ X= 1} Since I ( '11 1\ Tt) x =o} eog 2 "P { x = o} - tog~ = H ( 11) P{X = 1} - H ( 11 ~ I~ ) , h (A). (3. 7) (3. 1) and the estimates (3. 5), (3. 6), (3. 7) give rise to I ( Tl " 1i ) ~ to h N - /..

F with a domain V such that (~)='ll} P {f (3. 2} == 1-A (0<4<1). ) Then, ( 3. 3) . 2. , > h(A) = (3. 4} 1 X Proof. 5} Since for denote the RV taking the value f ( ~ ) :fo 11 and 0 if H(Td~) X = value (name! I~) = H('r\lx,~)+ H(xiTf) . :E and f ( 1 1 T· ... S A tog/ N -1). (3. 6) Further, on account of (3. 2) and (3. £ H( x) = - 1=> { - P{ X= 1} Since I ( '11 1\ Tt) x =o} eog 2 "P { x = o} - tog~ = H ( 11) P{X = 1} - H ( 11 ~ I~ ) , h (A). (3. 7) (3. 1) and the estimates (3. 5), (3. 6), (3. 7) give rise to I ( Tl " 1i ) ~ to h N - /..

For arbitrary (simple or compound) communication channels ('Y1=1,2, ... <1); (3. 9) this implies and provided that C1 exists. ( 3. 10) Discrete Communication Channels 36 Proof. Let let B1 ' '\T 1 ,_ • • • , tr n ••• ' with~Noise be code words of length n and BN be decoding sets such that ~ =1 , •.. , N ; if a com- pound channel is considered, the latter inequalities should hold for each component simple channel, cf. definition 2. 4. Thus, lemma 3. 1 can be applied for the ndimensional observation channel f being defined so that f ( Y ", (tr) = Vn ,~ , if p ) 'V v with E.

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