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Chapter 5. Pilot Design and Optimization... > 5.D Proof of Theorem 5.1 - Pg. 231

5.D Proof of Theorem 5.1 231 0 ( ) = T p ( ). Therefore, we can express k ( ) in terms of T p ( ) from the above recursion as 1 1 2 = 1 + 2 + 1 2 k ( ) k-2 ( ) k = 1 k = 1 k 1 - 1 1 + 2 0 ( ) 1 - 1 k 1 - 1 1 + 2 , T p ( ) 1 - 1 1 k . (5.C.1) During data transmission ( + 1 k T p ), the updating recursion for M k ( ) is given by (5.7). Therefore, 2 k ( ) = 2(k- ) M ( ) + (1 - 2(k- ) ) h - M 2 = 2(k- ) ( ) + (1 - 2(k- ) )( h - M ) 2 = 2(k- ) ( ) + (1 - 2(k- ) ) h , k = + 1, . . . , T p . (5.C.2) From (5.C.1)­(5.C.2), ( ) and T p ( ) satisfy the following relations: