UCSD ECE 255 C Handout

Since the supremum is taken over a bigger set as B increases, C(B) is nondecreasing for B ≥ 0. To prove the concavity, let B1 and B2 be two cost constraints. Suppose that R1 is achievable under B1 and R2 is achievable under B2 i.e. R1 ≤ C(B1) and R2 ≤ C(B2). Let k = ⌊αn⌋, k ′ = n − k and α ∈ [0, 1]. We can construct a code by using a (21 , k) code for the first k transmissions and a (2 ′R2 , k) code for the rest of k transmissions. Hence, the resulting code can achieve rate αR1 + ᾱR2 with cost constraint E(b(X)) ≤ αB1 + ᾱB2. Therefore, C(B) is concave for B ≥ 0. (b) The information capacity–cost function is defined as