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## V. PROOF OF THEOREM 2 (THE CASE OF REALISTIC DUTY CYCLE CONSTRAINT) 

## A. Stationarity of the Capacity-achieving Input Distribution 

We first establish the fact that a stationary distribution achieves the capacity of the AWGN channel with the realistic duty cycle constraint and power constraint. 

Proposition 1: A stationary distribution 1 achieves 

Proof: Let T$_{k}$ ( · ) as a k -cyclic-shift operator on µ ∈ Λ $^{n}$( γ, q, c ) , defined as 

<img src='content_image/319252.jpg'>

<img src='content_image/319251.jpg'>

where k = 1 , · · · , n − 1 , and specifically T$_{0}$ ( µ ) = µ . For any distribution µ in Λ $^{n}$( γ, q, c ) , a distribution ν on X n can be defined as 

According the concavity of the mutual information I ( · ) , 

<img src='content_image/319250.jpg'>

<img src='content_image/319249.jpg'>

where I ( T$_{k}$ ( µ )) = I ( µ ) since the AWGN channel (1) is a memoryless and time-invariant. Obviously ν is a stationary distribution and satisfied the duty cycle constraint and power constraint, i.e., ν ∈ Λ $^{n}$( γ, q, c ) , hence Proposition 1 established. 

$^{1}$The stationarity of distribution ν on X n satisfies 

for any index s, t, k satisfied 

<img src='content_image/319247.jpg'>

<img src='content_image/319248.jpg'>

