Consider a discrete-time channel with continuous input and output alphabets given by XāR and YāR respectively, and described by a sequence of n-fold conditional pdfs fYnā£Xnā which govern receiving Yn=(Y1ā,ā¦,Ynā) at channel output when Xn=(X1ā,ā¦,Xnā) is sent at the input (X,Y,{fYnā£Xnā}i=1āā)The channel (without feedback) is called memoryless with a given marginal transition pdf fYā£Xā if fYnā£Xnā(ynā£xn)=i=1ānāfYā£Xā(yiāā£xiā)for all nā„1, (xn,yn)ā(XnĆYn).
Note
In practice, the real-valued input of a continuous channel must satisfy a certain limitation on its amplitude or power due to physical limitations of the transmitter. So we define the Average Cost Constraint to mitigate for this.