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Discrete-Time Continuous Memoryless Channels

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Definition
InfoTheory

Consider a discrete-time channel with continuous input and output alphabets given by XāŠ‚R\mathcal{X}\subset\mathbb{R} and YāŠ‚R\mathcal{Y}\subset\mathbb{R} respectively, and described by a sequence of nn-fold conditional pdfs fYn∣Xnf_{Y^{n}|X^{n}} which govern receiving Yn=(Y1,…,Yn)Y^{n}=(Y_{1},\ldots,Y_{n}) at channel output when Xn=(X1,…,Xn)X^{n}=(X_{1},\ldots,X_{n}) is sent at the input (X,Y,{fYn∣Xn}i=1āˆž)(\mathcal{X},\mathcal{Y},\{f_{Y^n|X^n}\}^{\infty}_{i=1})The channel (without feedback) is called memoryless with a given marginal transition pdf fY∣Xf_{Y|X} if fYn∣Xn(yn∣xn)=āˆi=1nfY∣X(yi∣xi)f_{Y^n|X^n}(y^{n}|x^{n})=\prod_{i=1}^{n}f_{Y|X}(y_{i}|x_{i})for all n≄1n\ge1, (xn,yn)∈(XnƗYn)(x^{n},y^{n})\in(\mathcal{X}^{n}\times\mathcal{Y}^{n}).

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.

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