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The limit will exist if the process is asymptotically mean stationary and the process has a discrete alphabet. It can be shown that if the process is stationary, then N- l H(XN) is nonincreasing in N and hence where the infimum is required since the minimum may not exist. 7 2': J(X;Y). Limiting Properties Information satisfies a form of ergodic theorem and this plays a key role in the proof of the source coding theorem. In this section we collect the appropriate definitions and state the prerequisite results without proof.

CODE PERFORMANCE if the limit exists, which it will if the pair process {X n , Xn} is asymptotically mean stationary. We shall usually assume that the limit exists. In fact, if the source is asymptotically mean stationary, then fixed rate code structures considered all yield asymptotically mean stationary pair processes. ) The limit will be a random variable, however, unless the pair process is also ergodic, in which case the limit is a constant. If the process is stationary, then the constant is the expectation of d(Xo, Xo).

5. A k-dimensional random vector X is coded as follows: First it is multiplied by a unitary matrix U to form a new vector Y = UX. (By unitary it is meant that U· = U- 1 , where U* is the complex conjugate of U). Each component Yi ; i = 0, I, ... ) k - 1 is separately quantized by a quantizer qi to form a reproduction Y; = qi(Yi). Let Y denote the resulting vector. This vector is then used to produce a reproduction X of X by the formula X = U- 1 Y. This code is called a transform code. (a) Suppose that we measure the distortion between input and output by the mean squared error k-l (X - X)t(X - X) =L IXi - X;l2.

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