## 3 Entropy Power/Entropy Rate Power  
For a continuous valued random variable \(X\) with probability density function \(p(x)\), we can write the differential entropy  
\[h(X)=-\int_{-\infty}^{\infty}p(x)\log p(x)dx \tag{3}\]  
where \(X\) has the variance \(\sigma^{2}\). In his original paper, [PERSON] [9] defined what he called the derived quantity of _entropy power_ corresponding to the differential entropy of the random variable \(X\). In particular, [PERSON] defined the entropy power as the power in a Gaussian random variable having the same entropy as the given random variable \(X\). Specifically, since a Gaussian random variable has the differential entropy, \(h(g)=\frac{1}{2}\log[2\pi e\sigma^{2}]\), solving for \(\sigma^{2}\) and letting \(\sigma^{2}=Q\), we have that the corresponding entropy power is  
\[Q=\frac{1}{2\pi e}\mathrm{e}^{2h(X)} \tag{4}\]  
where here \(h(X)\) is not Gaussian, but it is the differential entropy of the \"original\" random variable.  
Generally, we will be modeling our signals of interest as stationary random processes. If we let \(\mathbf{X}\) be a stationary continuous-valued random process with samples \(X^{n}=\{X_{k},k=1,2,\ldots,n\}\), then the differential _entropy rate_ of the process \(\mathbf{X}\) is [16]  
\[\overline{h}=\lim_{n\to\infty}\frac{1}{n}h(X^{n})=\lim_{n\to\infty}h(X_{n}|X^ {n-1}). \tag{5}\]  
We assume that this limit exists in our developments and we drop the overbar notation and use as before \(h=\overline{h}\). Thus, for random processes, we use the entropy rate in the definition of entropy power, which yields the nomenclature _entropy rate power_.  
We now consider a discrete-time stationary random process \(\mathbf{X}\) with correlation function \(\phi(k)=EX_{j}X_{j+k}\), and define its periodic discrete-time power spectral density \(\Phi(\omega)=\sum\limits_{k=-\infty}^{\infty}\phi(k)e^{-j\omega k}\) for \(|\omega|\leq\pi\). For an \(n\)-dimensional random process with correlation matrix \(\Phi_{n}\), we know that the determinant of \(\Phi_{n}\) is the product of its eigenvalues, \(\lambda_{k}^{(n)}\), so \(|\Phi_{n}|=\prod_{k=1}^{n}\lambda_{k}^{(n)}\), and we can write  
\[\lim_{n\to\infty}\log|\Phi_{n}|^{1/n}=\lim_{n\to\infty}\frac{1}{n}\log\left( \prod_{k=1}^{n}\lambda_{k}^{(n)}\right)=\lim_{n\to\infty}\frac{1}{n}\sum_{k=1} ^{n}\log\lambda_{k}^{(n)}. \tag{6}\]Using the Toeplitz Distribution Theorem [7; 12] on (6), it can be shown that [6; 12; 13]  
\[\lim_{n\rightarrow\infty}\log\lvert\Phi_{n}\rvert^{1/n}=\lim_{n\rightarrow\infty }\frac{1}{n}\sum_{k=1}^{n}\log\lambda_{k}^{(n)}=\frac{1}{2\pi}\int\limits_{- \pi}^{\pi}\log\Phi(\omega)d\omega. \tag{7}\]  
To obtain an expression for the entropy rate power, we note that the differential entropy of a Gaussian random process with the given power spectral density, \(\Phi(\omega)\), and correlation matrix \(\Phi_{n}\) is \(h(\mathbf{g})=(n/2)\log(2\pi e\lvert\Phi_{n}\rvert^{1/n})\), then solving for \(\lvert\Phi_{n}\rvert^{1/n}\) and taking the limit as in (7), we can write the entropy rate power \(Q\) as [8; 9]  
\[Q=\exp\Biggl{\{}\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\log\Phi(\omega)d\omega \Biggr{\}}. \tag{8}\]  
One of the primary applications of \(Q\) has been for developing a lower bound to the rate distortion function [9; 12; 13]. Note that, in defining \(Q\), we have not assumed that the original random process is Gaussian. The Gaussian assumption is only used in [PERSON]'s definition of entropy rate power. In this paper, we expand the utility of entropy rate power to digital signal processing, and in particular, in our examples, to speech codec performance evaluation and codec design.