Insights on coding gain and its properties for principal component filter banks

Abstract

Principal Component Filter Bank (PCFB) is considered optimal in terms of coding gain for specific conditions. P P Vaidyanathan stated that coding gain does not necessarily always increase with the increase in the number of bands. However, very few attempts are made in the literature to go beyond the confines of work done by P P Vaidyanathan. We present the analytic proofs for the monotonicity of specific shapes of PSDs. This papers also derives properties of coding gain of PCFBs, which brings the new insights on the coding gain of Principal Component Filter Banks.

Appendix: Subband variance of triangular PSD

Consider a triangular PSD with the subbands designed for PCFB. For M subbands, the frequency support of each band will be $\frac{\pi }{M}$. For $M=5$, these subbands will appear as shown in figure 7. The first subband will be from $\omega _4$ to $\omega _5$. The second subband will be from $\omega _3$ to $\omega _4$ and $\omega _5$ to $\omega _6$. The third subband will be from $\omega _2$ to $\omega _3$ and $\omega _6$ to $\omega _7$, and so no. Then,

$\text {Subband variance} = \frac{1}{\pi } \times \text {Area under the PSD for subband}.$

Thus,

$$\begin{aligned} \sigma _{1}^{2}&= \frac{1}{\pi } ( \text {Area of triangle} + \text {Area of rectangle} ) \nonumber \\&= \frac{1}{\pi } \left[ \frac{1}{2} \cdot \frac{\pi }{M} \cdot \frac{a}{M} + \frac{\pi }{M} \cdot \frac{a(M-1)}{M} \right] \nonumber \\&= a \left[ \frac{1+2(M-1)}{2M^2} \right] . \end{aligned}$$

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Figure 7

Subbands of triangular PSD.

The area of the second subband can be computed by finding the area of parts numbered 1 and 2 combined and then subtracting the area of the first subband. Thus,

$$\begin{aligned} \sigma _{2}^{2}&= \frac{1}{\pi } \left[ \frac{1}{2} \cdot \frac{2\pi }{M} \cdot \frac{2a}{M} + \frac{2\pi }{M} \cdot \left( a - \frac{2a}{M} \right) \right] - a\left[ \frac{1+2(M-1)}{2M^2}\right] \nonumber \\&= a \left[ \frac{1+2(M-2)}{2M^2} \right] . \end{aligned}$$

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The area of subsequent subbands can be found in a similar manner. In general,

$$\begin{aligned} \sigma _{M-k}^{2}&= a \left[ \frac{1+2k}{2M^2} \right] \text {where } k=0,1,\ldots ,M-1. \end{aligned}$$

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