Priority Based Error Correction Code (ECC) for the Embedded SRAM Memories in H.264 System

Abstract

With aggressive supply voltage scaling, SRAM bit-cell failures in the embedded memory of the H.264 system result in significant degradation to video quality. Error Correction Coding (ECC) has been widely used in the embedded memories in order to correct these failures, however, the conventional ECC approach does not consider the differences in the importance of the data stored in the memory. This paper presents a priority based ECC (PB-ECC) approach, where the more important higher order bits (HOBs) are protected with higher priority than the less important lower order bits (LOBs) since the human visual system is less sensitive to LOB errors. The mathematical analysis regarding the error correction capability of the PB-ECC scheme and its resulting peak signal-to-noise ratio(PSNR) degradation in H.264 system are also presented to help the designers to determine the bit-allocation of the higher and lower priority segments of the embedded memory. We designed and implemented three PB-ECC cases (Hamming only, BCH only, and Hybrid PB-ECC) using 90 nm CMOS technology. With the supply voltage at 900 mV or below, the experiment results delivers up to 6.0 dB PSNR improvement with a smaller circuit area compared to the conventional ECC approach.

Appendix A

The $MSE_{\mathrm {err}}$ in Eq. (13) is derived as follows: In the one luma pixel, the numerical value of the pixel data (Y) and the memory bit-cell failures (F) are expressed as: $Y=\sum\limits _{i=0}^{7} y_{i} \cdot 2^{i}$ And $F=\sum\limits _{i=0}^{7} f_{i} \cdot 2^{i}$, respectively.

Using the expected error values due to the memory failures, the $MSE_{\mathrm {err}}$ can be expressed as:

$$\begin{array}{@{}rcl@{}}MSE_{\mathrm{err}} & = & E\Big[\big\{(Y-F) \cdot C \big\}^{2} \Big] \\ & = & E\left[\left\{\sum\limits_{i=0}^{7} (y_{i}-f_{i}) \cdot 2^{i} \cdot C_{i} \right\}^{2}\right]. \end{array} $$

(20)

where $C_{i}$ is the distortion ratio of the $i_{th}$ order of memory bit-cell failures.

Considering the fact that the memory bit-cell failures are independent to each other and the probabilities of the flip directions $(p_{0\rightarrow 1} = p_{1\rightarrow 0} = \frac {1}{2})$are generally the same, we can change the above equation to:

$$\begin{array}{lll} && E\left[ \left\{ \sum\limits_{i=0}^{7} (y_{i}-f_{i}) \cdot 2^{i} \cdot C_{i} \right\}^{2} \right] \\&&\quad =E\left[ \sum\limits_{i=0}^{7} (y_{i}-f_{i}) \cdot 2^{2i} \cdot C_{i}^{2} \right]\\ &&\quad\quad - E\left[ \sum\limits_{i=0}^{7} \sum\limits_{j \neq i} (y_{i}-f_{i}) (y_{j}-f_{j}) \cdot C_{i} \cdot C_{j} \cdot 2^{i+j}\right]\\ &&\quad=\sum\limits_{i=0}^{7} E[(y_{i}-f_{i})]^{2} \cdot 2^{2i} \cdot C_{i}^{2}\\ &&\quad\quad- \sum\limits_{i=0}^{7} \sum\limits_{j \neq i} E[(y_{i}-f_{i}) (y_{j}-f_{j}) ] \cdot C_{i} \cdot C_{j} \cdot 2^{i+j}. \end{array} $$

(21)

Assuming that each bit in the luma pixel data are independent each other, the second term of the above equation can be written as:

$$\begin{array}{lll} &&\sum\limits_{i=0}^{7} \sum\limits_{j \neq i} E[(y_{i}-f_{i}) (y_{j}-f_{j}) ] \cdot C_{i} \cdot C_{j} \cdot 2^{i+j}\\ &&\quad\;=\sum\limits_{i=0}^{7} \sum\limits_{j \neq i} E[(y_{i}-f_{i})] \cdot E[(y_{j}-f_{j})] \cdot C_{i} \cdot C_{j} \cdot 2^{i+j}.\\ \end{array} $$

(22)

where

$$\begin{array}{lll} && E\big[(y_{j}-f_{j})\big]\\ &&\quad=0 \cdot (1-p_{i})+1\cdot p_{i}\cdot p_{0\rightarrow 1} +(-1)\cdot p_{i}\cdot p_{1\rightarrow 0} =0.\\ &&E\Big[(y_{j}-f_{j})^{2}\Big]\\ &&\quad=0^{2}{}\cdot{}(1{}-{}p_{i})\,+\,1^{2}\cdot p_{i}\cdot p_{0\rightarrow 1}\,+\,(-1)^{2}\cdot p_{i}\cdot p_{1\rightarrow 0} =p_{i}.\\ \end{array} $$

(23)

$p_{i}$ is the failure probability of a memory bit-cell. Therefore, $MSE_{\mathrm {err}}$ is expressed as:

$$ MSE_{\mathrm{err}}=\sum\limits_{i=0}^{7} p_{i} \cdot 2^{2i} \cdot C_{i}^{2}. $$

(24)

When the PB-ECC is used, the $p_{i}$ of the HPP and LPP are equal to $P_{b_{HPP}} (E,k_{HPP})$ and $P_{b_{LPP}} (E,k_{LPP})$, respectively. $MSE_{\mathrm {err}}$ is divided into two parts as:

$$\begin{array}{rll} MSE_{\mathrm{err}}&=&\alpha(h) \cdot P_{b_{HPP}}(E,8h) \,+\,\beta(h) \cdot P_{b_{LPP}}(E,64-8h) \\ \alpha(h)&=&\sum\limits_{i=8-h}^{7} 2^{2i} \cdot C_{i}^{2},\beta(h)\,=\,\sum\limits_{i=0}^{7-h} 2^{2i} \cdot C_{i}^{2}. \end{array} $$

(25)

where, h is the number of bit allocation. $\alpha (h)$ and $\beta (h)$ are the sum of the expected numerical square value of the errors in HPP and LPP, respectively.