## 4 Evaluation
### Evaluation for Entropy Coding Ability  
Figure 5 shows comparison between the entropy calculation of the [PERSON]'s (Equation (1)) and ASE coding's (Equation (2)). This evaluation is performed by calculating the entropies by using an eight-bit width and 10K-byte data stream. The values in the data stream are randomly generated. The horizontal axis of the graph shows the limited number of the random frequency. For example, when the number is 16, the random numbers from 0 to 15 in eight bits are generated and used for the entropy calculation. In the case of ASE coding, we plotted the average by accumulating the entropy calculated at every compression operation. Note that the entropy of the ASE coding does not include Cmark bit. The graph shows that the entropy of ASE coding maintains slightly lower than the [PERSON]'s. Let us analyze this result below. Focusing on a part in a data stream from \(t_{0}\)th to \(t_{1}\)th data, according to Equation (1), we can have the equation of the [PERSON]'s information entropy \(S^{\prime}\), as follows:  
\[S^{\prime}=-\sum_{k=t_{0}}^{t_{1}}P_{k}\log_{2}P_{k}.\]  
Figure 5: Entropy comparison between [PERSON]’s and ASE coding’s. Because the part \(P_{k}\log_{2}P_{k}\) is always negative value regarding any \(k\), the sum of the part is accumulated and linearly decreasing when the interval between \(t_{0}\) and \(t_{1}\) is getting expanded. When we consider \(S^{\infty}\) where \(t_{0}=0\) and \(t_{1}=\infty\), the relationship \(S^{\prime}\leq S^{\infty}\) is obvious. Additionally, when \(S^{\prime\prime}\) is defined as an entropy calculated by a part from \(t_{0}^{\prime}\) to \(t_{1}^{\prime}\) where \(t_{0}<t_{0}^{\prime}\) and \(t_{1}>t_{1}^{\prime}\), \(S^{\prime\prime}<S^{\prime}\). According to these relationships of the information entropy, we found that ASE coding's entropy plotted in the graph results lower than the Shannon's. This means that ASE coding has the possibility to encode data to shorter code than the Shannon's, because its instantaneous entropy calculation in a small time window works to assign adaptive codes to symbols. This is mainly caused by the entropy culling. It contributes to reduce the number of bits in the codes by purging the occupied entries in the look-up table. However, the Cmark bit must be added to the codes to identify it as whether a compressed code or an original symbol. Thus, the actual data patterns can result a little higher entropy than the Shannon's. Let us focus on a theoretical analysis of ASE coding, according to the discussion above. First, we define the compression ratio of the stream-based compression. In a data communication path of a system, the compressor and the decompressor are connected by a physical media. In this setting, we categorize the compression ratio into two types; a micro and a macro compression ratio. The former one is defined at the symbol level. The minimal size of the compressed symbol is only the Cmark bit, which is a single bit. Therefore, the best compression ratio is \((1/s)\), where \(s\) is the number of bits in a symbol. The worst length of the compressed symbol is \((s+Cmark)\) bits when it is not hit in the look-up table. The latter case, the macro compression ratio, is defined as a part of data stream between timings \(t_{0}\) and \(t_{1}\) when the terminal systems process the part. For example, in a case of a video stream, the decompressor side displays a video frame at a refresh rate. The refresh rate is presented by \((t_{1}-t_{0})\) for a frame with \(L\) symbols. In this case, the worst bit length of the part is \((s+1)\times L\). Therefore, the best and worst micro compression ratios are defined as \((1/s)\) and \(\{(s+1)/s\}\), respectively. However, the dynamic performance of the macro compression ratio is not simple, because we need to apply a frequency of data pattern (i.e., data entropy). In the ASE coding case, we have confirmed that the entropy is presented by the number of occupied entries in the look-up table as we discussed above in this section.