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--- tanszek:oktatas:techcomm:bcd_encoding [2024/09/30 18:31] – created knehez
+++ tanszek:oktatas:techcomm:bcd_encoding [2025/10/27 18:53] (current) – [Applications of BCD] knehez
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 ==== BCD (Binary-Coded Decimal) Encoding ====
-Binary-Coded Decimal (BCD) is a class of binary encodings in which each decimal digit is represented by its own binary sequence. In BCD, the binary form of a decimal number is encoded such that a 4-bit binary number represents each digit.
+Binary-Coded Decimal (BCD) is a class of binary encodings in which each decimal digit is represented by its own binary sequence. In BCD, the binary form of a decimal number is encoded such that a **4-bit binary number represents each digit**.
 Each decimal digit (0-9) is represented using 4 bits, as follows:
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   * **Digital clocks** (old ones): These devices often display numbers directly in decimal form, so BCD simplifies the process.
   * **Financial application**s: BCD can be used in systems requiring precise decimal representation, such as in currency and banking systems, to prevent rounding errors.
+=== How much is the redundancy of this encoding? ===
+Equation of redundancy is as follows: \(  R = \frac{H_{\text{max}} - H}{H_{\text{max}}} \)
+where:
+  * \(H_{\text{max}}\) is the maximum entropy,
+  * \(H\) is the actual entropy.
+Each decimal digit (0–9) is represented by a 4-bit binary code. However, 4 bits can represent 16 possible combinations (0000–1111), of which only 10 are used.
+Maximum entropy:
+$$ H_{\text{max}} = \log_2(16) = 4\ \text{bits per symbol} $$
+Actual entropy:
+Since only 10 symbols are used and they are equally likely:
+$$ H = \log_2(10) \approx 3.3219\ \text{bits per symbol} $$
+Substitute these values into the equation of redundancy:
+$$ R = \frac{4 - 3.3219}{4} = 0.1695 $$
+Thus, the redundancy of the BCD code is approximately 16.95%.
+This means that about 17% of the information capacity of the 4-bit representation is not used effectively due to the limited number of valid BCD codes.