show that the dynamic range of a signal increase by 6dB for each additional bit used to represent its value

Sure, let's go through this step by step.

1. The dynamic range of a signal is the ratio between the largest and smallest possible values of a changeable quantity, such as in sound and light. It is often measured in decibels (dB).

2. In digital systems, the dynamic range is often quantified in bits. Each bit represents a power of 2 (since a bit can have two values - 0 or 1). So, if we have n bits, we can represent 2^n different values.

3. The dynamic range in dB is calculated using the formula: Dynamic Range = 20 * log10(max value / min value). In the case of digital signals, the max value is 2^n and the min value is 1 (since we're considering positive values only).

4. Substituting these values into the formula, we get: Dynamic Range = 20 * log10(2^n / 1) = 20 * n * log10(2).

5. The value of log10(2) is approximately 0.3. So, the dynamic range becomes approximately 6n dB.

6. Therefore, for each additional bit used to represent the signal value, the dynamic range increases by approximately 6 dB. This is because each additional bit doubles the number of possible values, which increases the dynamic range by about 6 dB.