show that the dynamic range of a signal increase by 6dB for each additional bit used to represent its value
Question
show that the dynamic range of a signal increase by 6dB for each additional bit used to represent its value
Solution
Sure, let's go through this step by step.
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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).
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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.
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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).
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Substituting these values into the formula, we get: Dynamic Range = 20 * log10(2^n / 1) = 20 * n * log10(2).
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The value of log10(2) is approximately 0.3. So, the dynamic range becomes approximately 6n dB.
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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.
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