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A digital system is required to amplify a binary-encoded audio signal. The user should be able to control the gain of the amplifier from a minimum to a maximum in 100 increments. The minimum number of bits required to encode, in straight binary, is
- a)8
- b)6
- c)5
- d)7
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
A digital system is required to amplify a binary-encoded audio signal...
Explanation:
To calculate the minimum number of bits required to encode the gain of the amplifier, we need to consider the range of gain values (from minimum to maximum) and the number of increments required (100).
Range of Gain Values:
The user should be able to control the gain of the amplifier from a minimum to a maximum. Let's assume the minimum gain value is 1 and the maximum gain value is 100. Therefore, the range of gain values is:
Maximum Gain Value - Minimum Gain Value = 100 - 1 = 99
Number of Increments:
The user should be able to control the gain of the amplifier in 100 increments. Therefore, the step size between two adjacent gain values is:
Range of Gain Values / Number of Increments = 99 / 100 = 0.99
Binary Encoding:
To encode the gain values in binary, we need to find the number of bits required to represent the maximum gain value (100) in binary. The maximum gain value in binary is:
10010
Therefore, we need at least 7 bits to represent the gain values in binary. However, we need to consider the number of increments required (100) and the step size between two adjacent gain values (0.99).
Since the step size (0.99) is less than 1, we can use the 7th bit to represent the step size. Therefore, we can represent the gain values in binary using 6 bits and the 7th bit to represent the step size.
Therefore, the minimum number of bits required to encode the gain of the amplifier in straight binary is 7.
Conclusion:
The correct option is 'D' (7 bits).
To calculate the minimum number of bits required to encode the gain of the amplifier, we need to consider the range of gain values (from minimum to maximum) and the number of increments required (100).
Range of Gain Values:
The user should be able to control the gain of the amplifier from a minimum to a maximum. Let's assume the minimum gain value is 1 and the maximum gain value is 100. Therefore, the range of gain values is:
Maximum Gain Value - Minimum Gain Value = 100 - 1 = 99
Number of Increments:
The user should be able to control the gain of the amplifier in 100 increments. Therefore, the step size between two adjacent gain values is:
Range of Gain Values / Number of Increments = 99 / 100 = 0.99
Binary Encoding:
To encode the gain values in binary, we need to find the number of bits required to represent the maximum gain value (100) in binary. The maximum gain value in binary is:
10010
Therefore, we need at least 7 bits to represent the gain values in binary. However, we need to consider the number of increments required (100) and the step size between two adjacent gain values (0.99).
Since the step size (0.99) is less than 1, we can use the 7th bit to represent the step size. Therefore, we can represent the gain values in binary using 6 bits and the 7th bit to represent the step size.
Therefore, the minimum number of bits required to encode the gain of the amplifier in straight binary is 7.
Conclusion:
The correct option is 'D' (7 bits).
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