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The output signal to noise ratio (SNR) of a 10-bit PCM was found to be 30 dB. The desired SNR is 42 dB. It was decided to increase the SNR to the desired value by increasing the number of quantization levels. The percentage increase in the transmission bandwidth required for this SNR is
- a)20%
- b)15%
- c)10%
- d)5%
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
The output signal to noise ratio (SNR) of a 10-bit PCM was found to be...
Given, SNR = 30 dB, n = 10,
Desired value of SNR = 42 dB
With the increase in n by 1 bit, the value of SNR increases by 6 dB. Therefore, to increase the value of SNR by 12 dB, it is necessary to increase n by 2,
∴ n = 10 + 2 = 12
Now, BW of PCM system = 1/2 nfs
Therefore, BW with n = 10 is
and BW for n = 12 is
Desired value of SNR = 42 dB
With the increase in n by 1 bit, the value of SNR increases by 6 dB. Therefore, to increase the value of SNR by 12 dB, it is necessary to increase n by 2,
∴ n = 10 + 2 = 12
Now, BW of PCM system = 1/2 nfs
Therefore, BW with n = 10 is
and BW for n = 12 is
Most Upvoted Answer
The output signal to noise ratio (SNR) of a 10-bit PCM was found to be...
Given information:
- Output signal to noise ratio (SNR) of a 10-bit PCM = 30 dB
- Desired SNR = 42 dB
To increase the SNR to the desired value, we need to increase the number of quantization levels.
Concepts used:
- SNR formula: SNR = 1.76 + 6.02N, where N is the number of bits used for quantization
- Bandwidth formula: Bandwidth = 2 * f * log2L, where f is the highest frequency component of the signal and L is the number of quantization levels
Steps to solve the problem:
1. Find the number of bits required for the desired SNR using the SNR formula:
42 = 1.76 + 6.02N
N = (42 - 1.76) / 6.02
N ≈ 6.2
We need to use at least 7 bits for quantization to achieve the desired SNR.
2. Calculate the percentage increase in the number of quantization levels:
Percentage increase = ((New value - Old value) / Old value) * 100
= ((7 - 10) / 10) * 100
= -30%
3. Calculate the percentage increase in bandwidth using the bandwidth formula:
For 10-bit PCM:
Bandwidth = 2 * f * log2L
For 7-bit PCM:
Bandwidth = 2 * f * log27
= 2 * f * 2.81
Percentage increase = ((New value - Old value) / Old value) * 100
= ((2 * f * 2.81) - (2 * f * 2)) / (2 * f * 2) * 100
= 40.5%
4. Compare the percentage increase in bandwidth with the options given in the question:
Option (a) 20% < />
Option (b) 15% < />
Option (c) 10% < />
Option (d) 5% < />
Therefore, the correct option is (a) 20%.
Conclusion:
To increase the SNR from 30 dB to 42 dB by increasing the number of quantization levels, we need to use at least 7 bits for quantization. This results in a 30% decrease in the transmission bandwidth, which is not one of the options given in the question. The correct option is (a) 20%, which is the closest to the actual increase in bandwidth of 40.5%.
- Output signal to noise ratio (SNR) of a 10-bit PCM = 30 dB
- Desired SNR = 42 dB
To increase the SNR to the desired value, we need to increase the number of quantization levels.
Concepts used:
- SNR formula: SNR = 1.76 + 6.02N, where N is the number of bits used for quantization
- Bandwidth formula: Bandwidth = 2 * f * log2L, where f is the highest frequency component of the signal and L is the number of quantization levels
Steps to solve the problem:
1. Find the number of bits required for the desired SNR using the SNR formula:
42 = 1.76 + 6.02N
N = (42 - 1.76) / 6.02
N ≈ 6.2
We need to use at least 7 bits for quantization to achieve the desired SNR.
2. Calculate the percentage increase in the number of quantization levels:
Percentage increase = ((New value - Old value) / Old value) * 100
= ((7 - 10) / 10) * 100
= -30%
3. Calculate the percentage increase in bandwidth using the bandwidth formula:
For 10-bit PCM:
Bandwidth = 2 * f * log2L
For 7-bit PCM:
Bandwidth = 2 * f * log27
= 2 * f * 2.81
Percentage increase = ((New value - Old value) / Old value) * 100
= ((2 * f * 2.81) - (2 * f * 2)) / (2 * f * 2) * 100
= 40.5%
4. Compare the percentage increase in bandwidth with the options given in the question:
Option (a) 20% < />
Option (b) 15% < />
Option (c) 10% < />
Option (d) 5% < />
Therefore, the correct option is (a) 20%.
Conclusion:
To increase the SNR from 30 dB to 42 dB by increasing the number of quantization levels, we need to use at least 7 bits for quantization. This results in a 30% decrease in the transmission bandwidth, which is not one of the options given in the question. The correct option is (a) 20%, which is the closest to the actual increase in bandwidth of 40.5%.
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