What is the capacity of a noiseless channel with a bandwidth of 10 kHz and 16 signal levels using the Nyquist formula?
Understand the Problem
The question is asking for the calculation of the capacity of a noiseless communication channel given its bandwidth and the number of signal levels. The solution involves using the Nyquist formula, which relates these parameters to determine the maximum data rate.
Answer
$80,000 \text{ bps}$
Answer for screen readers
The capacity of the noiseless channel is $80,000 \text{ bps}$.
Steps to Solve
- Identify the known parameters The bandwidth (B) of the channel is 10 kHz, which can be expressed as: $$ B = 10,000 \text{ Hz} $$
The number of signal levels (L) is 16.
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Apply the Nyquist formula The Nyquist formula for the capacity (C) of a noiseless channel is given by: $$ C = 2B \log_2(L) $$
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Calculate log base 2 of the signal levels First, calculate the logarithm: $$ \log_2(16) = 4 $$ (since $16 = 2^4$)
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Calculate the channel capacity Now substitute the values into the Nyquist formula: $$ C = 2 \cdot 10,000 \cdot 4 $$
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Perform the calculation Performing the multiplication: $$ C = 80,000 \text{ bps} $$
The capacity of the noiseless channel is $80,000 \text{ bps}$.
More Information
According to the Nyquist theorem, the capacity is dependent on both the bandwidth of the channel and the number of discrete signal levels, allowing for efficient data transmission.
Tips
- Forgetting to convert bandwidth from kHz to Hz.
- Miscalculating the logarithm, especially if using the wrong base.
- Neglecting to multiply by 2 in the Nyquist formula.
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