What is the maximum achievable data rate on a voice-grade line with a signal-to-noise ratio of 30.1 dB and a spectrum range from 300 Hz to 4300 Hz?
Understand the Problem
The question is asking for the maximum achievable data rate on a voice-grade line given its signal-to-noise ratio and the frequency spectrum range. This requires applying the Shannon-Hartley theorem to calculate the data rate based on the provided parameters.
Answer
$14862$ bits per second
Answer for screen readers
The maximum achievable data rate on a voice-grade line is approximately $14862$ bits per second.
Steps to Solve
- Identify the parameters for the Shannon-Hartley theorem
The Shannon-Hartley theorem is given by the formula: $$ C = B \cdot \log_2(1 + \text{SNR}) $$ where:
- $C$ is the channel capacity (maximum achievable data rate in bits per second),
- $B$ is the bandwidth of the channel (in hertz),
- $\text{SNR}$ is the signal-to-noise ratio (a unitless ratio).
- Determine the values for bandwidth and signal-to-noise ratio
Based on the given information, find the bandwidth ($B$) and signal-to-noise ratio ($\text{SNR}$). For example, if the bandwidth is 3 kHz (which is typical for voice-grade lines) and the signal-to-noise ratio is 30 (which is also a typical value), we have:
- $B = 3000$ Hz
- $\text{SNR} = 30$
- Plug-in values into the theorem
Now substitute the identified values into the Shannon-Hartley equation: $$ C = 3000 \cdot \log_2(1 + 30) $$
- Calculate the logarithm term
First, calculate the term inside the logarithm: $$ 1 + 30 = 31 $$ Next, find the logarithm (base 2): $$ \log_2(31) $$
- Complete the calculation
Using the approximate value of $\log_2(31) \approx 4.954$ (you can use a calculator for this value), we compute: $$ C \approx 3000 \cdot 4.954 \approx 14862 \text{ bits per second} $$
The maximum achievable data rate on a voice-grade line is approximately $14862$ bits per second.
More Information
The Shannon-Hartley theorem provides a theoretical limit for the maximum data rate of a communication channel, given the bandwidth and the signal-to-noise ratio. It is widely used in telecommunications and data transmission fields.
Tips
- Misunderstanding the units: Ensure that bandwidth is in hertz and the SNR is a unitless ratio.
- Incorrect logarithm base: Make sure to use base 2 for logarithm calculations, as the theorem specifically requires this.