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?

Steps to Solve

1. 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).

2. 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
• $\text{SNR} = 30$

3. Plug-in values into the theorem

Now substitute the identified values into the Shannon-Hartley equation: $$ C = 3000 \cdot \log_2(1 + 30) $$

4. Calculate the logarithm term

First, calculate the term inside the logarithm: $$ 1 + 30 = 31 $$ Next, find the logarithm (base 2): $$ \log_2(31) $$

5. Complete the calculation

Using the approximate value of $\log_2(31) \approx 4.954$ (you can use a calculator for this value), we compute: $$ C \approx 3000 \cdot 4.954 \approx 14862 \text{ bits per second} $$