Question

A noiseless \(10-\mathrm{kHz}\) channel is sampled every \(1 \mathrm{msec}\). What is the maximum data rate?

Short answer

The maximum data rate is 20000 bps.

Step-by-step solution

Identify the Sampling Rate

The sampling rate is given by twice the highest frequency, according to the Nyquist theorem. Here, the channel frequency is \(10\, \text{kHz}\), so sampled every \(1\, \text{msec}\) means \(1000\, \text{samples/second}\).

Apply Nyquist Theorem

According to the Nyquist theorem, the maximum data rate (in bits per second) for a noiseless channel is \(2B \times \log_2(M)\), where \(B\) is the bandwidth in Hz and \(M\) is the number of discrete levels assumed to be \(2\) for binary signals. Hence, \(2 \times 10000 \times \log_2(2) = 20000\, \text{bps}.\)

Calculate Maximum Data Rate

The maximum data rate is then calculated as \(20000\, \text{bps}\) as determined by the Nyquist formula adjusted for binary signals.

Key concepts

Sampling Rate

The sampling rate is a fundamental concept in digital signal processing. It determines how often a signal is sampled to convert it from an analog to a digital form. According to the Nyquist theorem, to accurately capture all the information in a signal without losing any data, one must sample it at least twice the highest frequency of the signal. For our example, with a channel frequency of 10 kHz, the minimum necessary sampling rate is 20 kHz. However, in our exercise, the channel is sampled every 1 millisecond. This is equivalent to a sampling rate of 1000 times per second (1 kHz), significantly less than the Nyquist rate. Therefore, precision may be compromised, leading to possible data loss or inaccurate representation of the signal.

Channel Capacity

The channel capacity of a communication channel indicates the maximum amount of data that can be reliably transmitted through the channel per unit of time. The Nyquist theorem provides the theoretical foundation for understanding channel capacity in a noiseless environment. In a noiseless channel, the maximum possible data rate can be determined by the equation \(C = 2B \times \log_2(M)\), where \(C\) is the channel capacity, \(B\) is the bandwidth, and \(M\) represents the number of signal levels or 'symbols'.

• In digital communications, when using binary signals, \(M\) is 2, which simplifies the formula to \(C = 2B\). Thus, for a 10 kHz bandwidth, the maximum data rate would be 20,000 bits per second.

Data Rate Calculations

Data rate calculations involve determining how much data is transferred across a channel per unit of time. In our exercise, the Nyquist theorem is applied for a noiseless channel. The equation used is \(C = 2B \times \log_2(M)\).

• Given our 10 kHz bandwidth, applying this calculation with binary signaling (where \(M = 2\)), results in a maximum data rate of 20,000 bits per second.
• It's crucial to recognize that this is the maximum rate in an ideal context without noise interference. In real-world scenarios, noise often reduces the achievable rate.

Understanding these calculations allows us to predict communication effectiveness and ensure efficient data transmission. Properly estimating data rates is essential for designing communication networks that meet requirements for speed and reliability.