Question

If a binary signal is sent over a \(3-\mathrm{kHz}\) channel whose signal-to- noise ratio is \(20 \mathrm{~dB}\), what is the maximum achievable data rate?

Short answer

Maximum achievable data rate is approximately 19,974 bits per second.

Step-by-step solution

Understanding the Problem

We need to find the maximum achievable data rate of a binary signal transmitted over a given channel. To do that, we utilize the Shannon Capacity Formula, which relates the bandwidth, the signal-to-noise ratio (SNR), and channel capacity.

Identifying the Given Values

We have been given a channel bandwidth ( ) of 3 kHz and a signal-to-noise ratio (SNR) of 20 dB. We need to convert the SNR from decibels to a linear scale.

Converting SNR from decibels to linear scale

SNR in decibels can be converted to a linear scale using the formula: \[ SNR_{ ext{linear}} = 10^{rac{SNR_{ ext{dB}}}{10}} \]Substituting the given SNR (20 dB), we have:\[ SNR_{ ext{linear}} = 10^{rac{20}{10}} = 10^2 = 100 \]

Applying the Shannon Capacity Formula

The Shannon Capacity formula is given by:\[ C = B imes ext{log}_2(1 + SNR_{ ext{linear}}) \]where:- \( C \) is the channel capacity in bits per second,- \( B \) is the bandwidth in Hz.Substituting the known values:\[ C = 3000 imes ext{log}_2(1 + 100) \]

Calculating the Maximum Achievable Data Rate

Using the formula above, we first calculate \( \text{log}_2(101) \):\[ \text{log}_2(101) \approx 6.658 \]Thus, the channel capacity is:\[ C = 3000 imes 6.658 \approx 19974 ext{ bits per second} \]

Final Answer

Hence, the maximum achievable data rate is approximately 19,974 bits per second.

Key concepts

Understanding Channel Bandwidth

The concept of channel bandwidth is fundamental in understanding how data is transmitted over communication channels. Bandwidth refers to the difference between the highest and lowest frequencies that a channel can carry. In simpler terms, it is the range of frequencies the channel can manage without distortion. For example, a channel with a bandwidth of 3 kHz can support frequencies up to 3,000 Hz.

The channel bandwidth directly influences how much data can be sent over a communication channel within a given period. So, when we're discussing data rates and transmission capacity, we'll typically refer to Shannon's Capacity Formula, where bandwidth plays a vital role. This formula helps us see how bandwidth, along with other factors, determines the maximum data rate achievable in any channel, thereby shaping efficient communication systems. Keeping this in mind will help you grasp more complex concepts easily!

• Higher bandwidth means more data per second
• It determines the amount of information transmitted
• Measured in hertz (Hz)

Signal-to-Noise Ratio (SNR) in Communication

Signal-to-Noise Ratio, or SNR, is another key factor affecting data transmission. It describes the power ratio between the signal's strength and the noise level in the channel. Essentially, SNR tells us how "loud" the signal is compared to the background noise.

SNR is measured in decibels (dB), but for many calculations, it needs to be converted to a linear scale. This conversion is crucial as it simplifies the application of the Shannon Capacity Formula. In more practical terms, a high SNR means a clearer and stronger signal, leading to more accurate data transmission. Conversely, a low SNR indicates the presence of noise overpowering the signal, resulting in potential errors.

Understanding SNR is vital because it directly influences the quality and speed of communication:

• Higher SNR means less noise interference
• Indicates signal clarity
• Inverse relationship with potential data errors

Data Rate Calculation with Shannon's Formula

Calculating the data rate using Shannon's Capacity Formula is a fundamental aspect of understanding communications and networks. The formula essentially provides a theoretical maximum data rate for a channel given its bandwidth and SNR. This calculation is crucial when designing systems that need high data throughput.

The equation used is: \[ C = B \times \log_2(1 + SNR_{\text{linear}}) \]where \( C \) is the capacity, \( B \) is the bandwidth, and \( SNR_{\text{linear}} \) represents the linear scale SNR. Through this calculation, we determine the number of bits per second that can be effectively transmitted over the channel.

Employing this formula not only helps in determining current capacities but also in optimizing them:

• Produces maximum data rates
• Helps in channel efficiency evaluation
• Useful in system design and improvement

Binary Signal Transmission Basics

Binary signal transmission is the simplicity behind the screens, a fundamental concept in digital communication. In this process, complex data is broken down into binary code, consisting of 1s and 0s. These binary numbers are then transmitted as electrical signals over channels.

Whenever we talk about high data rates and efficient communications, we're also discussing how well binary signals are transmitted. The speed and quality can be greatly improved by minimizing noise, maximizing bandwidth, and applying thorough calculations like the Shannon Capacity.

This understanding is key in fields like data communications and information technology:

• Translates complex data into binary form
• Ensures data integrity during transmission
• Forms the backbone of digital communication systems