Channel Capacity and Shannon’s Theorem Quiz Quiz

Explore the core principles of channel capacity and Shannon’s Theorem in digital communications, including calculations, theoretical limits, and key definitions. Strengthen your understanding of bandwidth, noise, and information transmission with these thoughtfully crafted questions.

1. Shannon’s Capacity Theorem Application

   A communication channel with a bandwidth of 3 kHz transmits signals with a signal-to-noise ratio (SNR) of 30 dB. According to Shannon’s Theorem, which formula will correctly determine the maximum data rate?

   1. C = B × S/N
   2. C = B × log2(1 + S/N)
   3. C = B × log2(1 + SNR^2)
   4. C = B × log10(1 + S/N)

   Explanation: Shannon’s Capacity Theorem states that the maximum channel capacity, C, is calculated as C = B × log2(1 + S/N), where B is bandwidth and S/N is the linear signal-to-noise ratio. The log2 base is used, not log10, so option B is incorrect. Option C incorrectly squares SNR, which is not part of the theorem. Option D omits the logarithmic function, which is essential according to Shannon’s formula.

2. Interpreting Signal-to-Noise Ratios

   If a channel’s signal-to-noise ratio (S/N) is given as 20 dB, which of the following correctly converts this value to its linear (numeric) form for use in the Shannon capacity formula?

   1. S/N = 200
   2. S/N = 20
   3. S/N = 100
   4. S/N = 10

   Explanation: To convert dB to the linear S/N value, use the formula S/N = 10^(dB/10). For 20 dB: 10^(20/10) = 10^2 = 100, making S/N = 100 correct. S/N = 20 and S/N = 200 are common miscalculations, while S/N = 10 represents 10 dB. Using the correct conversion is essential for accurate channel capacity computation.

3. Understanding the Effect of Bandwidth

   In a scenario where the signal-to-noise ratio remains constant, what happens to the maximum channel capacity if the bandwidth is doubled?

   1. The channel capacity is halved.
   2. The channel capacity is also doubled.
   3. The channel capacity is reduced by a factor of log2(2).
   4. The channel capacity remains unchanged.

   Explanation: When bandwidth doubles and S/N remains unchanged, the channel capacity also doubles, as capacity (C) is directly proportional to bandwidth (B) in Shannon’s formula. Halving or keeping capacity the same would only occur if bandwidth were unchanged or halved. The term 'reduced by a factor of log2(2)' actually equals 1, so it does not represent a reduction.

4. Impact of Noise on Channel Capacity

   If external noise in a channel significantly increases, which of the following best describes the impact on the channel’s maximum theoretical data rate?

   1. The maximum data rate initially decreases and then increases.
   2. The maximum data rate remains constant.
   3. The maximum data rate decreases.
   4. The maximum data rate increases.

   Explanation: An increase in noise reduces the signal-to-noise ratio, thereby decreasing the maximum data rate as per Shannon’s theorem. The data rate cannot increase with more noise; thus, option B is incorrect. Option C ignores the effect of noise. The statement in option D is inaccurate, as there is no scenario where more noise eventually increases data rate.

5. Practical Implications of Shannon’s Limit

   Which statement accurately reflects the practical significance of the Shannon-Hartley theorem for real-world communication systems?

   1. It predicts actual data rates achieved in all real-world systems.
   2. It guarantees any data rate below the capacity without the need for error-correcting codes.
   3. It assumes signals can be transmitted without any power consumption.
   4. It defines an upper bound for error-free data transmission at a given bandwidth and SNR.

   Explanation: The Shannon-Hartley theorem mathematically sets a theoretical maximum (upper bound) on error-free data transmission for a given bandwidth and SNR. Option B is wrong because achieving close to the bound typically requires sophisticated coding. Option C is unrelated to the theorem. Option D is incorrect as real systems often fall short of the theoretical limit due to practical imperfections.