Test: Channel Capacity - Electronics and Communication Engineering (ECE) MCQ

It states the channel capacity C, i.e. the theoretical highest upper bound on the information rate of data that can be communicated at an arbitrarily low error rate using an average received signal power S through an analog communication channel that is subject to additive white Gaussian noise (AWGN) of power N.
Mathematically, it is defined as:

C = Channel capacity
B = Bandwidth of the channel
S = Signal power
N = Noise power
Bayes’ theorem
It states that the conditional probability of an event, based on the occurrence of another event, is equal to the likelihood of the second event given the first event multiplied by the probability of the first event.
Parseval's Theorem:
For continuous-time, periodic signal, the energy is given by:

Where a_k is the Fourier series coefficient of x(t), and T is the period of the signal.
For average power in one period of the periodic signal x(t), we write:

∴ |a_k|² is the average power in the kth harmonic of x(t).
∴ Parseval's relation states that the total average power in a periodic signal equals the sum of the average powers in all of its harmonic components.

Shannon–Hartley theorem: 
It states the channel capacity C, i.e. the theoretical highest upper bound on the information rate of data that can be communicated at an arbitrarily low error rate using an average received signal power S through an analog communication channel that is subject to additive white Gaussian noise (AWGN) of power N.
Mathematically, it is defined as:

C = Channel capacity
B = Bandwidth of the channel
S = Signal power
N = Noise power

For a binary channel as shown below

P(y_n) = P(x_m)P(y_n/x_m) , n = 1,2 and m = 1,2
Where,
P(y_n) → Probability of receiving output y_n
P(x_m) → Probability of the transmitting signal
P(y_n/x_m) → Probability of the received output provided x was transmitted
P(x_m/y_n) = [P(y_n/x_m)× P(x_m)]/P(y_n)
Where;
P(x_m/y_n) → Probability of x_m transmitted given that y_n was received
Calculation of P(z₁);

The marked line shows the path from x₁ to z₁ (abc and aec) and x₂ to z₁ (dec and dbc)
Following these paths, we can calculate P(z₁)
P(z1) = [P(x1) {(0.6× 0.4) or(0.4 × 0.7}] or [P(x2){(0.3 × 0.4) or (0.7 × 0.7)}]
⇒   P(z1) = [0.7 {(0.6× 0.4) + (0.4 × 0.7}] + [0.3{(0.3 × 0.4) + (0.7 × 0.7)}]
∴ P(z₁) = 0.547
Similarly, P(z₂) can be calculated;
P(z₂) = [P(x1) {(0.6× 0.6) +(0.4 × 0.3}] + [P(x2){(0.3 × 0.6) + (0.7 × 0.3)}] = 0.453
Calculation of P(x₂/y₁):
P(y₁) = (0.7 × 0.6) + (0.3 × 0.3) = 0.51

• Channel capacity is the maximum rate at which the data can be transmitted through a channel without errors.
• The capacity of a channel can be increased by increasing channel bandwidth as well as by increasing signal to noise ratio.
• Channel capacity (C) is given as,
  

Where,
B: Bandwidth
S/N: Signal to noise ratio
Entropy:
The entropy of a probability distribution is the average or the amount of information when drawing from a probability distribution.
It is calculated as:

p_i is the probability of the occurrence of a symbol.

Shannon–Hartley theorem: 
It states the channel capacity C, meaning the theoretical highest upper bound on the information rate of data that can be communicated at an arbitrarily low error rate using an average received signal power S through an analog communication channel that is subject to additive white Gaussian noise (AWGN) of power N.
Mathematically, it is defined as:

C = Channel capacity
B = Bandwidth of the channel
S = Signal power
N = Noise power
∴ It is a measure of capacity on a channel. And it is impossible to transmit information at a faster rate without error.

Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature.
The modifiers denote specific characteristics: Additive because it is added to any noise that might be intrinsic to the information system.
The capacity of an additive white Gaussian noise channel by Shanon's formula:
 
Where B refers to the bandwidth of the channel
SNR means Signal to Noise Ratio : can be defined as the ratio of relevant to irrelevant information in an interface or communication channel.
SNR now can be calculated as,

In the above problem, S is the signal power which is equivalent to 10 W.
Bandwidth = 1 MHz
Noise power spectral density of No/2 = 10⁽⁻⁹⁾ W/Hz

Shannon–Hartley theorem: 
It states the channel capacity C, i.e. the theoretical highest upper bound on the information rate of data that can be communicated at an arbitrarily low error rate using an average received signal power S through an analog communication channel that is subject to additive white Gaussian noise (AWGN) of power N.
Mathematically, it is defined as:

C = Channel capacity
B = Bandwidth of the channel
S = Signal power
N = Noise power
∴ It is a measure of capacity on a channel. And it is impossible to transmit information at a faster rate without error.
So that Shannon’s Theorem sets limit on the maximum capacity of a channel with a given noise level.
The Shannon–Hartley theorem establishes that the channel capacity for a finite-bandwidth continuous-time channel is subject to Gaussian noise.
It connects Hartley's result with Shannon's channel capacity theorem in a form that is equivalent to specifying the M in Hartley's line rate formula in terms of a signal-to-noise ratio, but achieving reliability through error-correction coding rather than through reliably distinguishable pulse levels.
Bandwidth and noise affect the rate at which information can be transmitted over an analog channel.
Bandwidth limitations alone do not impose a cap on the maximum information rate because it is still possible for the signal to take on an indefinitely large number of different voltage levels on each symbol pulse, with each slightly different level being assigned a different meaning or bit sequence.
Taking into account both noise and bandwidth limitations, however, there is a limit to the amount of information that can be transferred by a signal of a bounded power, even when sophisticated multi-level encoding techniques are used.
In the channel considered by the Shannon–Hartley theorem, noise and signal are combined by addition. That is, the receiver measures a signal that is equal to the sum of the signal encoding the desired information and a continuous random variable that represents the noise.

Channel capacity:

• Channel capacity is the maximum rate at which the data can be transmitted through a channel without errors.
• The capacity of a channel can be increased by increasing channel bandwidth as well as by increasing signal to noise ratio.
• Channel capacity (C) is given as, 

Where,
B: Bandwidth
S/N: Signal to noise ratio

Entropy:
The entropy of a probability distribution is the average or the amount of information when drawing from a probability distribution.
It is calculated as:

p_i is the probability of the occurrence of a symbol.

Shannon-Hartley Theorem- It tells the maximum rate at which information can be transmitted over a communications channel of a specified bandwidth in the presence of noise.

Where C is the channel capacity in bits per second
B is the bandwidth of the channel in hertz
S is the average received signal power over the bandwidth
N is the average noise
S/N is the signal-to-noise ratio (SNR)
For transmitting data without error R ≤ C where R = information rate
Assume information Rate = R
Form Shanon hartley theorem
R_max = C

When channel Band width (B) = B₁
Signal to noise ratio = SNR₁

Shannon’s channel capacity is the maximum bits that can be transferred error-free. Mathematically, this is defined as:

B = Bandwidth of the channel
S/N =Signal to noise ratio
Note: In the expression of channel capacity, S/N is not in dB.
Calculation:
Given B = 4 kHz and SNR = 255
Channel capacity will be:

C = 32 kbits/sec