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NET & GATE: Communications (EC) Study Material
What is an error in a communication system?
In wireless communication, we sometimes receive the wrong bit, i.e., the transmitter sends binary '1', but we're receiving binary '0' on the receiver side. That is called a bit of error. Now, we'll tell you why this error occurs. We are all aware of signal attenuation and additive noise in wireless communication. You also know that we use a threshold level at the receiver to detect '0' or '1'. Anyhow if the signal is much affected by attenuation or noise, then we receive binary '0' instead of '1' and vice versa.
We commonly use the term 'bit error rate' to measure bit error. Bit error rate tells us how many bits are affected among the total number of bits transmitted.
What are the possible remedies to reduce the bit error rate?
Channel Coding
Question
There is a digital communication system that sends a symbol or block of N bits. We expect the error probability in decoding to be 0.0001. But there is N number of bits in a symbol or block. And here, the occurrence of a mistake of any bit is independent of others. If we came to know at least one bit in the block/symbol has been decoded wrongly. Then what probability will the received symbol/block be erroneous?
Answer
Error probability of a bit = 0.0001
So, the probability of being decoded correctly= is 1-0.0001
As there are several bits, so correct probably = (1-0.0001)(1-0.0001)(1-0.0001)...Up To N Times
=(1-0.0001)^N
Erroneous Probability = 1 - correct probability
=1 - (1-0.0001)^N
Maximum Likelihood Decoding or ML Decoding
The decision boundary between two adjacent signal points will be their arithmetic mean.
Question
The S symbol is randomly selected from the S1, S2, S3, and S4 and communicated through a digital communication system. S1=-3, S2=-1, S3=+1, and S4=+2 are given. Y = S + W is the received symbol on the receiver side. W stands for "zero mean unit variance." When the transmitted symbol S = Si, the conditional probability of symbol error for maximum likelihood (ML) decoding is P. P is a Gaussian Random Variable independent of S. The index i with the highest conditional symbol error probability Pi is -----
Answer
As an ML detector is used, the decision boundary between two adjacent signal points will be their arithmetic mean.
For S1= -3, the probability of error, P1,
As the ML decoder first receives the symbol -3 and then -1, so -2 becomes the decision boundary, as shown in the figure below.
If the signal value lies between -∞ to -2, it is decoded correctly as -3. Otherwise, an error occurs.
Now, the probability of error, P1 = (1 - yellow-colored area)
For S2, probability of error is P2 (say)
For S3, probability of error is P3 (say)
P3=(1-yellow-colored area)
For S4, probability of error is P4 (say)
P4 = (1 - yellow-colored area)
In the concussion of the above four graphs, the probability of correctness is less for S3 among the four symbols. So, the possibility of error for S3 is more significant, or P3 is more considerable.
Probability & Information
When the probability of an event is less, then information about that event will be more.
I(x) is inversely proportional to p(x)
When probability = 1, the information will be zero, and vice versa.
.We commonly use the term 'entropy' in information theory. Entropy denotes the average number of bits required per symbol to transfer information.
For example:
The probability of receiving bit '1' is 0.5 & probability of receiving bit '0' is 0.5 on the receiver side. Then the entropy H(x) is going to be -0.5*log(0.5) -0.5*log(0.5) = 1 bit/symbol
Electronic Devices
Networks, Signal, & Systems
Superposition Theorem
In the superposition theorem, we calculate the individual response of each independent source on an element or branch. Then we sum up the voltage and current.
Thevenin's Theorem
In Thevenin's theorem, we basically find the Vth and Rth. Procedure for thevenin's theorem
1. Firstly, we open the circuit, the load
2. Then we find Vth across the load from the circuit
3. Then, we open the circuit's current source and short-circuit the voltage sources. Remember this step is only applicable to independent sources.
4. Then, we find Rth from the circuit.
RL circuit with source:
i(t) = [ i(0+) - i(∞)]*exp(-Rt/L) + i(∞)
v(t) = [ v(0+) - v(∞)]*exp(-t/RC) + v(∞)
The main functions of the inductor and capacitor in a circuit are to prevent the sudden change of current and voltage, respectively.
Question:
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A similar rule is applicable for inductors also. When the switch is transformed from an off to an on state, the voltage across the inductor will be exact, but the current direction will be reversed.
Question:
Find the rate of rise of voltage across the capacitor at t = 0+
