**Q1. what is hamming distance? mention it’s significance.**

**Ans:**Hamming weight:- number of non zero elements in a code word.

Hamming distance:- It is defined as the number of bits in which two code vectors differ

i.e, X = [1 0 1 0 1] and Y=[1 1 1 0 1] then the hamming distance between code vectors X and Y is ‘ 1 ‘.

i.e, X_{1} = [1 0 1 0 ] and X_{2} =[1 1 1 1] then the hamming distance between code vectors X_{1} and X_{2} is ‘ 2’.

Significance:- It is important because error detection and correction is possible if

where t is the hamming weight and d_{min} is the minimum hamming distance between two code words.

**Q2. what is Bit Error rate?**

**Ans:** In telecommunication transmission, the bit error rate (BER) is the percentage of bits that have errors relative to the total number of bits received in a transmission, usually expressed as ten to a negative power.

Bit Error Rate (BER) = (number of bits transmitted in error/ total number of bits transmitted).

For example, a transmission might have a BER of 10 to the minus 4, meaning that, out of 1,0,000 bits transmitted, one bit was in error. The BER is an indication of how often a packet or other data unit has to be re transmitted because of an error.

**Q3. What is Probability of error in Digital Communications?**

**Ans:**In digital Communications data to be transmitted is digital either a ‘0’ or ‘1’, when a digital signal is being transmitted during the transmission the signal being corrupted by some sort of random process( noise), these processes can be quantified by their Probability Density Function (pdf), such as Gaussian, Uniform, Rayleigh etc. Depending on the medium through which a signal travels, it is attacked by these random process, akin to adding a random signal to the one transmitted, Now the bit that was of say amplitude V(‘1’) may be changed to either V+(‘1’) or V-(‘0’). This may or may not cause an error at the receiver. This depends on the way the bit is mapped to symbols, a process called de-modulation.

Now probability of error is defined as the probability of receiving the bit incorrect at the receiver.

i.e, when a ‘0’ is transmitted at the transmitter if that bit ‘0’ is received at the receiver there is no error occurred during the transmission , instead of receiving a ‘0’ if a ‘1’ has been received then there is an error occurred in the transmission. Now, Probability of error gives the extent to which a ‘1’ has to be received in place of a ‘0’ and vice-versa. P_{e} gives the rate of making a bad decision.

By Baye’s rule of probability, P_{e} states the overall rate of making a bad decision

Overall Probability of error

Probability of receiving ‘1’ when a ‘0’ has been transmitted also known as Conditional probability.

Probability of receiving ‘0’ when a ‘1’ has been transmitted also known as Conditional probability.

Probability of ‘0’ and probability of ‘1’

If equal number of 0’s and 1’s are sent (0’s and 1’s are equi probable)

then , which gives , To evaluate this expression , an assumption is needed that is provided that the threshold is exactly in the middle then

**Q3. What is the need for MSK(Minimum Shift Keying) in Digital Communications?**

**Ans: Need for MSK:-**

Linear modulation schemes without memory like QPSK, OQPSK, DPSK and

FSK exhibit phase discontinuity in the modulated waveform. These phase transitions

cause problems for band limited and power-efficient transmission especially in an

interference limited environment. The sharp phase changes in the modulated signal cause relatively prominent side-lobe levels of the signal spectrum compared to the main lobe. In a cellular communication system, where frequency reuse is done extensively, these side lobe levels should be as small as possible. Further in a power-limited environment, a nonlinear power amplifier along with a band pass filter in the transmitter front-end results in phase distortion for the modulated signal waveform with sharp phase transitions. The abrupt phase transitions generate frequency components that have significant amplitudes. Thus the resultant power in the side-lobes causes co-channel and inter-channel interference.

Consequently, in a practical situation, it may be necessary to use either a linear power amplifier or a non-linear amplifier using extensive distortion compensation or selective pre-distortion to suppress out-of-band frequency radiation. However, high power amplifiers may have to be operated in the non-linear region in order to improve the transmission power. Continuous phase modulation schemes are preferred to counter these problems.

Continuous Phase Frequency Shift Keying (CPFSK) refers to a family of

continuous phase modulation schemes that allow use of highly power-efficient non-linear power amplifiers. Minimum Shift Keying (MSK) modulation is a special subclass of CPFSK modulation and MSK modulation is free from many of the problems mentioned above.

Minimum Shift Keying is a form of Continuous Phase -Frequency Shift Keying that is used in a number of applications, A version of MSK modulation known as GMSK – Gaussian filtered Minimum Phase Shift Keying is used for a number of radio Communication applications including GSM cellular Tele Communication systems.

In MSK output wave form is continuous in phase here there are no phase discontinuities because the frequency changes occur at the carrier zero crossing points. hence there are no abrupt changes in Amplitude. The side lobes of MSK are very small hence Band pass filtering is not required to avoid inter channel interference.

MSK Ouput looks like no phase distortion existing.