Communication - Topic 3/4

Question 1

What is the setup of a Baseband Digital Reciever?

Answer 1

Question 2

What does white noise mean?

Answer 2

noise containing many frequencies with equal intensities.

Question 3

Baseband Digital Reciever: How is the signal component of a Matched Filter determined?

Answer 3

Question 4

The response of a linear filter to any signal is going to be given by…

Answer 4

convolution

Question 5

Explain how this input signal x(t) is created

Answer 5

What this is saying is that x(t) is equal to a sequence of pulses who’s ampliude is being modulated in accordance to the different symbols that are encoding the bits.

In other words, p(t) is just a generic set of pulses generated over time and by multiplying it with the sequence of symbols X_k we get the desired signal.

Question 6

Baseband Digital Reciever: How is the noise component of a Matched Filter determined?

Answer 6

Convolution

Question 7

What does the switch in the middle do?

Answer 7

The switch says that every Ts seconds you are going to observe the output of the filter to make a decision.

Question 8

How is thermal noise generated and what shape can it be modeled by?

Answer 8

Thermal noise is generated in electric devices by the vibration (thermal agitation) of electrons.

Thermal noise can be modelled by a zero-mean stationary white Gaussian process with spectral density

S_N(f) = N₀/2 = kT/2

Question 9

Baseband Digital Reciever: Given a matched filter, what is the maximum posible output SNR?

Answer 9

maximum possible SNR is equal to 2*Ɛ_p/N₀

which is the energy of the signal divided by the noise spectral density

Question 10

Correlation and Dump Filter Setup

Answer 10

Question 11

Baseband Digital Reciever: What is the signal component output of a correlation and dump filter?

Answer 11

Question 12

Baseband Digital Reciever: What is the noise component output of a correlation and dump filter?

Answer 12

Note how the noise component of a Correlation and Dump filter is equivalent to a Matched Filter

Question 13

What is the probability density function of Noise N_k ?

Answer 13

Question 14

What is the probability density function of Y_k given X_k ?

What is it useful for?

Answer 14

You can determine the distribution of Y_k for all the values of X_k .

Question 15

MAP Detector/Decision Rule

What is it ‘optimal’ and what are its equations?

Answer 15

MAP decision rule is optimal in the sense that it minimizes the (symbol) error probability.

Question 16

MAP Decision Rule: What does the function mean and how does it work?

Answer 16

The argmax() function is going to select the symbol (taken from the available symbols) that maximises the specific function it is acting on. In this case, the argmax() is applied to conditional distribution of X_k occurring given Y_k .

Each possible transmitted symbol needs to be substituted in the function, and whatever outputs the largest value is to be chosen.

For example, in the binary case, for X_k = -1 you evaluate the result, for X_k = 1 you evaluate, and then you select the one which maximises the function.

In essence, we are looking at whether the value is due to noise or not. Seeking to maximise the probability that a value might be correct, reduces the probability of an error.

Question 17

ML Decision Rule: What is it and what is its function?

Answer 17

You use ML when you do not know the individual probabilities of the symbols being transmitted.

It is not optimal as it does not minimise the error probability in general.

When all the symbols are equally likely using MAP, the values obtained are equal to ML and means the ML decision rule is optimal in this case.

Question 18

Given an alphabet A = {-A,A} how do you determine the decision regions/thresholds associated with these symbols?

In other words, under what conditions does the Decision Device declare that symbol A has been transmitted?

Answer 18

It produces the estimate X_k = A.

Whenever the output of the matched filter Y_k is greater than the RHS it declares that the symbol A has been transmitted.

The equation for Y_k is determined by equating the argmax of A and -A with a greater than or equal to symbol.

Question 19

What is the probability density function of a conditional

Answer 19

Question 20

How do you determine the probability of error when the symbol X_k = A is transmitted?

Answer 20

It involves taking the integral of the probability density over the range of the Gaussian distribution which does not lie in the decision region. This is the equivalent to finding the area under the distribution.

Question 21

Error Probability: What does this graph show us?

Answer 21

This is a graph of the signal-to-noise ratio vs the probability of an error.

It shows that the unipolar scheme needs roughly 3db more to maintain the same error probability.

Similarly, the quaternary scheme needs a larger signal-to-noise ratio to get the same error probability as the unipolar.

Question 22

Baseband Digital Reciever: What does a Matched Filter do?

Answer 22

It maximizes the SNR when you sample its output at a certain instant of time.

Question 23

What is the essence behind filters in Baseband Digital Recievers?

Answer 23

Each pulse creates an impulse response, this impulse response is sampled periodically to reconstruct the signal without noise.

Question 24

What is the decision region?

Answer 24

The decision region partitions the output space into M regions. Whenever the output is in region M, the estimate of the device is going to be D_M .

Question 25

Error Probability: What is the issue regarding the number of levels?

Answer 25

The more levels to the scheme the more it will be affected by noise, so the scheme will require more energy to maintain the same error probability.

However, the advantage of more levels is a better bandwidth efficiency.

So there needs to be a compromise between energy efficiency (related to error probability) and bandwidth efficiency.

Question 26

Why is it that you can encode 2 bits with 4 symbols?

What is the name of this mapping scheme?

What is the advantage of using this scheme over a Binary Polar Scheme?

Answer 26

There are four different values you can get with 2 bits when binary is used.

00, 01, 10, 11

Each of these values can have an associated symbol.

Thus 2 bits can be encoded in one signal interval.

Name: Quaternary Scheme

The advantage is that the Quaternary scheme can use less bandwidth to send the same signal, thus it has higher bandwidth efficiency.

Question 27

Matched Filter: What do you need to do to minimise noise as much as possible?

Answer 27

To minimise noise as much as possible the SNR needs to be maximised.

To maximies SNR, P² needs to be at its maximum an the upper bound of the cauchy-schwartz inquality needs to be hit.

It can be satisfed by setting the response of the filter to… (see image)

From this we can see that the filter needs to be proportional to the frequency response of the signal/pulse you’re trying to detect.

The impulse response of the filter can be determined by taking the inveres FT of this equation.

Question 28

What is the concept of the Matched Filter?

Answer 28

The matched filter maximises SNR (thus minimising noise) by matching the frequency of the filter to the frequency of the input signal (the pulse you are trying to detect).