X Channel, Z Channel and Interference Channel

When we talk about multi-user communication, there are many users transmitting at the same time and same frequency. Based on the way how the signals are transmitted and received, different channels are defined in the literature.

Let us consider a simple case when there are two transmitters and two receivers. At any time instance, both the transmitters transmit and both the receivers receive. However, all the data streams that have been transmitted may not be desired by both the receivers. Also, all the channels linking a particular transmitter and the receiver may not be of the equal strength. Those channels that carry the unwanted data streams or those channels that are linked between the receiver and the undesired transmitters are called the undesired channels or the interference channels.

In the figure shown below,Transmitter T1 has message m1 intended to Receiver R1 and Transmitter T2 has message intended to receiver R2 and they are transmitting at the same time and with the same frequency.  This causes interference to the Receiver R1 from T2 and and interference to the Receiver R2 from T1, these are indicated by the dotted lines and the solid lines show the channels between the desired transmitters and the receivers in the figure. Such a channel is referred to as Interference Channel.

Interference channel

In the Z channel, the Transmitter T1 has message intended to Receiver R1only thus interfering to the receiver R2 but the Transmitter T2 has the messages intended to both the Receivers R1 and R2 or can be the other way. In the figure below, the interfering channel is shown by the dotted line and there is only one interference channel while remaining 3 are the desired channels which are shown by the solid lines. Here Receiver R1 should decode two messages from T1 and T2 but the receiver R2 should decode only one message from T2.

Z channel

The other channel is the X channel. In the X channel, both the Transmitters T1 and T2 have messages intended for both the receivers R1 and R2. It means R1 has to decode both the messages from T1 and T2 and R2 also has to decode both the messages from T1 and T2. It seems pretty interesting to have X channel, if we can send the messages at the same time to different users and also the receivers can receive the messages at the same time from different users, unlike broadcast and multiple access channels. X channel has special significance because of its higher and non-integer degrees of freedom, and hence higher capacity. The X channel is shown in the diagram below, here mij is the message from transmitter j to receiver i.

X channel

Ergodicity

The term ergodicity is vague in itself and possess different meanings in different fields. The word is derived from the Greek word  'ergon' (work) and 'odos' (path), and  was coined by Austrian physicist Ludwig Boltzmann.  There is a separate course in Sigma algebra to deal with ergodicity in detail. In plain words, ergodicity is a term that describes the system which possess same behavior as averaged over time and space.

The main purpose of this article is to make the readers acquainted with the term ergodic and ergodicity, statistically and literally. 

Let us consider an example to understand the concept of ergodicity. This example is given in http://news.softpedia.com/news/What-is-ergodicity-15686.shtml . I  am mentioning the same here because it explain the concept of ergodicity in simple words. 

Suppose we are concerned with determining the most visited parks in the city. One possible way to do so is to follow a single person for a month and see which park does he visit most in that period of time and the other way of figuring out is to take a momentary snapshot of all the parks at a time and see which parks got most of the people at that particular instant of time.  Here, we have got two analyses, one is the statistical data of an individual for a certain period of time and the other is the statistical data of an entire ensemble of people at a particular moment in time. 

So, we see that the situation one is for only a single person,which may not be valid for a large number of people and the statistics depend only on the single individual to whom we considered. While in the situation two it may not be valid for a longer period of time. We are considering only a short span and in the long run, the result may differ. So it is not sure that the statistical results we obtain from both the observations should be the same or alike.

We say that an ensemble is ergodic if both the situations we considered here gives the same statistical results i.e  when the temporal and spatial statistics of an event or a number of samples are same then the event or samples are called ergodic. In simple words, ergodic refers to the dynamical system that has the same behavior averaged over time and averaged over space.

The initial concept in ergodicity is the Poincaré recurrence theorem in statistics which states that certain systems after a sufficiently long time, return to a state very close to the initial state. It means after the long time, the time average exist almost everywhere in the space and is almost the same as the space average.

In signal processing the stochastic process is said to be ergodic if its statistical properties can be deduced from a single long sample or realization of the process (Source wikipedia). If we would like to determine the mean of a stationary stochastic process, we observe a large number of samples and use their ensemble average. However if we have access to only their single sample over a certain period of time, can this time average be used as an estimate for the sample mean. For the ergodic process, the time average should be the estimate of the sample mean. 

 A random process X(t) is said to be ergodic in the mean, i.e., first-order ergodic if the mean of sample average asymptotically approaches the ensemble mean:

 i.e.  lim T tends to infinity E(µxTau) =µx(t)

and lim T tends to infinity var(µxTau) =0      

Conditional Probability

The concept of conditioning in Statistics is of great importance in many fields of applications, such as, information theory, finance, random signal processing etc. In most of the cases, when the events are random and dependent and we want to specify the probability of occurrence of one given the other event, we use the concept of conditional probability.

Consider there are red and black balls in a box. We are picking the balls at random form the box and we do not put back in again, then the two events i.e. picking a red ball and picking a black ball at random are dependent. When we pick the first ball, the probability does not depend on the previous event as it is the initial pick and probability of picking any one ball is equally likely or in other words, they are uniformly distributed.

But when we pick the  second ball, there is now one ball less in the box and the probability of occurrence of the second event is dependent on the first event. In other words, we can condition the occurrence of event II on the known result of event one. We can now determine "what is the probability of occurrence of event II given the event I has already occurred?". This approach of determining the probability of occurrence of any event is called the conditional probability.

Truly speaking, the concept of conditioning is the heart of probability theory and we can use this concept to define the law of total probability based on the Bayes' rule.

Let us consider a combined experiment in which two events $A$ and $B$ occur with joint probabilities $P(A,B)$. Joint probability is the probability of occurring both the events. The probability of occurrence of  $B$ given $A$  is the conditional probability and is represented as $P(B|A)$, which is expressed in terms of the joint probability as :

                               $P(B|A)=\frac{P(A,B)}{P(A)}$.                     

 Similarly, the probability of occurrence of A conditioned on B is mathematically expressed as

                              $P(A|B)=\frac{P(A,B)}{P(B)}$.             

In each case $P(A), P(B) > 0$, because the value of probability always lies between 0 and 1 (axiomatic definition). Combining above two conditions, we can write 

                             $P(B|A)P(A)=P(A|B)P(B)$ 

                            $\Rightarrow P(B|A)=\frac{P(A|B)P(B)}{P(A)}.$                       

This is called the famous Bayes' Rule in statistics.

Say $A=\{1,2,3\}$ and $B=\{3,4,5\}$ are the two events occurred when rolling a die. It means $P(A)=3/6=1/2$ and $P(B)=3/6=1/2$, they are independent events and $P(A,B)=1/6$

                         so, $P(B|A)=\frac{P(A,B)}{P(A)}=\frac{1/6}{1/2}=\frac{1}{3}$.

and for mutually exclusive events, $P(A,B)=\phi=0$. So conditional probability is zero in such case.

If $P(A,B)=P(A)$ i.e A is a subset of B, then $P(B|A)=P(A)/P(A)=1.$

and if $P(A,B)=P(B)$ i.e. B is a subset of A, then $P(B|A)=P(B)/P(A).$

Conditional probability is also useful in checking the statistical independence. If the occurrence of A does not depend on the occurrence of B then

              $P(A|B)=P(A)$ 

       and
           $P(A,B)= P(A)P(B)$

Law of total probability states that if there are $n$ mutually exclusive events represented as $E_1, E_2 \cdots E_n$, the sum of the probabilities of which sum to 1 and $F$ is some other arbitrary event, then the probability of $F$ is expressed in terms of the conditional probability as:
$P(F)=P(F|E_1)P(E_1)+ P(F|E_2)P(E_2) + \cdots + P(F|E_n)P(E_n) $.