Wireless Indeed: March 2015

Thursday 26 March 2015

Optimal Receiver for Additive White Gaussian Noise Channel

Sometimes students get confused while studying receivers design in Digital communication system. I have just tried to explain things in a concise and simple method.

Lets consider a receiver model as:

so received signal is the additive sum of transmitted symbols and white noise added by the channel.

suppose we received a symbol r and the transmitted symbol was s.

Probability of correct decision given symbol r is received can be written as P(s/r), mathematically

$\begin{array}{l} p(Correct - decision/r - received) = p(s - sent/r - received)\\ \Rightarrow p(correct - decision) = \int {p(correct - decision/r).p(r).dr} \end{array}$
$= \int {p(s - sent/r).p(r).dr}$

where , p(r) , probability density function of received vector which is always positive.

Thus , The optimal receiver will be one which maximizes ${p(s - sent/r).p(r)}$ .

The receiver knows the set of symbols which are being used by the transmitter to send data.To decide on s , it is designed to select s among all possible values of s, such that the conditional probability ${p(s - sent/r)}$ is maximum. Mathematically find $s\limits^\^$ such that,

$s\limits^\^ = {{\rm{argument }}\max }\limits_{1 \le s \le M} {\rm{ p(s/r)}}{\rm{.p(r)}}$

here , we can drop p(r) as the value of p(r) is always positive. Thus the decision rule equation becomes

$s\limits^\^ = {{\rm{argument }}\max }\limits_{1 \le s \le M} {\rm{ p(s/r)}}$
From the above rule equation we design system for MAP(Maximum aposterior probability) RULE and also ML(Maximum Likelihood) RULE.

Sunday 22 March 2015

Channel Equalizers

What are channel equalizers ?

If you search the word equalizers on google, it says "a thing that has an equalizing effect" . The same applies in relation to communication theory. When the transmitted signal pass through a dispersive channel , the symbols interfere between them resulting in Information loss. We call this as ISI or inter symbol interference. Equalizers are filters that help us reduce the ISI and equalize the symbols.

Linear Equalizer:

Zero Forcing(ZF) Equalizer
Minimum Mean Square Error(MMSE) Equalizer

Decision Feedback Equalizer :

Improves the Linear Equalizers by adding a new feedback filter which uses the feedback from prior decisions to cancel the interference due to past symbols. This in turn reduce the noise enhancement as compared to ZF and MMSE Linear Equalizers.

Lets try to understand it mathematically.

we can model a linear modulated signal over a dispersive channel as below:

where,
${B_n}$ - Transmitted bits
${G_{transmitter}}$ - Transmitter filter
${G_{Channel}}$ - Channel filter
AWGN- Additive white gausian noise n(t)
Rx- Received Symbols $r[n]$

We can write the receiver model as, $r[n] = Ub[n] + W[n]$

where ,
   $r[n]$   - received symbol vector
$U$ - Vector , whose each column gives the response on each
symbol from all the filters through which it passes. mathematically
$U = {G_{transmitter}} \otimes {G_{Channel}} \otimes {G_{{{\rm Re}\nolimits} ceiver}}$ .
G-Corresponding filters Models of Transmitter, Channel and Receiver.
All the elements of U Matrix are shifted in acyclic so that it gives contribution to the corresponding transmitted vector.
B[n]- transmitted symbols Vector. $(b[n - {k_1}],.........b[n],b[n + 1],..........b[n + {k_2}])$
W[n]- AWGN noise with Power Spectral density $\sigma _w^2$

we can write received symbols r[n] as,
   $r[n] = b[n]{u_0} + \sum\limits_{j \ne 0}^n {b[n + j]{u_j}} + w[n]$
Now the idea of Linear Equalizer is to design a filter such that the second term $\sum\limits_{j \ne 0}^n {b[n + j]{u_j}}$ is significantly smaller as compared to $b[n]{u_0}$ . To understand lets expand the Model stated above,

$r[n] = \left( {\begin{array}{ccccccccccccccc} {{a_{32}}}&{{a_{11}}}&0\\ 0&{{a_{22}}}&0\\ 0&{{a_{32}}}&{{a_{11}}} \end{array}} \right)b[n] + {w_n}$
$\Rightarrow r[n] = \left( {\begin{array}{ccccccccccccccc} {{a_{32}}}&{{a_{11}}}&0\\ 0&{{a_{22}}}&0\\ 0&{{a_{32}}}&{{a_{11}}} \end{array}} \right){[b[n - 1],b[n],b[n + 1]]^T} + w[n]$
$\Rightarrow r[n] = {[{a_{32}}b[n - 1] + {a_{11}}b[n],{a_{22}}b[n],{a_{32}}b[n] + {a_{11}}b[n + 1]]^T} + w[n]$
$\Rightarrow {r_1}[n] = {a_{32}}b[n - 1] + {a_{11}}b[n] + {w_1}[n]$ , similarly ${r_2}[n]$ and ${r_3}[n]$ .

The work of equalizer filter is to give us $b[n]$   from ${r_1}[n]$   and remove the influence of $b[n - 1]$ .

ZF : In case of Zero forcing equalizer , we design the ZF filter such that the term

$\sum\limits_{j \ne 0}^n {b[n + j]{u_j}}$ = 0 and $b[n]{u_0}$ has only contribution to $r[n]$ plus the additive noise for sure.

MMSE: The above explanation shows that the ZF Equalizer ignores the effect of noise at the output, thus to have a trade off between ISI and effect of Noise is the MMSE Equalizer. We design the equalizer filter such that Mean Square Error(MSE) , $MSE = E[|{F_{MMSE}}r[n] - b[n]{|^2}]$ is minimized to get the Filter ${F_{MMSE}}$ .

Sunday 15 March 2015

Poles and Stability of System

Why do we say that poles should be within unit circle for the system to be stable ?

let the Transfer function which is the designed filter in our system have a impulse response h(n), the system is stable only if h(n) tends to zero as n tends to infinity.

In terms of Z transform, consider a causal impulse response of the form

$h(n) = {r^n}{e^{jwnT}}$ ,n=0,1,2...

If r>1 , then the amplitude envelope will increase exponentially as ${r^n}$ ,

the Signal $h(n)$ has the Z-Transform,

$H(z) = \sum\limits_{n = - \infty }^{n = \infty } \begin{array}{l} \\ {r^n}{e^{jwnT}}{z^{ - n}}\\ \end{array} = \frac{1}{{1 - {r^n}{e^{jwT}}{z^{ - 1}}}}$

taking in , $|{r^n}{e^{jwT}}{z^{ - 1}}| < 1$ or $r < |z|$

thus the Transfer function consists of a single pole at $z = r{e^{jwT}}$ when $r < |z|$ , now consider $r \ge 1$ , the poles of H(z) moves out of unit circle and the impulse response has exponential increasing amplitude or the definition of stability fails. Thus for $r \ge 1$ , the Z transform no longer exists on the unit circle.

Above we considered the case with single pole, it will give the same result with multiple pole transfer function. To summarize all poles of an LTI System should lie withing the unit circle for a stable system.

Saturday 14 March 2015

Orthogonal and OrthoNormal Vectors

Orthogonal and OrthoNormal Vectors : When the dot product of two vectors are zero , we say it as orthogonal vectors. If the vectors are normalised to a value 1, the these same orthogonal vectors are called OrthoNormal vectors. mathematically,

any orthonormal set is orthogonal but reverse is not true.
any orthogonal set corresponds to a unique orthonormal set but an orthonormal set may correspond to different orthogonal sets.