# 0.9 Error analysis of digital communications

 Page 1 / 1
In this module, we first introduce the eye diagram and constellation diagram as qualitative ways of evaluating the symbol error probability of a digital communication system. We discuss various symbol alphabets, such as QAM, PAM, and PSK, and their associated decision regions. Finally, we derive the symbol error probability for PAM and QAM in additive white Gaussian noise, using the Q and erfc functions, and discuss Gray coding.

Recall the figure below, from the module Discrete-Time Implementation of Digital Communication . When the channel is trivial and noiseless and thepulses satisfy the Nyquist criterion (i.e., $g\left[k\right]*q\left[k\right]=\delta \left[k\right]$ ), the digital comm system will work perfectly, yielding $y\left[n\right]=a\left[n\right]$ .

In practice, however,

• the pulses $g\left[k\right]$ and $q\left[k\right]$ will be truncated to finite length,
• the channel will not be trivial (i.e., $\stackrel{˜}{h}\left[k\right]\ne \delta \left[k\right]$ ), and
• the channel will not be noiseless (i.e., $\stackrel{˜}{w}\left[k\right]\ne 0$ ),

leading to $y\left[n\right]\ne a\left[n\right]$ , in which case we must infer the value of $a\left[n\right]$ from the received samples ${\left\{y\left[m\right]\right\}}_{m=-\infty }^{\infty }$ . For now, we consider using only the single sample $y\left[n\right]$ to infer $a\left[n\right]$ .

Key question: What are the mechanisms by which errors are made?

To better understand error behavior, we can plot the “eye diagram” or the “constellation diagram” and calculate the symbol error rate (SER).

## Eye diagrams

Usually used when $a\left[n\right]\in \mathbb{R}$ , the eye diagram is a plot which superimposes T -second segments of $Re\left\{y\left(t\right)\right\}$ over the time intervals $t\in \left[nT-\frac{T}{2},nT+\frac{T}{2}\right)$ for many integers n .

In MATLAB, the eye diagram can be made by superimposing P -sample segments of $Re\left\{{y}_{↑}\left[k\right]\right\}$ corresponding to intervals $k\in \left\{nP-\frac{P}{2},\cdots ,nP+\frac{P-1}{2}\right\}$ for many n . (Usually $P\ge 8$ .)

If $a\left[n\right]\in \mathbb{C}$ , eye diagrams can be plotted for both the “I” and “Q” channels using $Re\left\{y\left(t\right)\right\}$ and $Im\left\{y\left(t\right)\right\}$ , respectively.MATLAB for digital mod/demod with eye diagram:

## Constellation diagrams

The constellation diagram is a plot of $Im\left\{y\left[n\right]\right\}$ vs. $Re\left\{y\left[n\right]\right\}$ for many integers n . When the comm system is working well, the points cluster around thesymbol alphabet values:

Recall that $y\left[n\right]\in \mathbb{C}$ due to the complex-baseband channel model, regardless of whether $a\left[n\right]\in \mathbb{R}$ or $a\left[n\right]\in \mathbb{C}$ .

Sometimes it is instructive to superimpose a plot of $Im\left\{{y}_{↑}\left[n\right]\right\}$ vs. $Re\left\{{y}_{↑}\left[n\right]\right\}$ , which approximates the trajectory of $y\left(t\right)$ in the complex plane:

MATLAB for digital mod/demod with constellation diagram:

PAM : “pulse amplitude modulation”
PSK : "phase shift keying"

Note that: "QPSK" = 4-QAM = 4-PSK and "BPSK" = 2-PAM = 2-PSK

When the alphabet entries are spaced by Δ and picked with equal probability, the symbol variance ${\sigma }_{a}^{2}={E\left\{|a\left[n\right]|}^{2}\right\}$ obeys:

 alphabet M 2 -QAM M -PAM M -PSK σ a 2 $\frac{{\Delta }^{2}}{6}\left({M}^{2}-1\right)$ $\frac{{\Delta }^{2}}{12}\left({M}^{2}-1\right)$ $\frac{{\Delta }^{2}}{4{sin}^{2}\left(\pi /M\right)}$

## Decision regions

A reasonable way to infer the transmitted symbol $a\left[n\right]$ from the received sample $y\left[n\right]$ is to decide that $a\left[n\right]$ was the alphabet element nearest to $y\left[n\right]$ .

Nearest-element decision making is equivalent to using decision regions whose boundaries are equidistant from the two nearest alphabet elements:

When $a\left[n\right]=a$ , the symbol error rate (SER) equals the probability that $y\left[n\right]$ lies outside the decision region corresponding to alphabet member a . Writing $y\left[n\right]=a\left[n\right]+e\left[n\right]$ , we represent the cumulative effect of noise and ISI by the error $e\left[n\right]$ . Usually we model $e\left[n\right]$ as a Gaussian random variable with mean 0 and variance σ e 2 .

## Symbol error rate (ser) for M -pam

Let's first consider an M -PAM alphabet, where $a\left[n\right]\in \mathbb{R}$ . Since the decision regions show that $Im\left\{y\left[n\right]\right\}$ is not useful, we'll consider only the real parts of $y\left[n\right]$ and $e\left[n\right]$ .

When $a\left[n\right]=a$ , we have $y\left[n\right]=a+e\left[n\right]$ , implying that $y\left[n\right]$ is Gaussian with mean a and variance σ e 2 , abbreviated as “ $\mathcal{N}\left(a,{\sigma }_{e}^{2}\right)$ ”. This is illustrated below for the case of 4-PAM:

Formally, we say that ${p}_{y\left[n\right]|a\left[n\right]}\left(y|a\right)$ , the probability density function (pdf) of $y\left[n\right]$ conditioned on $a\left[n\right]=a$ , obeys

${p}_{y\left[n\right]|a\left[n\right]}\left(y|a\right)=\underset{\mathcal{N}\left(a,{\sigma }_{e}^{2}\right)}{\underbrace{\frac{1}{\sqrt{2\pi {\sigma }_{e}^{2}}}exp\left(-,\frac{{\left(y-a\right)}^{2}}{2{\sigma }_{e}^{2}}\right)}}.$

Basically, ${p}_{y\left[n\right]|a\left[n\right]}\left(y|a\right)$ tells us how likely it is that $y\left[n\right]=y$ given that $a\left[n\right]=a$ .

Consider first the case where a is an “interior” (not an “edge”) element of the symbol alphabet.Given that $a\left[n\right]=a$ , we make an error when $y\left[n\right] or when $y\left[n\right]>a+\frac{\Delta }{2}$ . To find the probability of the latter error event, i.e.,

$Pr\left\{y\left[n\right]>a+\frac{\Delta }{2}\phantom{\rule{3.33333pt}{0ex}}\left|\phantom{\rule{3.33333pt}{0ex}},a,\left[,n,\right],=,a\right},$

we integrate ${p}_{y\left[n\right]|a\left[n\right]}\left(y|a\right)$ over $y\in \left(a+\frac{\Delta }{2},\infty \right)$ :

${\int }_{a+\frac{\Delta }{2}}^{\infty }\underset{\mathcal{N}\left(a,{\sigma }_{e}^{2}\right)}{\underbrace{{p}_{y\left[n\right]|a\left[n\right]}\left(y|a\right)}}dy={\int }_{a+\frac{\Delta }{2}}^{\infty }\frac{1}{\sqrt{2\pi {\sigma }_{e}^{2}}}exp\left(-,\frac{{\left(y-a\right)}^{2}}{2{\sigma }_{e}^{2}}\right)dy.$

The integral represents the shaded area below:

This integral is often solved via

${\int }_{x}^{\infty }\underset{\mathcal{N}\left(\mu ,{\sigma }^{2}\right)}{\underbrace{\frac{1}{\sqrt{2\pi {\sigma }^{2}}}exp\left(-,\frac{{\left(y-\mu \right)}^{2}}{2{\sigma }^{2}}\right)}}dy=Q\left(\frac{x-\mu }{\sigma }\right),$

using the “Q function”:

While the Q function is not represented in MATLAB, it can be calculated using the “complementary error function” $erfc$ :

$Q\left(x\right)=\frac{1}{2}erfc\left(\frac{x}{\sqrt{2}}\right)$

In any case, the latter error event occurs with probability

$\begin{array}{ccc}\hfill Pr\left\{y\left[n\right]>a+\frac{\Delta }{2}\phantom{\rule{3.33333pt}{0ex}}\left|\phantom{\rule{3.33333pt}{0ex}},a,\left[,n,\right],=,a\right}& =& Q\left(\frac{\left(a+\frac{\Delta }{2}\right)-a}{{\sigma }_{e}}\right)\hfill \\ & =& Q\left(\frac{\Delta }{2{\sigma }_{e}}\right).\hfill \end{array}$

By symmetry, the former error event probability is also

$\begin{array}{ccc}\hfill Pr\left\{y\left[n\right]

Since these two events are disjoint, the probability of making a decision error on an interior symbol equals their sum:

$Q\left(\frac{\Delta }{2{\sigma }_{e}}\right)+Q\left(\frac{\Delta }{2{\sigma }_{e}}\right)=2Q\left(\frac{\Delta }{2{\sigma }_{e}}\right).$

For edge symbols, we experience half the decision error probability, since there is only one decision boundary to cross.

Finally, we average over the conditional error probabilities:

$\begin{array}{ccc}\hfill Pr\left\{\text{error}\right\}& =& \sum _{a\phantom{\rule{0.166667em}{0ex}}\in \phantom{\rule{0.166667em}{0ex}}\text{alphabet}}Pr\left\{\text{error}|a\left[n\right]=a\right\}\underset{=1/M\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}a}{\underbrace{Pr\left\{a\left[n\right]=a\right\}}}\hfill \\ & =& Q\left(\frac{\Delta }{2{\sigma }_{e}}\right)·\frac{2}{M}+2Q\left(\frac{\Delta }{2{\sigma }_{e}}\right)·\frac{M\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}2}{M}\hfill \\ & =& 2\left(\frac{M\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}1}{M}\right)Q\left(\frac{\Delta }{2{\sigma }_{e}}\right),\hfill \end{array}$

Using ${\sigma }_{a}^{2}=\frac{{\Delta }^{2}}{12}\left({M}^{2}-1\right)$ , we can finally write

${\text{SER}}_{M\text{-PAM}}=2\left(\frac{M\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}1}{M}\right)Q\left(\sqrt{\frac{3}{\left({M}^{2}-1\right)}\frac{{\sigma }_{a}^{2}}{{\sigma }_{e}^{2}}}\right).$

## Symbol error rate (ser) for M 2 -qam

With QAM, we have complex-valued $y\left[n\right]$ , $a\left[n\right]$ , $e\left[n\right]$ . We'll assume that $Re\left\{e\left[n\right]\right\}$ and $Im\left\{e\left[n\right]\right\}$ are uncorrelated and equal variance.To calculate SER, we can re-use the PAM approach with a few modifications:

1. integration is done on the complex plane,
2. σ e 2 -variance $e\left[n\right]$ $⇒$ $\frac{{\sigma }_{e}^{2}}{2}$ -variance $Re\left\{e\left[n\right]\right\}$ & $Im\left\{e\left[n\right]\right\}$ ,
3. M 2 -QAM has 4 corner points, $4\left(M-2\right)$ edge points, and ${M}^{2}-4M+4$ interior points,
4. calculate $Pr\left\{\text{error}|a\left[n\right]=a\right\}$ via $1-Pr\left\{\text{correct}|a\left[n\right]=a\right\}$ , since the regions of integration are simpler:

After a bit of algebra, we find

$\begin{array}{ccc}\hfill {\text{SER}}_{{M}^{2}\text{-QAM}}& =& 1-{\left[1,-,2,\left(\frac{M-1}{M}\right),Q,\left(\sqrt{\frac{3}{\left({M}^{2}-1\right)}\frac{{\sigma }_{a}^{2}}{{\sigma }_{e}^{2}}}\right)\right]}^{2}.\hfill \end{array}$

## Bit error rate (ber) and gray coding

With an M -ary alphabet, there are ${log}_{2}M$ bits per symbol, so 1 symbol error could cause up to ${log}_{2}M$ bit errors.

Gray coding is a clever way of mapping bits to symbols so that neighboring symbols differ by only a single bit.Since the vast majority of errors occur when $y\left[n\right]$ falls into a neighboring decision region, Gray coding yields BER $\approx$ SER.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
Got questions? Join the online conversation and get instant answers! By Michael Sag By Richley Crapo By Brianna Beck By OpenStax By Anh Dao By OpenStax By OpenStax By Madison Christian By Katie Montrose By Melinda Salzer