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.,
), the digital comm system
will work perfectly, yielding
.
In practice, however,
the pulses
and
will be truncated to finite length,
the channel will not be trivial (i.e.,
), and
the channel will not be noiseless (i.e.,
),
leading to
, in which case
we must
infer the value of
from the received samples
.
For now, we consider using only the single sample
to infer
.
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
,
the eye diagram is a plot which superimposes
T -second segments of
over the time intervals
for many integers
n .
In MATLAB, the eye diagram can be made by
superimposing
P -sample segments of
corresponding
to intervals
for many
n . (Usually
.)
If
, eye diagrams can be plotted for both the
“I” and “Q” channels using
and
,
respectively.MATLAB for digital mod/demod with eye diagram:
Constellation diagrams
The constellation diagram is a plot of
vs.
for many integers
n .
When the comm system is working well, the points cluster around thesymbol alphabet values:
Recall that
due to the complex-baseband channel model,
regardless of whether
or
.
Sometimes it is instructive to superimpose a plot of
vs.
,
which approximates the trajectory of
in the complex
plane:
MATLAB for digital mod/demod with constellation diagram:
When the alphabet entries are spaced by
Δ and
picked with equal probability, the symbol variance
obeys:
alphabet
M
2 -QAM
M -PAM
M -PSK
σ
a2
Decision regions
A reasonable way to infer the transmitted symbol
from the received
sample
is to decide that
was the alphabet element
nearest to
.
Nearest-element decision making is equivalent to using
decision
regions whose boundaries are equidistant from the two nearest alphabet
elements:
When
, the
symbol error rate (SER) equals the
probability that
lies outside the decision region
corresponding to alphabet member
a .
Writing
,
we represent the cumulative effect of noise and ISI by the error
.
Usually we model
as a Gaussian random variable with mean 0
and variance
σ
e2 .
Symbol error rate (ser) for
M -pam
Let's first consider an
M -PAM alphabet, where
.
Since the decision regions show that
is not useful,
we'll consider only the real parts of
and
.
When
, we have
, implying that
is Gaussian with mean
a and variance
σ
e2 ,
abbreviated as “
”.
This is illustrated below for the case of 4-PAM:
Formally, we say that
, the probability density
function (pdf) of
conditioned on
, obeys
Basically,
tells us how likely it is that
given that
.
Consider first the case where
a is an “interior” (not an “edge”) element
of the symbol alphabet.Given that
, we make an error when
or when
.
To find the probability of the latter error event, i.e.,
we integrate
over
:
The integral represents the shaded area below:
This integral is often solved via
using the “Q function”:
While the Q function is not represented in MATLAB, it can
be calculated using the “complementary error function”
:
In any case, the latter error event occurs with probability
By symmetry, the former error event probability is also
Since these two events are disjoint, the probability of making a decision
error on an interior symbol equals their sum:
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:
Using
, we can finally write
Symbol error rate (ser) for
M
2 -qam
With QAM, we have complex-valued
,
,
.
We'll assume that
and
are uncorrelated
and equal variance.To calculate SER, we can re-use the PAM approach with a few modifications:
integration is done on the complex plane,
σ
e2 -variance
-variance
&
,
M
2 -QAM has 4 corner points,
edge points, and
interior points,
calculate
via
, since the
regions of integration are simpler:
After a bit of algebra, we find
Bit error rate (ber) and gray coding
With an
M -ary alphabet, there are
bits per symbol,
so 1 symbol error could cause up to
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
falls into a
neighboring decision region, Gray coding yields BER
SER.
Questions & Answers
differentiate between demand and supply
giving examples
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
When MP₁ becomes negative, TP start to decline.
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Economic growth as an increase in the production and consumption of goods and services within an economy.but
Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has
The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50.
A,Calculate quantities of x and y which maximize utility.
B,Calculate value of Lagrange multiplier.
C,Calculate quantities of X and Y consumed with a given price.
D,alculate optimum level of output .
the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema
suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product
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Source:
OpenStax, Introduction to analog and digital communications. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10968/1.2
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