1- Studying Components of a Communication System

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Studying Components of a Communication System

Section Overview

Modulating a Random Signal

Plotting Signal Constellations

Pulse Shaping Using a Raised Cosine Filter

Using a Convolutional Code

Section Overview

Communications Toolbox software implements a variet of communications!related

tas"s# Man of the functions in the toolbox perform computations associated with aparticular component of a communication sstem$ such as a demodulator or e%uali&er#

Other functions are designed for visuali&ation or analsis#

'hile the later chapters of this document discuss various toolbox features in more depth$this section builds an example step b step to give ou a first loo" at the toolbox# This

section also shows how Communications Toolbox functionalities build upon the

computational and visuali&ation tools in the underling M(T)(* environment#

Modulating a Random Signal

This first example addresses the following problem+

Problem Process a binar data stream using a communication sstem that

consists of a baseband modulator$ channel$ and demodulator# Compute thesstem,s bit error rate -*.R/# (lso$ displa the transmitted and received signals in

a scatter plot#

The following table indicates the "e tas"s in solving the problem$ along with relevantCommunications Toolbox functions# The solution arbitraril chooses baseband 01!2(M

-%uadrature amplitude modulation/ as the modulation scheme and ('34 -additive white

3aussian noise/ as the channel model#
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Task Function or Method

3enerate a random binar data stream randint

Modulate using 01!2(M modulatemethod on modem.qammodob5ect

(dd white 3aussian noise awgn

Create a scatter plot scatterplot

6emodulate using 01!2(M modulatemethod on modem.qamdemodob5ect

Compute the sstem,s *.R biterr

Solution of Problem

The discussion below describes each step in more detail$ introducing M!code along the

wa# To view all the code in one editor window$ enter the following in the M(T)(*

Command 'indow#

edit commdoc_mod

! "enerate a Random #inary $ata Stream! The conventional format for representing

a signal in M(T)(* is a vector or matrix# This example uses the randintfunction to

create a column vector that lists the successive values of a binar data stream# The length

of the binar data stream -that is$ the number of rows in the column vector/ is arbitrarilset to 78$888#

%ote The sampling times associated with the bits do not appear explicitl$ and

M(T)(* has no inherent notion of time# For the purpose of this example$

"nowing onl the values in the data stream is enough to solve the problem#

The code below also creates a stem plot of a portion of the data stream$ showing the

binar values# 9our plot might loo" different because the example uses random numbers#

4otice the use of the colon -:/ operator in M(T)(* to select a portion of the vector# Formore information about this sntax$ see The Colon Operatorin the M(T)(*

documentation set#
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%% Setup% Define parameters.M = 16; % Sie of signal constellation

! = log"#M$; % umber of bits per s&mboln = 'e(; % umber of bits to processnsamp = 1; % )*ersampling rate+Mod = modem.qammod#M$; % ,reate a 16-/M modulator

%% Signal Source% ,reate a binar& data stream as a column *ector.0 = randint#n1$; % 2andom binar& data stream

% 3lot first (4 bits in a stem plot.stem#0#1:(4$5filled5$;title#52andom its5$;0label#5it 7nde05$; &label#5inar& 8alue5$;


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&! Prepare to Modulate! The modem.qammodob5ect implements an M!ar 2(M

modulator$ M being 01 in this example# :t is configured to receive integers between 8 and

0; rather than

of values from 0across a row of a matrix$ using the res+apefunction in M(T)(*$ and

then appl the bi"defunction to convert each

characters after the res+apecommand form the uncon5ugated arra transpose operator in

M(T)(*# For more information about this and the similar 5operator$ see Reshaping a

Matrixin the M(T)(* documentation set#/

%% it-to-S&mbol Mapping% ,on*ert t+e bits in 0 into !-bit s&mbols.0s&m = bi"de#res+ape#0!lengt+#0$9!$.55left-msb5$;

%% Stem 3lot of S&mbols% 3lot first 14 s&mbols in a stem plot.figure; % ,reate new figure window.stem#0s&m#1:14$$;title#52andom S&mbols5$;0label#5S&mbol 7nde05$; &label#57nteger 8alue5$;
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'! Modulate (sing )*+,M! =aving defined 0s&mas a column vector containing

integers between 8 and 0;$ ou can use the modulatemethod of the modem.qammod

ob5ect to modulate 0s&musing the baseband representation# Recall that Mis 01$ the

alphabet si&e#

%% Modulation& = modulate#modem.qammod#M$0s&m$; % Modulate using 16-/M.

The result is a complex column vector whose values are in the 01!point 2(M signal

constellation# ( later step in this example will show what the constellation loo"s li"e#

To learn more about modulation functions$ see Modulation#(lso$ note that themodulate

method of the modem.qammodob5ect does not appl an pulse shaping# To extend this

example to use pulse shaping$ see Pulse Shaping Using a Raised Cosine Filter#For an

example that uses rectangular pulse shaping with PS> modulation$ see basicsimdemo#

-! ,dd .hite "aussian %oise! (ppling the awgnfunction to the modulated signal adds

white 3aussian noise to it# The ratio of bit energ to noise power spectral densit$ .b?48$

is arbitraril set at 08 d*#

The expression to convert this value to the corresponding signal!to!noise ratio -S4R/

involves !$ the number of bits per smbol -which is < for 01!2(M/$ and nsamp$ the

oversampling factor -which is 0 in this example/# The factor !is used to convert .b?48to

an e%uivalent .s?48$ which is the ratio of symbolenerg to noise power spectral densit#

The factor nsampis used to convert .s?48in the smbol rate bandwidth to an S4R in the

sampling bandwidth#

%ote The definitions of &t0and &r0and the nsampterm in the definition of snr

are not significant in this example so far$ but will ma"e it easier to extend the

example later to use pulse shaping#

%% ransmitted Signal&t0 = &;

%% ,+annel% Send signal o*er an / 14?log14#!$ - 14?log14#nsamp$;&nois& = awgn#&t0snr5measured5$;

%% 2ecei*ed Signal&r0 = &nois&;

To learn more about awgnand other channel functions$ see Channels#
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/! Create a Scatter Plot! (ppling the scatterplotfunction to the transmitted and

received signals shows what the signal constellation loo"s li"e and how the noise distorts

the signal# :n the plot$ the hori&ontal axis is the in!phase component of the signal and thevertical axis is the %uadrature component# The code below also uses the title$ legend$

and a0isfunctions in M(T)(* to customi&e the plot#

%% Scatter 3lot% ,reate scatter plot of nois& signal and transmitted% signal on t+e same a0es.+ = scatterplot#&r0#1:nsamp?@e'$nsamp45g.5$;+old on;scatterplot#&t0#1:@e'$145!?5+$;title#52ecei*ed Signal5$;legend#52ecei*ed Signal55Signal ,onstellation5$;a0is#A-@ @ -@ @B$; % Set a0is ranges.

+old off;


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To learn more about scatterplot$ see Scatter Plots#
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)! $emodulate (sing )*+,M! (ppling the demodulatemethod of the

modem.qamdemodob5ect to the received signal demodulates it# The result is a column

vector containing integers between 8 and 0;#

%% Demodulation% Demodulate signal using 16-/M.s&m = demodulate#modem.qamdemod#M$&r0$;

0! Convert the 1nteger*2alued Signal to a #inary Signal! The previous step produced

s&m$ a vector of integers# To obtain an e%uivalent binar signal$ use the de"bifunction

to convert each integer to a corresponding binar

use the res+apefunction to arrange all the bits in a single column vector rather than a

four!column matrix#

%% S&mbol-to-it Mapping% Cndo t+e bit-to-s&mbol mapping performed earlier. = de"bi#s&m5left-msb5$; % ,on*ert integers to bits.% ,on*ert from a matri0 to a *ector. = res+ape#.5numel#$1$;

3! Compute the System4s #5R! (ppling the biterrfunction to the original binar

vector and to the binar vector from the demodulation step above ields the number of bit

errors and the bit error rate#

%% 2 ,omputation% ,ompare 0 and to obtain t+e number of errors and

% t+e bit error rate.Anumber_of_errorsbit_error_rateB = biterr#0$

The statistics appear in the M(T)(* Command 'indow# 9our results might var

because the example uses random numbers#

number_of_errors =

1

bit_error_rate =

4.44"(

To learn more about biterr$ see Performance Results via Simulation#
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Plotting Signal Constellations

The example in the previous section created a scatter plot from the modulated signal#

(lthough the plot showed the points in the 2(M constellation$ the plot did not indicatewhich integers between 8 and 0; the modulator mapped to a given constellation point#

This section addresses the following problem+

Problem Plot a 01!2(M signal constellation with annotations that indicate themapping from integers to constellation points#

The solution uses the scatterplotfunction to create the plot and the te0tfunction in

M(T)(* to create the annotations#

Solution of Problem

To view a completed M!file for this example$ enter edit commdoc_constin the

M(T)(* Command 'indow#

! Find ,ll Points in the )*+,M Signal Constellation! The ,onstellationpropert

of the modem.qammodob5ect contains all points in the 01!2(M signal constellation#

M = 16; % umber of points in constellation+=modem.qammod#M$; % Modulator obEectmapping=+.S&mbolMapping; % S&mbol mapping *ectorpt = +.,onstellation; % 8ector of all points in constellation


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&! Plot the Signal Constellation! The scatterplotfunction plots the points in pt#

% 3lot t+e constellation.scatterplot#pt$;

'! ,nnotate the Plot to 1ndicate the Mapping! To annotate the plot to show therelationship between mappingand pt$ use the te0tfunction to place a number in the plot

beside each constellation point# The coordinates of the annotation are near the real and

imaginar parts of the constellation point$ but slightl offset to avoid overlap# The text ofthe annotation comes from the binar representation of mapping# -The dec"binfunction

in M(T)(* produces a string of digit characters$ while the de"bifunction used in thelast section produces a vector of numbers#/

% 7nclude te0t annotations t+at number t+e points.te0t#real#pt$>4.1imag#pt$dec"bin#mapping$$;a0is#A-( ( -( (B$; % ,+ange a0is so all labels fit in plot.


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#inary*Coded )*+,M Signal Constellation

56amining the Plot

:n the plot above$ notice that 8880 and 8808 correspond to ad5acent constellation points

on the left side of the diagram# *ecause these binar representations differ b two bits$the ad5acenc indicates that the modem.qammodob5ect did notuse a 3ra!coded signal

constellation# -That is$ if it were a 3ra!coded signal constellation$ then the annotations

for each pair of ad5acent points would differ b one bit#/

* contrast$ the constellation below is one example of a 3ra!coded 01!2(M signalconstellation#


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"ray*Coded )*+,M Signal Constellation

The onl difference$ compared to the previous example$ is that ou configure

modem.qammodob5ect to use a 3ra!coded constellation#

%% Modified 3lot it+


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Pulse Shaping (sing a Raised Cosine Filter

This section further extends the example b addressing the following problem+

Problem Modif the 3ra!coded modulation example so that it uses a pair of

s%uare root raised cosine filters to perform pulse shaping and matched filtering atthe transmitter and receiver$ respectivel#

The solution uses the rcosinefunction to design the s%uare root raised cosine filter and

the rcosfltfunction to filter the signals# (lternativel$ ou can use the rcosflt

function to perform both tas"s in one command@ see Filtering with Raised Cosine Filters

or the rcosdemodemonstration for more details#

Solution of Problem

This solution modifies the code from commdoc_gra&.m# To view the original code in an

editor window$ enter the following command in the M(T)(* Command 'indow#

edit commdoc_gra&

To view a completed M!file for this example$ enter edit commdoc_rrcin the M(T)(*

Command 'indow#

! $efine Filter*Related Parameters! :n the Setupsection of the example$ replace the

definition of the oversampling rate$ nsamp$ with the following#

nsamp = (; % )*ersampling rate

(lso$ define other "e parameters related to the filter b inserting the following after the

Modulationsection of the example and before the ransmitted signalsection#

%% Filter Definition% Define filter-related parameters.filtorder = (4; % Filter order

dela& = filtorder9#nsamp?"$; %


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&! Create a S7uare Root Raised Cosine Filter! To design the filter and plot its impulse

response$ insert the following commands after the commands ou added in the previous

step#

% ,reate a square root raised cosine filter.rrcfilter = rcosine#1nsamp5fir9sqrt5rolloffdela&$;

% 3lot impulse response.figure; imp#rrcfilter1$;

'! Filter the Modulated Signal! To filter the modulated signal$ replace the

ransmitted Signalsection with following#

%% ransmitted Signal% Cpsample and appl& square root raised cosine filter.

&t0 = rcosflt#&1nsamp5filter5rrcfilter$;

% ,reate e&e diagram for part of filtered signal.e&ediagram#&t0#1:"444$nsamp?"$;

The rcosfltcommand internall upsamples the modulated signal$ &$ b a factor of

nsamp$ pads the upsampled signal with &eros at the end to flush the filter at the end of the

filtering operation$ and then applies the filter#


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The e&ediagramcommand creates an ee diagram for part of the filtered noiseless

signal# This diagram illustrates the effect of the pulse shaping# 4ote that the signal shows

significant intersmbol interference -:S:/ because the filter is a s%uare root raised cosinefilter$ not a full raised cosine filter#

To learn more about e&ediagram$ see .e 6iagrams#

-! Filter the Received Signal! To filter the received signal$ replace the 2ecei*ed

Signalsection with the following#

%% 2ecei*ed Signal% Filter recei*ed signal using square root raised cosine filter.&r0 = rcosflt#&nois&1nsamp5Fs9filter5rrcfilter$;&r0 = downsample#&r0nsamp$; % Downsample.&r0 = &r0#"?dela&>1:end-"?dela&$; % /ccount for dela&.

These commands appl the same s%uare root raised cosine filter that the transmitter usedearlier$ and then downsample the result b a factor of nsamp#
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The last command removes the first "?dela&smbols and the last "?dela&smbols in

the downsampled signal because the represent the cumulative dela of the two filtering

operations# 4ow &r0$ which is the input to the demodulator$ and &$ which is the output

from the modulator$ have the same vector si&e# :n the part of the example that computes

the bit error rate$ it is important to compare two vectors that have the same si&e#

/! ,d8ust the Scatter Plot! For variet in this example$ ma"e the scatter plot show thereceived signal before and after the filtering operation# To do this$ replace the Scatter

3lotsection of the example with the following#

%% Scatter 3lot% ,reate scatter plot of recei*ed signal before and% after filtering.+ = scatterplot#sqrt#nsamp$?&nois:nsamp?@e'$nsamp45g.5$;+old on;scatterplot#&r0#1:@e'$145!05+$;title#52ecei*ed Signal efore and /fter Filtering5$;legend#5efore Filtering55/fter Filtering5$;

a0is#A-@ @ -@ @B$; % Set a0is ranges.

4otice that the first scatterplotcommand scales &nois&b sqrt#nsamp$when

plotting# This is because the filtering operation changes the signal,s power#


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(sing a Convolutional Code

This section further extends the example b addressing the following problem+

Problem Modif the previous example so that it includes convolutional codingand decoding$ given the constraint lengths and generator polnomials of the

convolutional code#

The solution uses the con*encand *itdecfunctions to perform encoding and decoding$

respectivel# :t also uses the pol&"trellisfunction to define a trellis that represents a

convolutional encoder# To learn more about these functions$ see Convolutional Coding#

See also *itsimdemofor an example of convolutional coding and decoding#

Solution of Problem

This solution modifies the code from Pulse Shaping Using a Raised Cosine Filter# To

view the original code in an editor window$ enter the following command in the


edit commdoc_rrc

To view a completed M!file for this example$ enter edit commdoc_codein the


! 1ncrease the %umber of Symbols! Convolutional coding at this value of bo

reduces the *.R mar"edl# (s a result$ accumulating enough errors to compute a reliable*.R re%uires ou to process more smbols# :n the Setupsection$ replace the definition

of the number of bits$ n$ with the following#

n = @e@; % umber of bits to process

%ote The larger number of bits in this example causes it to ta"e a noticeabllonger time to run compared to the examples in previous sections#

&! 5ncode the #inary $ata! To encode the binar data before mapping it to integers for

modulation$ insert the following after the Signal Sourcesection of the example and

before the it-to-S&mbol Mappingsection#

%% ncoder
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% Define a con*olutional coding trellis and use it% to encode t+e binar& data.t = pol&"trellis#A@ (BA"' '@ 4; 4 @ 1'B$; % relliscode = con*enc#0t$; % ncode.coderate = "9';

The pol&"trelliscommand defines the trellis that represents the convolutional codethat con*encuses for encoding the binar vector$ 0# The two input arguments in the

pol&"trelliscommand indicate the constraint length and generator polnomials$

respectivel$ of the code# ( diagram showing this encoder is in.xample+ ( Rate!A?7

Feedforward .ncoder#

'! ,pply the #it*to*Symbol Mapping to the 5ncoded Signal! The bit!to!smbol

mapping must appl to the encoded signal$ code$ not the original uncoded data# Replace

the first definition of 0s&m-within the it-to-S&mbol Mappingsection/ with the

following#

% . Do ordinar& binar&-to-decimal mapping.0s&m = bi"de#res+ape#code!lengt+#code$9!$.55left-msb5$;

Recall that !is 14?log14#!?coderate$-14?log14#nsamp$;

/! $ecode the Convolutional Code! To decode the convolutional code before

computing the error rate$ insert the following after the entire S&mbol-to-it Mapping

section and 5ust before the 2 ,omputationsection#

%% Decoder% Decode t+e con*olutional code.tb = 16; % racebac! lengt+ for decoding

= *itdec#ttb5cont55+ard5$; % Decode.

The sntax for the *itdecfunction instructs it to use hard decisions# The 5cont5

argument instructs it to use a mode designed for maintaining continuit when ou invo"ethe function repeatedl -as in a loop/# (lthough this example does not use a loop$ the

5cont5mode is used for the purpose of illustrating how to compensate for the dela in

this decoding operation# The dela is discussed further in More (bout 6elas#
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)! ,ccount for $elay .hen Computing #5R! The continuous operation mode of the

iterbi decoder incurs a dela whose duration in bits e%uals the tracebac" length$ tb$

times the number of input streams to the encoder# For this rate A?7 code$ the encoder hastwo input streams$ so the dela is "?tbbits#

(s a result$ the first "?tbbits in the decoded vector$ $ are 5ust &eros# 'hen computing

the bit error rate$ ou should ignore the first "?tbbits in and the last "?tbbits in the

original vector$ 0# :f ou do not compensate for the dela$ then the *.R computation is

meaningless because it compares two vectors that do not trul correspond to each other#

Therefore$ replace the 2 ,omputationsection with the following#

%% 2 ,omputation% ,ompare 0 and to obtain t+e number of errors and% t+e bit error rate. a!e t+e decoding dela& into account.decdela& = "?tb; % Decoder dela& in bitsAnumber_of_errorsbit_error_rateB = ... biterr#0#1:end-decdela&$#decdela&>1:end$$

More ,bout $elays

The decoding operation in this example incurs a dela$ which means that the output of thedecoder lags the input# Timing information does not appear explicitl in the example$ and

the duration of the dela depends on the specific operations being performed# 6elas

occur in various communications!related operations$ including convolutional decoding$

convolutional interleaving?deinterleaving$ e%uali&ation$ and filtering# To find out theduration of the dela caused b specific functions or operations$ refer to the specific

documentation for those functions or operations# For example+

The *itdecreference page

6elas of Convolutional :nterleavers

6elas from .%uali&ation

.xample+ Compensating for 3roup 6elas in 6ata (nalsis

Fading Channels

The .ffect of 6elas on Recover of Convolutionall :nterleaved 6atadiscussion also

includes two tpical was to compensate for delas#
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1- Studying Components of a Communication System