ii baseband modulation schemes 020206[1]

8/6/2019 II Baseband Modulation Schemes 020206[1]

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TSTE91 System Design

Communications System Simulation Using Simulink

Part II Baseband Modulation schemes

Sebastian Prot, Kent Palmkvist

Electronic Systems, Dept. EE, LiTH

020206


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1 Abstract .............................................................................................. 1

2 Theory................................................................................................ 1

2.1 Signal constellation ....................................................................... 1

2.2 Basic mapping schemes ................................................................ 2

2.2.1 M-ary PSK mapping.................................................................. 3

2.2.2 M-ary ASK mapping................................................................. 6

2.3 Quadrature modulation................................................................ 7

2.3.1 QASK mapping......................................................................... 8

2.3.1.1Properties of QASK mapping............................................... 9

2.4 Arbitrary-map QAM.................................................................. 11

3 Block descriptions............................................................................ 12

3.1 Modulator banks......................................................................... 13

3.1.1 Amplitude modulation............................................................. 13

3.1.1.1MASK modulator............................................................... 13

3.1.1.2S-QAM modulator ............................................................. 14

3.1.2 Phase modulation .................................................................... 15

3.1.2.1MPSK modulator ............................................................... 16

3.1.2.2Arbitrary-map QASK......................................................... 16

3.2 Sampled read from workspace multilevel data source .......... 173.2.1 Randint() function ................................................................... 17

3.3 Measurement tools...................................................................... 17

3.3.1 Scattered plot .......................................................................... 17

4 The system basics............................................................................. 18

4.1 The system setup parameters..................................................... 18

4.1.1 Initial commands..................................................................... 19

4.2 The signals flow........................................................................... 20

4.3 Transmission analyses ................................................................ 20

5 Bibliography .................................................................................... 22


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1

M-ary QAM and M-ary PSK modulations

1 Abstract

This manual introduces the basics of digital amplitude (MASK M-ary

Amplitude Shift Keying / MQAM M-ary Quadrature Amplitude Modulation)

and phase (MPSK M-ary Phase Shift Keying / MQPSK M-ary Quadrature

Phase Shift Keying) modulation.

2 Theory

Like analog modulation, digital modulation alters an analog carrieraccording to an information signal. However, in digital modulation, the

information signal is a discrete-time signal that can assume an finite number of

different values.

The simulink library includes both baseband and passband modulators.

Baseband, lowpass modulation methods requires less computation, producing

only complex envelope signal.

Passband modulation, on the other hand, alters an analog, time continues

carrier according to the baseband modulation envelope.

2.1 Signal constellation

Constellation, in this case, doesnt have anything to do with astronomy. It

means: a two-dimensional space of possible values assigned to binary data

during the process of signal mapping.

The adjective - two-dimensional describes a complex number domain,

where the Xaxis defines the real part and the Yaxis defines the imaginary part

of a complex envelope. For example, in Fig.1 point X indicates the real part

equal to 1 and imaginary part equal to j0,33(3), where j signifies the imaginary

component.

The number of points, M, in signal constellation is adequate to the number

of possible values assigned to a binary data stream. Furthermore constellation

map shape, identifies the mapping scheme used in the process of signal mapping

(compare e.g., fig.1 and fig.3 ).


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Fig.1. The square constellation map of 16QAM signal;

X{Re=1;Im=j0.33(3)}.

2.2 Basic mapping schemes

The main difference between MPSK and MASK modulation is theconstellation map used in the process of data mapping.

Fig.2. a). Phase modulator; b). Amplitude modulator.


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In both systems is the modulator output signal in polar form:

Zej( +)

,

where is the signal angle altered during PSK (Phase Shift Keying)mapping; is constant initial offset, andZsignifies the magnitude altered during

ASK mapping.

In the case of PSK can the modulation process be described as a function of

the angle:

f()=ej( +)

where is a constant phase offset,

and in AM (Amplitude Modulation) can the same process be characterized

as a function of the magnitude:

f(Z)=Ze-j

where is also a constant phase offset.

2.2.1 M-ary PSK mapping

The M-ary PSK constellation is circle formed as the example in fig.3

shows.

Fig.3. Circle constellation map of a 16 PSK signal;

X{Re=1,Im=0j}.


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The circle radius is independent of the modulation M parameter, since it is

equivalent to the signal magnitude and is constant. Thus, when Mincreases, the

distance between neighboring points in the signal constellation a (1) decrease

(compare fig.3 and fig.5).

a = 2Rsin (/M) (1)

where a is the distance between the two points on the circle of R radius, and

Mis the number of points in the constellation used for the mapping.

Equation 1 was derived according to fig.4.

Fig.4. Calculation of the distance between points

in the circle-map constellation.

Each points position in the circle-map of PSK can be calculated (see

Example 1) according to expression (2):

X(m) = e(j +j2m/M)

(2),

where is the phase offset; m is an integer between 0 and M-1, and M

defines the total number of points in signal constellation.

The signal magnitude and phase that are presented in the constellation maps

are encoded in rectangular coordinates and can be found by performing

conversion to polar notation by equation (3) and (4).


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(4))Re(

)Im(arctan

(3))(Re)(Im 22

=

+=

X

X

XXMag

Fig.5. 64 PSK signal constellation.

Example.1.

BPSKmodulation is a two values digital modulation, M= 2. Thus, input

data m defined as integer number in the range [0M-1], can take on only

two values 0 and 1. Assuming = 0, the modulated signal constellation

points can be calculated from (1) as follows:

Input value m = 0;X(0) = e

(0j +j20/2)= e

(0)= 1;

m = 1;

X(1) = e(0j +j2/2)

=ej

= cos( ) + jsin( ) = -1

Figure 6 shows several points of the 16PSK constellation map, in both polar

and rectangular notation. In all cases, the magnitude is equal to 1. The same

situation can be observed in fig. 5 where the 64PSK constellation map ispresented.


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These constellation maps with M being large are not practical for

transmission over noisy channels, because of the small distances between the

points in signal constellation map (compare fig.1 and 3).

Other modulation scheme that alters one of the parameters of the complex

envelope in polar notation, includes the already mentioned MASK.

Fig.6. 16 PSK constellation map. Some points are converted from rectangular to

polar notation, revealing magnitude and phase of the signal.

2.2.2 M-ary ASK mapping

In the PSK case, all the points lay on a circle with a constant radius

defining constant signal magnitude. On the other hand, in MASK all the points

form a line, which slope with relation to X axis is defined by the initial phase

parameter.

Figure 7 presents such a constellation map, where initial phase is equal to

zero. Observing the constallation map a conclusion can be drawn that in simple

ASK modulation, the phase is also modulated and it takes on two differentvalues: 0 and(actually and +, where is initial phase).

Right and Wrong! Why? The answer is simple. An AM modulator

multiplies a cosine carrier by a purely real modulating signal. When it takes on

values smaller then 0, carrier is simply inverted causing phase shift of , or -

(fig.8).

The key information here, is that phase shifts in ASK modulation are not

the results of phase modulation, since the modulator alters only the carriers

magnitude. Even so, MASK modulation gives the same results as MPSK when

Mis equal to 2.


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Fig.7. 16 ASK constellation map.

Fig.8. ASK modulation scheme.

2.3 Quadrature modulation

Baseband, quadrature modulation is based upon a binary signal mapping to

complex numbers in rectangular notation. Rectangular notation, means here

separate in-phase (I) and quadrature (Q) components.

In passband modulation, both of them modulate different carriers, which arethen summed before transmitting (its equivalent to the I and Q components

summing in the baseband model, fig.9).

Fig.9. Baseband quadrature modulator.


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The parameters of the resulting signal can be calculated according to the

expressions (3) and (4), after replacing Im(x) with the quadrature Q component

andRe(x) with the in-phaseIcomponent.

Setting different constellation maps changes the type of modulation. For

example, in QAM, BPSK, and QPSK, according to IEEE802.11a standard, is thebinary data signal mapped to a Gray-coded constellation.

Gray coding maps an integer number into a set of integers where the binary

representation of a given number only differs by one bit from its neighbors

(Example 2).

Example.2.

Example of Binary to Gray and Gray to Binary Mappings

Binary to Gray Mode Gray to Binary Mode

Input Output Input Output0 0 (000) 0 (000) 0

1 1 (001) 1 (001) 1

2 3 (011) 2 (010) 3

3 2 (010) 3 (011) 2

4 6 (110) 4 (100) 7

5 7 (111) 5 (101) 6

6 5 (101) 6 (110) 4

7 4 (100) 7 (111) 5

2.3.1 QASK mapping

The QASK modulator supplied in simulink library performs Gray-coded

mapping with an additional multiplication by a special normalization factor

KMOD (5), which assures approximately constant average power, regardless of

the value ofM.

d=(I + Q)KMOD, (5)

where d is an output signal, I and Q are in-phase and quadrature

components before normalization and KMOD is a normalization factor.

Thus, the points form square constellation in the range [-1:1] of both axes

(Fig.10) regardless of the value of the parameterM.

The distances between those points are however much larger then in

equivalent PSK modulation for the same value ofM (compare fig.1 with fig.3,

and fig.5 with fig.10), and can be calculated according to expressions in (6).


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Fig.10. 4-QAM and 64-QAM constellation maps.

(6)1

22Magint,even42

;24

;22

==>=

====

MaKM

aM

aM

MAX

K

2.3.1.1Properties of QASK mapping

QASK properties depend strongly onMvalue.

M = 2

When M = 2, a binary QAM mapping scheme is applied. In other words,

the carrier is multiplied by two opposite values1, depending on the data sequence

to be transmitted (Fig.11). This operation gives the same results as BPSK

modulation, and BPSK can thus be realized using a QASK modulator in the

physical applications.

Mathematical proof of that property can be derived by comparing example

1 with equation 7, where eq.7 depicts the binary QAM mapping process.

(7).10sin0cos1

10sin0cos10

0

====+===

=

=

jeZ

jeZZe

j

jj

where, Z is signal magnitude and is initial phase.

The only differences between BPSK and QASK is the inverse signs of the

calculated values in the constellation maps. In PSK m=0 generates 1 and m=1

generates1, while in binary QASK m=0 generates1 and m=1 generates 1.

1 In general 1 or -1.


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Fig.11. BPSK and square-map binary QAM constellation.

However, in Std 802.11a, the BPSK map is defined as presented in fig.12,

and it follows the rules of binary QAM mapping scheme, described in this

section, thus binary QASK is usually called BPSK in physical systems.

Fig.12. BPSK constellation bit encoding.

Fig.13. QPSK constellation bit encoding.

M = 4

Similar situation can be observed when M = 4 (4-QAM is applied). In this

case, the points in the QAM are, again, assigned according to the gray-coded

map defined in Std 802.11a as presented in figure 13, where the input bit b0, is

the earliest in the stream.

Figure 14 presents both the 4-QAM and 4-PSK constellation maps

generated by the Simulink model. Notice, that in both cases, the magnitude is

constant regardless of the constellation point taken under consideration.Although in QAM, transmitted energy is higher since a square map is applied.


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Thus, a conclusion can be drawn, that in the case of M = 4, again, both

modulation schemes produce phase modulated signal. Thus, QPSK according to

Std 802.11a is realized by an QASK modulator.

Fig.14. 4-PSK and 4-QAM constellation maps.

M > 4

If the value ofMis larger then 4, both magnitude and phase alters in QAM.

Although phase shifts are not a straightforward effect of phase modulation. They

appear due to summation of sinusoids in the passband model, or corresponding,

Iand Q components in the baseband model.

Taking all into account, BPSK, QPSK and M-ary QAM modulation

schemes with Gray-coded mapping specified in Std802.11a, can be performed

using a single QAM modulator.

Additionally, if the quadrature amplitude modulator has an option that

allows for constellation map definition, it is possible to define it in such a way,

that the system acts like a phase modulator, regardless of the value ofM.

2.4 Arbitrary-map QAM

If a quadrature AM modulator allows the user to set the constellation map to

an arbitrary form2, the MPSK can be effectively modeled with the use of the

Eulers relation (8), which is actually the key equation for using complex

numbers in science and engineering. In this case it allows the user to calculate

the constellation points from equation (1) receiving separate in-phase and

quadrature definition (9) and (10).

2 Separate for in-phase and quadrature component.


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8sincos )(xjxejx +=

)(M

mee M

Mj

jx 92cosRe)Re()

]1:0[2(

=

=

)(M

mee M

Mj

jx 102sinIm)Im()

]1:0[2(

=

=

where m defines modulator input as an integer value.

The resulting constellation map is identical to that of a PSK modulation

(fig.15), although it is produced by a quadrature amplitude modulator.

Fig.15. PSK constellation map, generated by

an Arbitrary-map QAM modulator.

3 Block descriptions

The system consists of three main sections (fig.16):

Amplitude modulation Phase modulation Measurement tools

The blocks outside the marked regions are not important for understanding

the theory, but are important from a simulink point of view, since they are used

for stimuli generation and signal format conversion.


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Fig.16. The Simulink system introducing basic and quadrature baseband digital

mapping schemes such as PSK, ASK, QASK and QPSK.

3.1 Modulator banks

3.1.1 Amplitude modulation

3.1.1.1MASK modulator

Basic digital amplitude modulation is performed by the MASK modulator.

It generates a one dimensional constellation map.

The parameters are the same as in the S-QAM modulator (see the next

section below).


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Fig.17. MASK block setup mask.

3.1.1.2S-QAM modulatorSquare-map amplitude modulation is performed by the S-map basebandQAM modulator which requires the user to define a few basic mapping

parameters (fig.18).

M-ary number

The M-ary number parameter sets the mapping constellation of the signals.

The description in brackets defines the QAM modulator input signal, whose

values need to be in the range of[0:M-1]. If a binary data source is used in the

system, the signal needs to be converted by an Integer vector to scalar

converter from the Communication blockset/ Utility functions library.

Alternatively, a properly configured Read from workspace block can be

applied (see section 3.4).

Symbol interval and offset

When the symbol interval is a two-element vector, the second element is the

offset (default value is 0).

Fig.18. S-QASK model setup dialog box.


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Initial phase

Initial phase is applied in the second stage of modulation (fig.10).

Sample time

The sample time parameter was probably supposed to define the modulatedoutput signal sampling rate, but model designer have forgotten to apply such

functionality into the subsystem. Thus, the sample time parameter does not have

any influence on the modulator operation and can be set, e.g., to zero.

If all the parameters are set correctly (fig.19), the QAM modulator first

maps the input signal according to square constellation definition, and in second

step it mixes in-phase and quadrature components to form a single complex

signal and optionally applies an initial phase offsets (fig.20).

Fig.19. S-QAM modulator subsystem.

Fig.20. Quadrature modulation stage.3.1.2 Phase modulation

The phase modulation section introduces two methods of performing

MPSK. One of them performs a baseband MPSK modulation, and the other one

performs an Arbitrary-map QASK modulator.

In both cases, binary input data needs to be converted to a M-level signal,

since in both cases is the constellation defined by the same value of the M

parameter.


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3.1.2.1MPSK modulator

Like in almost all simulink models, the MPSK mod block setup mask

allows the user to specify basic parameters (fig.21). All of these are described in

section 3.1 above.

Fig.21. The MPSK model setup dialog box.

The baseband MPSK Mod alters the signal phase according to expression

(1), and outputs a baseband envelope of the modulated cosine signal.

3.1.2.2Arbitrary-map QASK

The Arbitrary-map baseband quadrature modulator also generates a

modulated carrier envelope, but it alters both magnitude and phase of the input

signal (fig.22) according to constellation definition in rectangular notation.

In-phase and quadrature components

The two vectors define the values that are assigned to input data in the

process of data mapping.

Fig.22. A-QASK setup mask.


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Sample Time

Like in the QASK modulator case, the sample time does not apply to this

system.

3.2 Sampled read from workspace multilevel data source

Due to the modulators specifications, the input binary data stream needs to

be replaced with a M-level signal. That can be performed in several ways, but

the simplest is to replace the binary data source with a multilevel one.

The Sampled Read From Workspace block (Fig.23) was found to be

useful in this situation. This block allows a system to read data from workspace

or to create a data vector by executing a Matlab function. In this case, actual

data is generated with the use of the RANDINT() function random integer

matrix generator which is described in next section.

Fig.23. Data signal source.

3.2.1 Randint() function

The first and the second parameter specified in the call to this function

defines that a N-by-M matrix of random integers is to be generated.

The third parameter RANGE - specifies the minimum and maximum

output integer. When RANGE is a scalar, positive integer, the output range is

[0, RANGE-1].

3.3 Measurement tools

The modulators output signals are to be measured and analyzed usingmeasurement tools collected in the Measurement tools section, which includes

Scattered plot and the already known Scope.

3.3.1 Scattered plot

Plots generated by this block illustrate signal constellations imaginary and

real component in the same two-dimensional space. The displayed color

definitions and axes settings can be changed in the model mask (fig.24).


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Fig.24. Eye-diagram scattered plot setup mask.

Color codes and line properties can be found, by typing in theMatlab command window or simplyplotin Matlab Help Window.

Plot update sample time

The plot update sample time should be at least two times shorter than the

symbol period.

The key knowledge about scattered plot is that it displays two multiplexed

signals in the same 2-dimensional space. The more precise scattered plot, and

eye diagram description can be found in the third part of the manuals which

introduces passband modulation.

4 The system basics

Before starting the simulation, you should analyze the system carefully and

try to understand dependencies between particular blocks, parameters, etc.

The following description explains the signals flow and tasks that are

performed in different system stages.

4.1 The system setup parameters

The main mask allows the user to change the number of points in the signal

mapping constellation (Fig.25). Thus, different values ofM causes different

mapping schemes to be simulated. Table 1 presents some popular and practical

system specifications.


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Fig.25. The system setup dialog box.

Table 1. Popular modulation schemes realized by the simulink system.

M MASK MPSK QAM A-MAP QAM

2 BPSK BPSK BPSK BPSK

4 4-ASK C-map QPSK S-map QPSK C-map QPSK

8 8-ASK /4-PSK 8-QAM /4-QPSK16 16-ASK 16-PSK 16-QAM 16-QPSK

64 64-ASK 64-PSK 64-QAM 64-QPSK

128 128-ASK 128-PSK 128-QAM 128-QPSK

Etc.

Where, yellow color distinguishes phase modulations.

4.1.1 Initial commandsThe system needs one parameter to be set during the initialization process

(fig.26). That parameter specifies the input data bit duration Tbit,, and was

applied in the system for easier future development.

Fig.26. System mask definition with initialization parameter setup.


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4.2 The signals flow

The complete system consist of two basic sections:

modulator banks, measurement tools.

All modulators are fed by the multilevel data signal generated using the

randint() function and Read from Workspace block.

The modulator banks perform two types of signal modulation:

AM:o MASK,o QAM,

PM:o MPSK,o QPSK.

Next, in-phase and quadrature components of the modulated signals are

separated in order to generate scatter plots (fig.27) using the supplied tools

(fig.16). Additionally, mapped signal envelopes are also converted from

rectangular to polar form, producing separated magnitude and angle signals,

which are further cumulatively presented by the Scope block.

4.3 Transmission analyses

The Measurement Tools section, described above, allows the student toanalyze the signal constellations resulting from AM and PM modulation.

It is advised to set the M value in ascending order, according to the Table 1,

and compare the generated constellations from simulation to simulation.

WhenMis set to 2, binary PSK is applied by all the modulators. There are

no differences between the constellations.

When M = 4, it results in 4-ary PSK being applied, but differences when

identifying different maps can be noticed.

M = 8, results in /4-QPSK modulation realized by the quadraturemodulator.

Further increases of the value ofMcauses the modulators to generate:

MASK modulated signal in MASK mod case, MPSK modulated signal in MPSK mod case, M-QAM modulated signal in MQAM case, and M-QPSK modulated signal in A-map QAM case.

Plots generated by the Scope presents all modulators output signals on the

same screen, allowing the user to compare the magnitude and phase of digital

PM and AM modulated signals (fig.28).


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Fig.27. Scattered plots depicting constellation maps of 16-MASK(left top),

16-QAM (left bottom), 16-PSK (right top) and 16-QPSK (right bottom).

This plot shows that PSK modulation actually only alters the signals phase

and that quadrature amplitude modulator generates phase modulated and/or

amplitude modulated signals, depending on the specified number of

constellation point.

Additionally, the system introduces alternative way of performing phase

modulation with the use of a standard QAM modulator and properly set

constellation map. In such case is the modulation called QPSK Quadrature

Phase Shift Keying.


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Fig.28. Magnitude and phase components of modulated signals. Characteristics

generated for 16 level mapping.

Fig.28 presents complex envelopes of the modulated signals for M = 16. Inthe case of phase modulated signals, for M higher then 4, which was described

in chapter 2.3 Normalization, very small errors occurs causing magnitude

variation around its nominal value of . The Simulink block Scope

automatically enlarges those variations so they can be observed on the screen.

5 Bibliography

[1] Steven W. Smith. The Scientist and Engineer's Guide to Digital Signal

ii baseband modulation schemes 020206[1]

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