This document discusses generating a QPSK signal using MATLAB. It begins with an introduction to PSK modulation techniques and QPSK. It then describes the simulation design process in MATLAB, including representing the QPSK signal using I and Q components, and generating the signal using formulas programmed in MATLAB code. The code generates a QPSK modulated signal from binary input data along with plots of the original data and modulated signal. It also describes representing the generated QPSK signal using scatter plots in MATLAB to visualize the constellation.

1. Generation of PSK Signal Using Non linear
Devices via MATLAB®
Youness Lahdili 1*
1
Faculty of Electrical Engineering, Universiti Teknologi Malaysia, Malaysia
* corresponding author: y.lahdili@gmail.com
Abstract— This paper is intended to test the ability
of MATLAB to generate a Phase-Shift Keying
(PSK) signal, with all necessary parameters being
considered for processing by mean of non linear
devices.
Phase-Shift Keying (PSK), is widely used these days
within a whole raft of communications systems. It is
particularly well suited to the growing area of
computer data communications. PSK enables data
to be carried on in a more efficient manner than
other forms of modulation.
With more forms of communications transferring
from analogue formats to digital formats, PSK is
growing in importance, and along with it the
various forms of simulations and researches that
pertain to it.
Our choice for the implementation of the PSK
technique will be more oriented towards the
Quadrature Phase-Shift Keying (QPSK), which is a
more flexible, though simple type of PSK
modulation, and which is a predilection for
Bluetooth, and RFID designers. Our paper will
demonstrate the feasibility of such a modulation
technique in the latter applications among others
where it is already in use, such as in the low- range
Ethernet transmission and wireless LAN.
A MATLAB simulation is going to be performed,
comprising an analysis from different outlooks,
aided also by the powerful capabilities of MATLAB
language coupled with Simulink© design IDE.
In MATLAB, the PSK signal and digital stimulus
are translated into computer algorithms and
mathematical shells, assimilated by the MATLAB
Compiler machine. On the other hand, by using
Simulink© another approach of PSK signal
analysis is engaged by making use of the built-in
function and modulation blocks in the graphical
environment.
This paper is a theoretical and experimental study,
which can nurture the prospects of students with a
vested interest in PSK simulation and its
benchmarking.
Keywords— Phase-Shift Keying, digital modulation,
MATLAB, Simulation, non-linear devices,
Quadrature Phase-Shift Keying, Simulink tools
I. LITERATURE REVIEW
In this literature review we are carrying out a
critical and in depth evaluation of previous research
and studies on PSK. It is a summary and synopsis
of this particular area of PSK modulation in relation
with MATLAB codification of it, allowing any
person reading this paper to establish why we are
pursuing this particular PSK study, and what are the
other scopes of PSK modulation that may need
further attention.
Having a look in previous technical, we have
noticed the lack of references for the MATLAB
programs generated for PSK modulation. And there
is a need for simple and clutter-free simulation
codes, that shows the PSK waveforms without
creating confusion to the novice readers, who are
not interested in others modulation techniques.
PSK modulation has gained in popularity among
designers, and the simulation of such a modulation
technique, is the ideal way to explore its
capabilities, and preview its limitations. But most of

2. the simulations of PSK signals that we came across,
are dealing with one method of simulation (i.e.
linear only), or they use other simulators, that may
not be suitable for mathematical accuracy or not
having a notoriety in academic circles, indeed we
found some PSK simulations using LabVIEW© or
Altera Quartus © which are typical industrial
implementation simulators, but not oriented for
mathematical plotting or academic experiments.
We came to know that phase modulation PSK is
also used for some analogue transmissions;
however, it is far more widely used as a digital form
of modulation, where it comes in many flavours.
Indeed, it is even possible to combine phase shift
keying and amplitude keying in a form of
modulation known as quadrature amplitude
modulation, QAM. A derivative of this latter
scheme is called QAM64, which is a popular and
economic transmission standard, as it is used not
only in computer transmission, but also in fibre
optical medium and high level transmissions. Other
spin-off of the PSK technique are the LTI and
WiMAX which can be dubbed “The next
generation modulating scheme” strongly employed
in wireless and 3G applications.
The list below gives some of the more commonly
used forms of phase shift keying, PSK, and related
forms of modulation that are extensively used in
conveying data:
 PSK - Phase Shift Keying
 BPSK - Binary Phase Shift Keying
 QPSK - Quadrature Phase Shift Keying
 O-QPSK - Offset Quadrature PSK
 8 PSK - 8 Point Phase Shift Keying
 16 PSK - 16 Point Phase Shift Keying
 QAM - Quadrature Amplitude Modulation
 16 QAM - 16 Point Quadrature Amplitude
 64 QAM - 64 Point Quadrature Amplitude
 MSK - Minimum Shift Keying
 GMSK - Gaussian filtered Minimum Shift
Keying
These are just some of the major forms of phase
shift keying, PSK, that are widely used in radio
communications applications today. But there is
still room for improvement in the way PSK is
integrated into computational systems like
MATLAB, and that is what our study is meant for.
II. INTRODUCTION
Phase-shift keying (PSK) is a digital modulation
scheme that conveys data by changing, or
modulating, the phase of a reference signal (the
carrier wave).
Any digital modulation scheme uses a finite number
of distinct signals to represent digital data. PSK
uses a finite number of phases, each assigned a
unique pattern of binary digits. Usually, each phase
encodes an equal number of bits. Each pattern of
bits forms the symbol that is represented by the
particular phase. The demodulator, which is
designed specifically for the symbol-set used by the
modulator, determines the phase of the received
signal and maps it back to the symbol it represents,
thus recovering the original data. This requires the
receiver to be able to compare the phase of the
received signal to a reference signal — such a
system is termed coherent (and referred to as
CPSK).
Alternatively, instead of operating with respect to a
constant reference wave, the broadcast can operate
with respect to itself. Changes in phase of a single
broadcast waveform can be considered the
significant items. In this system, the demodulator
determines the changes in the phase of the received
signal rather than the phase (relative to a reference
wave) itself. Since this scheme depends on the
difference between successive phases, it is termed
differential phase-shift keying (DPSK). DPSK can
be significantly simpler to implement than ordinary
PSK since there is no need for the demodulator to
have a copy of the reference signal to determine the
exact phase of the received signal (it is a non-
coherent scheme). In exchange, it produces more
erroneous demodulations.
In PSK, the phase is changed to represent the data
signal. There are two fundamental ways of utilizing
the phase of a signal in this way:

3. - By viewing the phase itself as conveying the
information, in which case the demodulator must
have a reference signal to compare the received
signal's phase against; or
- By viewing the change in the phase as conveying
information — differential schemes, some of which
do not need a reference carrier (to a certain extent).
The figure below illustrates these two concepts:
Fig. 1 BPSK signal against its binary data stream
A convenient way to represent PSK schemes is
on a constellation diagram. This shows the points in
the complex plane where, in this context, the real
and imaginary axes are termed the in-phase and
quadrature axes respectively due to their 90°
separation. Such a representation on perpendicular
axes lends itself to straightforward implementation.
The amplitude of each point along the in-phase axis
is used to modulate a cosine (or sine) wave and the
amplitude along the quadrature axis to modulate a
sine (or cosine) wave.
In PSK, the constellation points chosen are usually
positioned with uniform angular spacing around a
circle. This gives maximum phase-separation
between adjacent points and thus the best immunity
to corruption. They are positioned on a circle so
that they can all be transmitted with the same
energy. In this way, the moduli of the complex
numbers they represent will be the same and thus so
will the amplitudes needed for the cosine and sine
waves. Two common examples are "binary phase-
shift keying" (BPSK) which uses two phases (see
Fig. 2), and "quadrature phase-shift keying" (QPSK)
which uses four phases (see Fig. 3), although any
number of phases may be used. Since the data to be
conveyed are usually binary, the PSK scheme is
usually designed with the number of constellation
points being a power of 2.
Fig. 2 Constellation diagram for BPSK
Fig. 3 Constellation diagram for QPSK
As said in the Abstract, we are going to
investigate on the QPSK (Fig. 3) since it is of more
recurrent use in the markets, and due to its decently
large bandwidth of transmission compared to BPSK,
it is more likely to be chosen than BPSK. But it is
worth to say that BPSK (also sometimes called
PRK, Phase Reversal Keying, or 2PSK) is the
simplest form of phase shift keying (PSK). It uses
two phases which are separated by 180° and so can
also be termed 2-PSK. It does not particularly
matter exactly where the constellation points are

4. positioned, and in this figure they are shown on the
real axis, at 0° and 180°. This modulation is the
most robust of all the PSKs since it takes the
highest level of noise or distortion to make the
demodulator reach an incorrect decision. It is,
however, only able to modulate at 1 bit/symbol (as
seen in Fig. 1) and so is unsuitable for high data-
rate applications.
The general form for BPSK follows the equation:
This yields two phases, 0 and π. In the specific form,
binary data is often conveyed with the following
signals:
for binary ‘0’ and:
for binary ‘1’, but this is of course arbitrary, and it
depend on the convention on the receiver side.
III. SIMULATION DESIGN
Emulating the QPSK is a matter of programming.
So after some researches on the MATLAB syntax
rules, and thanks to the canonical formulas that
rules the PSK modulation (see Analysis and
Discussion part), we have came out with a method
of PSK signal simulation which is algorithm based,
and programmed through MALTAB code.
We are going to elaborate on the Algorithm-based
technique which is considered as a non-linear
method, since the variable entered must pass
through some MATLAB mathematical shells such
as square-root, and cosinus as shown in the
upcoming formulas of PSK.
A. Algorithm-based Generation of PSK in MATLAB
Similarly to the BSK equation, the following
formula si(t) is the governing law for locating the
PSK phase in the constellation:
It is having four possible states (n=1..4), each one
represent a two bits data combination.
This yields to four phases π/4, 3π/4, 5π/4 and 7π/4
as needed.
That si(t) formula can be rearranged, to give way to
two vector components I & Q, as follows:
I(t)
Q(t)
I(t) and Q(t) are offset by 90°, and their negative
summation will lead directly to the general QPSK
Reduced Form RF, as depicted in Fig. 4.
Fig. 4 Block Diagram of an I/Q Modulator in QPSK
The first function I(t) is used as the in-phase
component of the signal and the second Q(t) as the
quadrature component of the signal (see Fig. 5)
Fig. 5 I and Q Represented in Polar Form
So our task consists in converting these formulas
into MATLAB codified form, and to follow the
modulation steps prescribed as in Fig. 4 & 5, and
that is the backbone of our simulation in next parts.

5. After due transformations and program flow
charting, we have reached a program that is
producing the desired QPSK output:
% Data sequence to be sent through PSK
d = [0 1 0 0 1 1 0 0 1 1]
% Convert unipolar to bipolar
b = 2*d-1
% Bit duration
Tb = 1
% Carrier frequency
fc = 3/Tb
% This will result in unit amplitude waveforms
Es = Tb/2
% Discrete time samples between 0 and
10*Tb(1000 samples)
t = linspace(0, 10, 1000);
% Number of samples
N = length(t)
% Number of samples per bit
Nsb = N/length(d)
% Replicate each bit Nsb times
dd = repmat(d', 1, Nsb);
bb = repmat(b', 1, Nsb);
% Represent dw into a column vector(column by
column)
dw = dd';
% Represent dw into a row vector(row by row)
dw = dw(:)';
% Represent bw into a column vector(column by
column)
bw = bb';
% Data sequence samples
bw = bw(:)';
o = b(1:2:end) %separating odd bits
e = b(2:2:end) %separating even bits
oo = repmat(o', 1, Nsb)
ee = repmat(e', 1, Nsb)
ow = oo'
ow = ow(:)'
ew = ee'
ew = ew(:)'
tc = linspace(0, 10, 500);
% Inserting the PSK formulas
IPhaseOsc = 1/sqrt(2*Es/Tb)*cos(2*pi*fc*tc)
QPhaseOsc = 1/sqrt(2*Es/Tb)*sin(2*pi*fc*tc)
qpskModulated = ow.*IPhaseOsc + ew.*QPhaseOsc
% Plotting the waveforms
subplot(4,1,1);
plot(t,dw);axis([0 10 -1.5 1.5])
subplot(4,1,2);
plot(t,bw);axis([0 10 -1.5 1.5])
subplot(4,1,3);
plot(tc,qpskModulated);axis([0 10 -1.5 1.5])
And as an output of this program, we get our PSK
modulated signal waveform, along with the original
binary data transmitted:
Fig. 6 MATLAB generated QPSK Signal
What we just did
in the program,
is to take
advantage of the
fact that QPSK
can be viewed as
two independent
BPSK signals.
And so we
created two
channels
IPhaseOsc and
QPhaseOsc that
were processed
independently by
MATALB and
then conjoined
later, to produce
the final QPSK
signal. The
following Fig. 7
is representing
how it was done
in MATLAB.
Fig. 7 Block
Diagram tracing the
Path of Data Flow
in our MATLAB
Code

6. B. Simulate QPSK Signal Using Scatter Plots
In this part, we will observe the generated signal
for our QPSK modulated system. The output
symbols are pulse shaped, using a raised cosine
filter.
To create a QPSK modulator object. We typed the
following at the MATLAB command line:
hMod = modem.pskmod('M', 4, 'PhaseOffset',
pi/4);
We created an upsampling filter, with an upsample
rate of 16:
Rup = 16; % up sampling rate
hFilDesign = fdesign.pulseshaping(Rup,'Raised
Cosine','Nsym,Beta',Rup,0.50);
hFil = design(hFilDesign);
We created the transmit signal:
d = randi([0 hMod.M-1], 100, 1); % Generate
data symbols
sym = modulate(hMod, d); % Generate
modulated symbols
xmt = filter(hFil, upsample(sym, Rup));
We created a scatter plot and we set the samples per
symbol to the upsampling rate of the signal:
hScope = commscope.ScatterPlot
hScope.SamplesPerSymbol = Rup;
In this simulation, the absolute sampling rate or
symbol rate is not specified. Se we use the default
value for SamplingFrequency, which is 8000. This
results in 2000 symbols per second symbol rate.
We set the constellation value of the scatter plot to
the expected constellation, by typing in MATLAB:
hScope.Constellation = hMod.Constellation;
Since the pulse shaping filter introduces a delay, we
discarded these transient values by setting
MeasurementDelay to the group delay of the filter,
which is four symbol durations or 4/Rs seconds:
groupDelay = (hFilDesign.NumberOfSymbols/2);
hScope.MeasurementDelay = groupDelay
/hScope.SymbolRate;
We updated the scatter plot with transmitted signal:
update(hScope, xmt)
We have then displayed the ideal constellation and
evaluated how closely it matched the transmitted
signal. To display the ideal constellation:
hScope.PlotSettings.Constellation = 'on';
One way to create a better match between the two
signals is to normalize the filter. We normalized the
filter by typing the following in MATLAB:
hFil.Numerator = hFil.Numerator /
max(hFil.Numerator);
We refilter the signal using a normalized filter.
xmt = filter(hFil, upsample(sym, Rup));
We reset the scope before displaying the
transmitted signal. Resetting the scope also resets
the counter for measurement delay, discarding the
transient filter values. To reset the scope, we typed
the following at the MATLAB command line:
reset(hScope)
And finally we updated for the second time the
scatter plot so it displays the QPSK signal.
update(hScope, xmt)
Fig. 8 The match between the ideal constellation points and
the transmitted signal is nearly identical

7. To view the transmitted signal more clearly, we
turned off the ideal constellation by clicking on its
ratio button in the Fig. 8 window, and we selected
Signal Trajectory to display all the possible
trajectories and movement of the phase shifts when
the signal is being modulated in real-time:
Fig. 9 Scatter Plot of our final QPSK signal generated
IV.ANALYSIS AND DISCUSSION
For proceeding to the analysis of our generated
signal, we opted for two useful techniques used by
professional of telecommunication domain, namely:
- The function h = modem.pskmod(M) which
constructs a PSK modulator object h for M-ary
modulation. And analyse it inside a transmission
system, and
- The graphical block QPSK Modulator
Baseband in Simulink GUI, which comes with a set
of virtual spectral analysers and scatter plot
displayers, as it would be in a real life laboratory.
These two techniques are inherent MATLAB
functions, used to produce PSK signals, without the
need of an Algorithm-based program, like the one
we developed in the part III-A. We only need to plug
these functions in the correct space in MATLAB or
Simulink, in order to get the full analysis of the
QPSK generation system.
A. Analyse the noise generated with the transmitted
PSK signal using the function modem.pskmod(M)
Coming back to our last signal generated, we
have the possibility to emulate the Additive White
Gaussian Noise (AWGN), by the mean of a simple
command y = awgn(xmt,snr), this command
add white Gaussian noise to our QPSK signal by
passing xmt that we created previously through an
AWGN channel. We therefore type the following at
the MATLAB command line in part III-B.:
rcv = awgn(xmt, 20, 'measured'); % Add AWGN
MATLAB will return the following plot:
Fig. 10 Scatter Plot of QPSK signal with GWN added to it

8. This last plot, is illustrating the effect of
interference and noise when hitting a QPSK signal.
The noise will not only affect the amplitude of the
QPSK signal (represented in blue points, mapped in
the four coins), but it will also have an adverse
impact on the trajectory of the phase changes,
which changed from straight lines to crooked and
broken lines.
We do not have any concern if the QPSK signal
amplitude is altered, because the information is not
residing there, but the problem is the distortions that
are corrupting the trajectories, and this may
compromise the conveying of the data.
But to have the certitude that the noise will have no
harmful fallouts, we conducted a full transmission
simulation in order to get the signal QPSK on the
reception side, and to compare it with the original
signal QPSK in the transmission side, to see if there
was any change of data values due to the noise.
After this full transmission simulation, we have
obtained these MATLAB results plotted in Fig. 11:
Fig. 11 Transmission and Reception of QPSK signal with
GWN added to it, and no change of data is observed
As demonstrated in the six plots above, the input
Binary Data is perfectly matching with the detected
Binary Data after QPSK demodulation is done, and
the noise have absolutely no threat to the veracity of
the data transmission as they have been preserved
intact through the transmission channel.
That is an experiment that shows the reliability of
QPSK and how conservative it is for the signal
transmitted, despite presence of noise or
interferences that mix up with the original signal.
B. The graphical block QPSK Modulator Baseband
in Simulink GUI used to analyse QPSK
Before we start in examining the signal from
Simulink platform, it is useful to remind about the
Bit Error Rate (BER) in relation with signal fidelity
at reception and how SNR can be taken into
account for getting the most faithful signal possible.

9. At this stage of our paper, it is now clear how
QPSK digital data is represented by 4 points around
a circle which correspond to 4 phases of the carrier
signal. These points are also called symbols. Fig. 12
shows this mapping in Gray Code disposition. Gray
coding is used in this mapping so that no two
adjacent symbols differ by more than 1 bit. This
helps in reducing the Bit Error Rate (BER).
Fig. 12 The Constellation of our QPSK mapped in Gray Code
To analyse QPSK signal in term of its BER
consideration, we need:
-Input Data to transmit/modulate with carrier
-Noisy Medium to transfer the data
-Demodulation of transmitted data.
-Comparison of original signal, and the
demodulated signal to calculate the Bit Error Rate
(BER)
For the input signal we can use the following
command:
x_signal = randint(1000,1,4);
To modulate the data against the grey coded
constellation, we can used the
modem.pskmod(M)MATLAB function.
y = modem.pskmod (x_signal,constell_gray);
Then we made a ‘for’ loop, each iteration of the
loop represents the Signal to Noise Ratio (SNR).
for SNR=0:2:10
y_noisy = awgn(y,SNR,'measured');
From the below graphs generated, we can see that
the higher the SNR, the better the received signal:
Fig. 13 The Constellation of our QPSK with SNR=6
Fig. 14 The Constellation of our QPSK with SNR=10
Now we can start the proper analyze using the
Simulink interface.
The output of the Simulink machine provides more
insight into the QPSK modulation technique. Apart
from plotting the modulated and demodulated
signal it also shows the constellation at
transmitter/receiver, and the Error Rate in a very
organized and structured manner. These Simulink

10. models are extracted from the internal library, as
they are inbuilt functions from communication
toolbox of Simulink.
1) Analyze with respect to Error Rate
By simply
grabbing and
dragging the
models stated in
the screenshot
beside, we have
succeed to
determine the
Error Rate, of a
typical QPSK
signal transmitted
through an
AWGN noisy
channel and
demodulated at
reception.
This Error Rate is
displayed on the
Sink provided by
Simulink Library
of Tools, and it
has a figure of
0.752 which is
close to ½ as in
theory. This rate
represents also
the probability of
error occurrence,
in the field of
computing error
statistics
2) Analyze with respect to Possible Transitions
The model in the screenshot beside plots the output
of our QPSK signal using the intrinsic Modulator
Baseband block associated with Simulink.
The image in Fig. 15 shows the possible transitions
from each symbol in the QPSK signal constellation
to the next symbol. And since QPSK is made from
a constellation of four symbols, so we obtain the
shape of a square, with diagonals:
Fig. 15 The
Constellation of
our QPSK Signal
with Transitions
from four
symbols,
generated via the
Complex to
Real-Imaginary
module of
Simulink
3) Analyze with respect to the Scatter Plot
Fig. 16 The
Scatter Plot of
our QPSK Signal
with dispersion
of symbols in the
four coins,
similar to what in
Fig. 10, 14, but
here using the
Scatter Plot
Scope found in
Simulink Library
of Sinks

11. V. CONCLUSION
So far, we have established that PSK modulation
is a very reliable technique of modulation, since the
transmission of data was done in conformity with
the ranges that we have set-up in MATLAB
program code parameters. PSK is hence producing
rugged signals, as they are characterized with an
impressing immunity to noise, such as White
Gaussian Noise or Electromagnetic Noise that we
tried to add to the pure signal generated.
With all this said, we have asserted that the
practical use of PSK modulation is efficient, cost-
effective and simple. And from the simulation
waveforms extracted from MATLAB, we can
clearly see that the transmitted binary data is an
image of the received data, and that they perfectly
match, and no loss in gain is observed, which just
mean that PSK is performing the transmission of
digital signals successfully.
Also concluded, is that among the several flavors of
phase shift keying that are available for use, each
form has its own advantages and disadvantages, and
a choice of the optimum format has to be made for
each digital communications system that is
designed. To make the right choice it is necessary
to have a close-to-reality simulation on MATLAB
or Simulink, beside the knowledge and
understanding of the way in which PSK works.
From our simulations and our related research in
books and articles, we found that, in general the
higher order forms of modulation allow higher data
rates to be carried within a given bandwidth.
However the downside is that the higher data rates
require a better signal to noise ratio before the error
rates start to rise and this counteracts any
improvements in data rate performance. In view of
this balance many radio communications systems
are able to dynamically choose the form of
modulation depending upon the prevailing
conditions and requirements.
All-in-all, this study was very conclusive, and it had
opened our eyes on many facets of PSK and digital
modulation in general, that was ambiguous to us
before. The use of MATLAB and Simulink was
very intuitive and productive, and we have acquired
important skills in writing, analysing and debugging
the codes. In Simulink we have found a more
straight-forward approach which we valorised
tremendously, and we saw solution in it to many
recurrent engineering and communication
problems. MATLAB is definitely the best
alternative to solve modulation-based problems and
to get simulations and results that will help into
improving the communication system developed,
and making crucial decisions on which modulation
technique is to be employed.
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