CAD OF COMMUNICATION SYSTEMS

COMPUTER AIDED DESIGN OF COMMUNICATION SYSTEMS
Professor Marina Mondin
This report is provided by:
Makan Mohammadi Ostadkalayeh
Plitecnico Di Torino
S201977

Ivan Karamazov says that, more than anything else, the death of children
makes him want to give back his ticket to the universe. But he does not
give it back. He keeps on fighting and loving; he keeps on keeping on.'
(Marshall Berman, New York City, January 1981)
Marshall Berman, all that is solid melts into air ,the experience of modernity

Introduction
Radio link is so useful and popular way in communication systems, satellite transmission,
cellular network and also there are many application in medical instrument and many others
but there are different problems mainly due to condition of the propagation medium as a
channel.
The simulation of communication system extends new windows for studying on various topics
and same as all simulations it implements as a referenced model that executes in accelerator
mode.
There are many approach but here we use Monte Carlo method for simulation with focus on
random number. Monte Carlo simulation method does not require to use random number but
for many techniques is essential.
Following simulation is base on digital simulation of system in analog domain, in the other word
digital input generates a digital output but this is discretized of analog output. This is a key idea
behind simulation theorem.
𝑇𝑐
𝑇𝑐
An useful simulation has to verify this condition as well:
𝑦 𝑑[𝑛] = 𝑦𝑎(𝑛𝑇𝑐)
Monte Carlo Simulation is performed to understand a performance estimation from the
analysis of the digital communication system in reality. A large number of i.i.d. random binary
sequences are generated, so that the overall analysis, exploiting all the possible event space,
allows to evaluate reliable performance indices as referred in 𝑦 𝑑[𝑛] = 𝑦𝑎(𝑛𝑇𝑐)
X(t)
𝑦 𝑑[𝑛]ℎ 𝑑[𝑛]
ℎ 𝑎(𝑡) 𝑦𝑎(𝑛𝑇𝑐)
)
ℎ 𝑎(𝑡)

QPSK/OQPSK Modulator:
Quadrature-phase shift keying (QPSK) is a form of phase-shift keying in which four different
phase angles are used. In QPSK, the four angles are usually separated by 90° spacing. QPSK or
Quadrature PSK is other form of angle-modulated, constant-amplitude digital modulation.
01 11
00 10
4-QAM Constellation
Offset quadrature phase-shift keying (OQPSK) is a variant of phase-shift keying modulation
using 4 different values of the phase to transmit. Taking four values of the phase (two bits) at a
time to construct a QPSK symbol can allow the phase of the signal to jump by as much as 180°
at a time. This produces large amplitude fluctuations in the signal; an undesirable quality in
communication systems. By offsetting the timing of the odd and even bits by one bit-period, or
half a symbol-period, the in-phase and quadrature components will never change at the same
time. In the constellation diagram shown on the left, it can be seen that this will limit the
phase-shift to no more than 90° at a time. This yields much lower amplitude fluctuations than
non-offset QPSK and is often preferred in practice.
QPSK

OQPSK
In this report there is a MATLAB function, called qpskmod.m that obtains 4-QAM.
As you know, the single carrier transmission system is composed by a random bit generator
which emits a binary uniformly distributed random sequence that is modulated by QPSK/OQPSK
modulator in order to produce a symbol sequence, where each symbol carries 𝑁𝑏 bits.
Upsampling is required to create shaped digital sequence from the symbol sequence. We will
see this operation help us to achieve a continues time signal which describes the symbol
sequence .This operation is performed by running the function called upsampling.m The so
obtained sequence is then upsampled to obtain a sufficient number of samples for the
following pulse shaping stage which is performed with Root Raised Cosine Filter.
Pulse Shaping stage -----------------------------------------------------------------------
In RF communication, pulse shaping is essential for making the digital signal to a signal in
frequency band. For this project we used Raised Cosine Filter. Raise Cosine Filter describes by
some parameter also splitting factor(𝛼) , Roll-off factor (𝜌) and M (implementation parameter).
In order to implement a Raised Cosine, first one it is required to split in the two relative TX and
RX filters and also splitting factor help us to recall filter in TX and RX . When 𝛼 = 0.5 filter
called Square Root Raised Cosine. Effect of 𝛼 is important, low values of 𝛼 increase amount of
oscillations and decrease peak.
Roll-off factor (𝜌): It is key element in Raise Cosine Filter that control the bending of the fall
region in frequency domain , mathematical expression for relation between roll-off and
bandwidth is:
𝐵 = 2𝑇𝑠(1 + 𝜌)

Similar to 𝛼 , Low value of 𝜌 induce a smoother behavior more similar to Sinc(.) whereas large
values of 𝜌 induce a sharper time evolution.
Implementation parameter (M): defines as the additional support of the filter over time axes. It
describes the number of side-lobes that exceed the main Ns typical support of the shaping
filter. Result is a new map and new representation from upsampled results .Eye diagram after
TX pulse shaping is presented in figure:
Time
Eye Diagram for Quadrature Signal
This signal has to be amplitude by Non Linear Amplifier cause of being ensure it has sufficient
power to reach the transmitter side. NLA model works with normalized output power. The
average power is computed from:
𝑃0 =
1
𝑁𝑠𝑎𝑚
∑ |𝑋(𝑛𝑇𝑐)|2
=
𝑁 𝑠𝑎𝑚
𝑛=1
1
𝑁𝑠𝑎𝑚
∑ (𝑋𝐼(𝑛𝑇𝑐))2
+ (𝑋 𝑄(𝑛𝑇𝑐))2
𝑁 𝑠𝑎𝑚
𝑛=1
−10 −5 0 5 10
−1
−0.5
0
0.5
1
Eye Diagram for In−Phase Signal
−10 −5 0 5 10
−1
−0.5
0
0.5
1
Time

𝑋𝐼(𝑛𝑇𝑐) , 𝑋 𝑄(𝑛𝑇𝑐) Represent in-phase and quadrature samples of upsampling result and 𝑁𝑠𝑎𝑚
are samples of signal scaled by 𝑃0 .
Non Linear Amplification ---------------------------------------------------------------------
Nonlinearity is the behavior of an amplifier, in which the output signal strength does not vary in
direct proportion to the input signal strength. In a nonlinear device, the output-to-input
amplitude ratio (also called the gain) depends on the strength of the input signal. In an
amplifier that exhibits nonlinearity, the output-versus-input signal amplitude graph appears as
a curved line over part or all of the input amplitude range. However real devices do not satisfy
this behavior and they usually effect on amplitude and phase distortions which depend on their
proper I/O characteristic and parameters. Since the amplifiers are memoryless devices, the
effects on Amplitude and Phase are only dependent on input power at current time. The signal
amplification is necessary for reaching sufficient power in transmitted side.
There are some key points about NLA that we have to attend them during our simulation:
PAPR, Back-off and finally effect of NLA on power of the signal.
PAPR or Peak-To-Average Power ratio is the peak amplitude squared (peak power) divided by
the RMS value squared (average power). PAPR show us amount of variation of power. It is clear
for reducing unwanted effects of NLA, it is better PAPR tend to 1. PAPR also describe in this way
in our simulation:
𝑃𝐴𝑃𝑅 =
𝑎𝑟𝑔𝑚𝑎𝑥 𝑋(𝑛𝑇𝑐)(|𝑋(𝑛𝑇𝑐)|2
)
𝑃0
The amount of power that can be appear in output is limited, maximum power reachable in
out-put is saturation power. All devices work on specific saturation zone, It is possible to set the
parameter known as Back-off that the amplifier works far from saturation point. Output Back-
Off or 𝛽 𝑜𝑢𝑡 (OPBO)is the power level at the output of amplifier relative to maximum output
level possible using the amplifier and also Input Back-Off or 𝛽𝑖𝑛 (IPBO) is the power level at the
input of amplifier relative to input power which produces maximum output power.
Two important effects of NLA operates on the power of signal is time distortion and bandwidth
enlargement.
Input signals with wider dynamics, as said before, reduce the enhancement of sidelobes in
power spectral density (Bandwidth Enlargement) and also Time Distortion is the natural effect
on waveform induced by amplitude and phase distortion in Non-Linear behavior.
In MATLAB code result of Raised Cosine Shaping Filter implement on Non Linear Amplifier (NLA)
in a function it is named:

function [Z,AM,PM]=interp_twt(X,BKOFF,flag)
Input data
X Input complex signal
BKOFF Amplifier back-off (operational point) in dB
flag Kind of the amplifier:
1. TWT (Travelling Wave Tube), RAI model
2. SSPA (Solid State Power Amplifier), ESA model
3. TWT (Travelling Wave Tube), 2nd model
4. SSPA (Solid State Power Amplifier), UOY model
Output data
Z Output complex signal
AM Amplitude assigned to X
PM Extra phase rotation assigned to X
AM/PM (for flag No 3 , TWT (Travelling Wave Tube), 2nd model) figure and Eye-Diagram show us
effect of NLA on the signal :

In following figures you can see clearly effect of NLA on the original signal in Eye Diagram and PSD .
Time
−10 −5 0 5 10
−1
−0.5
0
0.5
1
−10 −5 0 5 10
−1
−0.5
0
0.5
1
Time

Green one is PSD of original signal and red one is NLA out-put and effect of NLS flag No3 on
original signal.
−100 −80 −60 −40 −20 0 20 40 60 80 100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
10

Channel Model: ----------------------------------------------------------------------------
Modeling of a channel is a representation of signal variation during travel from transmitted side
to receiver side. Channel effect in system can be model with focus on 2 factor: Attenuation and
Time delay.
Channel attenuation is an important factor in the use of each transmission medium.
Attenuation can be describe as:
𝐴𝑓𝑠 = (
𝜆
4𝜋𝑙
)
2
= (
𝑐
4𝜋𝑙
)
2
Also it is so easy to compute Time delay, just enough to divide distance to velocity of light.
𝐷 =
𝑙
𝑐
Depending on channel conditions, the bandwidth occupation of the signal or propagation
environment of the signal, the channel may derive deviation of the attenuation on signal’s
spectral components.
This is named “Fading” and there are two kind of Slow Fading and Fast Fading.
Slow fading refers to the case where the channel frequency response is almost flat over the
signal bandwidth exactly opposite of Fast Fading.
In order to implement the channel behavior, different models may be considered same as
AWGN Channel and Two-Ray Path ones. Both are treated as linear filters.
AWGN Channel :
It's a simple model of the imperfections that a communication channel consists of. When you
transmit a certain signal into space or atmosphere or copper line to be received at the other
end, there are disturbances present in the channel (space/atmosphere/copper line) due to
various reasons. One such reason is the thermal noise by the virtue of electrons' movement in
the electronic circuit being used for transmission and reception of the signal. This disturbance
or noise is modelled as Additive White Gaussian Noise.
Additive: Because the noise will get added to your transmitted signal not multiplied. So, the
received signal 𝑦(𝑡) = 𝑥(𝑡) + 𝑛(𝑡) , where 𝑥(𝑡) was the original clean transmitted signal, and
𝑛(𝑡) is the noise or disturbance in the channel.
Gaussian: This thermal noise is random in nature, of course noise can't be deterministic
otherwise you would subtract the deterministic noise from 𝑦(𝑡) as soon as you receive 𝑦(𝑡).
So, this random thermal noise has Gaussian distribution with zero mean and variance as the
Noise power.
White: meaning same amount of all the colors. Or same power for all the frequencies. Which
means that this noise is equally present with the same power at all the frequencies. So, in
frequency domain, Noise level is flat throughout at every frequency.

Since the signal samples of 𝐼/𝑄 modulation schemes are supposed to be complex, the noise
samples are modelled as Complex Gaussian random variable with zero mean and variance 𝑁0.In
this case, the frequency response of the channel is constant, the coefficient of the equivalent
FIR filter would be a single delta in the origin.
𝑦(𝑡) = 𝑥(𝑡) ∗ 𝛿(𝑡)
Two Ray channel:
2-ray Ground Reflected Model is a radio propagation model that predicts path loss when the
signal received consists of the line of sight component and multi path component formed
predominately by a single ground reflected wave.
𝑥(𝑡) 𝑙1 𝑦(𝑡)
𝑙2
Attenuation: 𝐴 𝑡𝑟 = 𝐴𝑓𝑠|𝛼 𝑐|2
and 𝛼 𝑐 is complex coefficient define as:
𝛼 𝑐 = |𝛼 𝑐|𝑒−𝑖𝜑 𝑐
Finally: 𝑦(𝑡) = 𝑥(𝑡) + 𝛼 𝑐. 𝑥(𝑡 − ∆𝐷) where ∆𝐷 =
𝑙2−𝑙1
𝑐
The channel response is:
𝑦(𝑡) = 𝑥(𝑡) ∗ [𝛿(𝑡) + 𝛼 𝑐. 𝛿(𝑡 − ∆𝐷)] = 𝑥(𝑡) ∗ ℎ 𝑐(𝑡)
ℎ 𝑐(𝑡) = 𝛿(𝑡) + 𝛼 𝑐. 𝛿(𝑡 − ∆𝐷)
𝐻𝑐(𝑓) = 1 + 𝛼 𝑐. 𝑒−𝑖2𝜋𝑓∆𝐷
|𝐻𝑐(𝑓)|2
= 1 + |𝛼 𝑐| cos(2𝜋𝑓∆𝐷 − 𝜑𝑐) + |𝛼 𝑐|2
Finally you can see result in MATLAB code that describe by a set of parameter:

A_2rays ……………………………………………. Channel FIR A coefficient
B_2rays ……………………………………………. Channel FIR B coefficient
A_2rays =1;
B_2rays=[1 zeros(1,(3*Ns)-1) 0.15*exp(1i*pi/4)];
SignalRX_total_int = filter(B_2rays,A_2rays,SignalTXA_total_int);
Receiving Filter: --------------------------------------------------------------------------
After channel, the signal is in receiver side. First operation in the transmitted side is Receiving
filter. As you remember, splitting factor help us to manage filter in receiving side. Receiving filter
splitting factor is 1 − 𝜶 . This filter Out-put carries sufficient condition for reconstructing the
signal in frequency domain to symbols RX throughout downsampling. Exactly similar to
transmitting filter.In this figure you can see Eye diagram of receiving filter in useful channel in the
standard co-channel interference system model in this simulation:
Time
An important point about shaping filter part is a transient that there are some sample which are
not part of the symbol sequence labelled on the original constellation. They are also a part of
−10 −5 0 5 10
−1.5
−1
−0.5
0
0.5
1
1.5
−10 −5 0 5 10
−1.5
−1
−0.5
0
0.5
1
1.5
Time

each states during simulation than we have to remove them .This samples are removable by
Transient Removal stage.
The length of transient is:
𝑁𝑡𝑟𝑎𝑛,𝑠𝑦𝑚 =
𝑁𝑡𝑟𝑎𝑛 − 1
𝑁𝑠
𝑁𝑡𝑟𝑎𝑛 = 𝑁𝑡𝑟𝑎𝑛,𝑠𝑦𝑚. 𝑁𝑠 + 1
As you can see after calculate 𝑁𝑡𝑟𝑎𝑛 , we can use this MATLAB code for doing Transient removal
:
RxfNoTrans = Rxf(NTran:end);
Down-sampling and Demodulation: -------------------------------------------------------------
---
This section carries two approach OQPSK, QPSK. It is important to consider the exact sample
shift adopted in the first context in order to realign correctly the sequence.
It is easy to see from the Eye diagram such as the one reported in this Figure that the optimum
sampling instant is 𝑁𝑠in both cases, but in the case of OQPSK the Quadrature component needs
to be shifted back by 𝑁𝑠 = 2.This is exactly what the simulation does, before the received signal
is downsampled by taking a sample every 𝑁𝑠. This figure is before down sampling we effect of
co-channel and adjacent channel are present.

Functions for I/Q demodulation have been implemented for 4-QAM modulation schemes in
order to reconstruct the original bit sequence and experiment the presence of binary errors.
Demodulated bits are less than modulated ones because of shaping filtering issues.
Performance Mapping ---------------------------------------------------------------------
Now we need to map our performance also in general performance reviews are a key
component of system development. In this simulation, the tools which are used to appreciate
system performance contain:Spectral Estimation (Power Spectral Density (PSD) Estimation) and
BER Estimator.
Power Spectral Density (PSD) Estimation:
For continued signals that describe, for example, stationary physical processes, it makes more
sense to define a power spectral density(PSD), which describes how the power of a signal or
time series is distributed over the different frequencies, as in the simple example given
previously. In this simulation It is performed by an average sliding window estimator Welch’s
periodogram.
Here there is an example wich you can see Power Spectral Density (PSD) of useful channel
signal:
MATLAB Code:
winbm = window(@blackman,N);
psdTXA_use=pwelch(SignalTXA_use,winbm,0,N);
plot(freq,10*log10(fftshift(psdTXA_use)));

The classical definition of BER or bit error rate is the number of bit errors per unit time. The bit
error ratio (also BER) is the number of bit errors divided by the total number of
transferred bits during a studied time interval. BER is a unitless performance measure. In our
simulation this part is so important because the 𝑃𝑏(𝑒) Estimator provides an estimate of bit
error probability working on received signal samples versus 𝜂 = (𝐸 𝑏/𝑁0). The main idea behind
the estimator is to compute an estimation for each value of 𝜎 𝑛
2
.
Recall that , received signal samples are affected by complex noise samples distributed as
𝑁𝑐(0, 𝜎 𝑛
2
)
𝑃𝑛
̂ (𝑒|𝜎 𝑛
2
) =
1
𝑁 𝑇𝑜𝑡𝑆𝑦𝑚
∑ 𝑃𝑠
𝑁 𝑇𝑜𝑡𝑆𝑦𝑚
𝑖=1
(𝑒|𝑆𝑖, 𝜎 𝑛
2
)
Also for QPSK Error Probability Estimator
−100 −80 −60 −40 −20 0 20 40 60 80 100
−70
−60
−50
−40
−30
−20
−10
0
10

𝑃𝑏(𝑒|𝑆𝑖, 𝜎 𝑛
2) =
1
4
𝑒𝑟𝑓𝑐
(
|𝑆𝐼,𝑖|
√2𝜎𝐼,𝑛
2
)
+
1
4
𝑒𝑟𝑓𝑐
(
|𝑆 𝑄,𝑖|
√2𝜎 𝑄,𝑛
2
)
𝜎𝐼,𝑛
2
= 𝜎 𝑄,𝑛
2
=
𝑃𝑥. 𝑇𝑏. 𝜖(ℎ 𝑟)
2. (
𝐸 𝑏
𝑁0
)
Some parameters and equations are important, first one 𝜎 𝑛
2
= 𝑁0 𝜖(ℎ 𝑟) and note that 𝜖(ℎ 𝑟)is
energy of filter. Also we know this one 𝐸 𝑏 = 𝑃𝑥. 𝑇𝑏 is energy per bit and 𝑃𝑥 is the average power
of the signal before passing filter in the receiver side.
In MATLAB code you can follow this approach:
% RxSam : Symbol decision metric (Received complex signal after subsampling
at 1/Ts)
% EbNo : Vector or the Eb/No values in dB at which the BER is evaluated
% Ps : Power of the received c. envelope measured before the received filter
% Tb : Bit interval in seconds
% Eh : Receive filter energy
% *****************************************************
variances = (Ps*Tb*Eh)./(2.*EbNoValues);
A = 1./sqrt(2*variances);
% this method avoids loops
matrix_real=A'*abs(real(RxSam));
matrix_imag=A'*abs(imag(RxSam));
BER=0.25*(erfc(matrix_real)+erfc(matrix_imag)); % matrix with all the
conditional error probabilities
EstimatedBER=mean(BER,2); % this mean is done row by row

It is easy to see in the figure the behavior of 𝑃𝑏(𝑒) estimator when there are effect of adjacent and co-
channel and also theoretical 𝑃𝑏(𝑒).
0

Multi Carrier System ----------------------------------------------------------------------
Before discussing about simulating the behavior of multi carrier of communication systems, let
me to offer a set of parameters modelling the available bandwidth utilization have been
adopted.
Frequency shift for side channels ∆𝑓
Frequency shift for Co-channels
∆𝑓
2
Attenuation of Co-channel 𝐴 𝑐𝑜𝑐ℎ
Delay for right Adjacent Channel 𝑑 𝑟
Delay for left Adjacent Channel 𝑑𝑙
Delay for Co-Channels 𝑑 𝑐𝑜𝑐ℎ
Adjacent Channel:
In RF, Adjacent channel noise occurs because no RF transmitter is perfectly "on frequency" -
there is a bandwidth used by the transmitter, and a tolerance band around each channel. In
theory it is assumed to observe the closest lateral frequency slots where two other transmitter
are sending data. Side frequency slots, also referred as Adjacent channels, are equally spaced
around the central frequency 𝑓0 of the useful channel of ∆𝑓. We can simulate effect of adjacent
channel mathematically in this way:
𝑋 𝑟(𝑡) = 𝑋 𝑟,0(𝑡 − 𝑑 𝑟 𝑇𝐶)𝑒 𝑗2𝜋∇𝑓𝑡
𝑋𝑙(𝑡) = 𝑋𝑙,0(𝑡 − 𝑑𝑙 𝑇𝐶)𝑒 𝑗2𝜋∇𝑓𝑡
Where 𝑇𝐶 is sampling time.
As an example in this simulation project we can follow represent of this topic in MATLAB code:
This part is related by implement delay:
c_chTx(2,:) = [zeros(1,Delay_right) c_chTx(2,1:end-Delay_right)];
s_chTx(2,:) = [zeros(1,Delay_right) s_chTx(2,1:end-Delay_right)];
c_chTx(3,:) = [zeros(1,Delay_left) c_chTx(3,1:end-Delay_left)];
s_chTx(3,:) = [zeros(1,Delay_left) s_chTx(3,1:end-Delay_left)];
And this part is related by shifting frequency:
SignalTXA_right_mod = SignalTXA(2,:).*exp(1i*2*pi*Df*m*Tc);
SignalTXA_left_mod=SignalTXA(3,:).*exp(-1i*2*pi*Df*m*Tc);
There is same story about Co-Channel. Cross-polarization co-channels usually do not affect
respective reciprocal channels, but since the correct projection of the orthogonal polarization
cannot be performed perfectly, due to polarization imperfect polarization filtering, some power
from their spectrum is added to reciprocal channel. Two Co-channel interferers are

implemented by generating i.i.d. random sequences, mapping them into waveforms, adding
time delays, attenuating of 𝐴 𝑐𝑜𝑐ℎ, then shifting their spectral components by ±∆𝑓/2
𝑋 𝑟,𝑐𝑜𝑐ℎ(𝑡) = 𝑋 𝑟,𝑐𝑜𝑐ℎ(𝑡 − 𝑑 𝑐𝑜𝑐ℎ 𝑇𝐶)𝐴 𝑐𝑜𝑐ℎ
𝑋𝑙,𝑐𝑜𝑐ℎ(𝑡) = 𝑋𝑙,𝑐𝑜𝑐ℎ(𝑡 − 𝑑 𝑐𝑜𝑐ℎ 𝑇𝐶)𝐴 𝑐𝑜𝑐ℎ
MATLAB Code:
c_chTx(4,:) = [zeros(1,Delay_cochannel) c_chTx(4,1:end-Delay_cochannel)];
s_chTx(4,:) = [zeros(1,Delay_cochannel) s_chTx(4,1:end-Delay_cochannel)];
And
SignalTXA_cochannel_mod=SignalTXA(4,:)*Axp.*exp(1i*pi*0.25);
Optimization Stage: -------------------------------------------------------------------------
Before discussing about optimizing system, we have attend about two important point. First
one is related to factors that during our simulation are not able represent or map our main
signal. In this case NLA and Raised Cosine Filter out put do not provide and describe input
signal.
Second point is related to set the parameters used by the communication system that help us
in convenient way, our simulation system works in optimal conditions. In order to do this, let us
consider the following parameters of interest these parameters directly are connected to NLA
and Raised Cosine Shaping Filter:
_ 𝜌 roll-off factor of the root-raised cosine shaping filter.
_ 𝛼 splitting factor of the transmitter’s shaping filter (and also for the receiver one has factor 1-
𝛼;
_ 𝛽𝑖𝑛 back-off factor of the Non Linear Amplifier.
By managing different values of above parameters, the system can work in best performance. If
we want to reach a 𝐵𝐸𝑅𝑡𝑎𝑟𝑔𝑒𝑡, we have to evaluate Cost Function and set parameter to achieve
minimum cost.
𝐹(𝛼, 𝜌, 𝛽𝑖𝑛) =
𝐸 𝑏
𝑁0
| 𝑑𝐵+𝛽𝑜𝑢𝑡

Note and define about some parameters
Back-off
In nonlinear amplifier we have 𝛽𝑖𝑛and 𝛽 𝑜𝑢𝑡,as the ratio, in dB ,between input and output power
at saturation and average input and output power respectively:
𝛽𝑖𝑛 = 10𝑙𝑜𝑔10(𝑝𝑖𝑛,𝑠𝑎𝑡/𝑝𝑖𝑛,𝑎𝑣𝑒)
𝛽 𝑜𝑢𝑡 = 10𝑙𝑜𝑔10(𝑝 𝑜𝑢𝑡,𝑠𝑎𝑡/𝑝 𝑜𝑢𝑡,𝑎𝑣𝑒)
We always want to work close to saturation point because of having power efficiency (small
𝛽𝑖𝑛) but this way the output signal is strongly distorted.
ISI is high
Signal spectrum is significantly enlarged (ACI)
Therefore worse BER !
On the other hand when working far from saturation point, it has linear characteristics, so the
average output power is lower than the saturation power
Low transmitted power>>low received NSR (low power efficiency) Therefore back‐off is critical
parameter and needs to be optimized.
Roll off
Roll off tells how wide is the transition region. With 𝜌 = 0, we have rectangular shape. In
frequency domain, when we have small 𝜌, bandwidth is small, the waveform has large
oscillation in time, and also we have large PAPR .With large 𝜌, band width is large so we have
large distortion .Therefore roll off is another critical point and needs to be optimized.
Alpha (splitting factor)
Alpha is splitting factor. It is exponent of the raised cosine characteristic. It makes our filter
more vertical or horizontal in frequency domain. In order to minimize the error probability, we
would like to minimize the variance.
We look for small energy of received signal to minimize the noise, in the same time we would
like to have small energy of transmitted signal to minimize the power consumption. So we

should find an optimum Matched filter condition is met when alpha is equal to 0.5.In the real
world alpha is given by manufacturer.
𝑓 𝑎 𝑇 ∗ 𝐻 𝑅(𝑓)1−𝑎 𝑇 = 𝑅(𝑓)
Where 𝑅(𝑓)stays for raised cosine. In the case 𝛼 𝑇 = 0.5 the two filters are matched and in this
case we will talk about the Square-Root Raised cosine (SSRC).
𝑁𝑠:
In order to guarantee a good precision in digital models, some oversampling must be
considered. In general a good precision of our digital models is guaranteed over a so called
simulation bandwidth 𝐵𝑠𝑖𝑚.
𝐵𝑎𝑛𝑑𝑤𝑖𝑑𝑡ℎ 𝑠𝑖𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 = 1/5𝑇𝑐
𝑁𝑠 defines our simulation bandwidth. We know that analog bandwidth is infinite, therefore
digital simulating (with finite frequency axis) of analog system is impossible. The more
compacted BW in frequency, the smaller is 𝑁𝑠 .For Raised Cosine filter, we can choose 𝑁𝑠 = 8.
𝑇𝐶:
Sampling interval
𝐵𝑎𝑛𝑑𝑤𝑖𝑑𝑡ℎ 𝑠𝑖𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 = 1/5𝑇𝑐
𝑀:
𝑀 has an impact on the length of the raised cosine impulse response in such a way that:
2𝑁𝑠(𝑀 + 1) By choosing small 𝑀, filter length is small and it does not work well, therefore we
will have ISI and also loss.

System Analyses By Simulation: -----------------------------------------------
The transmission system sketched in the figure. It transmits QPSK modulated symbols over four
independent channels with bit‐rate 40 Mbit/s, with independent non‐linear amplifiers. The
channels have a channel spacing of ∆𝑓 and transmit the same power, normalized to 1 W at the
input of each amplifier.
All the channels employ raised cosine shaping filters with splitting factor at the transmitter 𝛼 𝑇,
the same roll‐off factor 𝜌, and also we use the same back‐off 𝛽 for the non‐linear amplifiers,
but they are not synchronous. We use 𝑁𝑠 = 8.The saturation power of the amplifiers is also 1
W.
For generate asynchronous data channels, we insert a delay after the TX filter, delay has this
properties:
Delay_left = 7; % in number of samples
Delay_right = 2; % in number of samples
Delay_cochannel = 5; % in number of samples
Delay_cochannel_right = 1; % in number of samples
Delay_cochannel_left = 4; % in number of samples
𝑒 𝑗2 𝜋∆ 𝑓𝑚 𝑇 𝐶
𝑒−𝑗2 𝜋∆ 𝑓𝑚 𝑇 𝐶
Cross Polarization Attenuation 𝑒 𝑗𝜋 /4
source Raised
Cosine
Mapper
source Raised
Cosine
Mapper
source
Raised
Cosine
Mapper
source Raised
Cosine
Mapper
Receiver
channel
Cha.

First one we try to observer delay at the eye diagrams:
As you can see Useful channel eye diagram do not has any shift to left and right and also It is
clear and sharp shifting left and right in the eye diagrams. The below we have eye diagrams of
the signals after TX filtering and delay insertion and before NLA.
Useful Channel
−10 −5 0 5 10
−1
−0.5
0
0.5
1
Time
−10 −5 0 5 10
−1
−0.5
0
0.5
1
Time

After implement Delay Right
−10 −5 0 5 10
−1
−0.5
0
0.5
1
Time
−10 −5 0 5 10
−1
−0.5
0
0.5
1
Time

After implement delay left :
−10 −5 0 5 10
−1
−0.5
0
0.5
1
Time
−10 −5 0 5 10
−1
−0.5
0
0.5
1
Time

Delay Co-Channel
−10 −5 0 5 10
−1
−0.5
0
0.5
1
Time
−10 −5 0 5 10
−1
−0.5
0
0.5
1
Time

This is Non Linear Amplifier behavior:
As you remember – we discussed about that – after normalization of the signal we need to use
NLA for reaching sufficient power to send signal via channel. In MATLAB code this part is related
to NLA :
SignalTXA(index,:) = interp_twt(SignalTX(index,:),BKOFF,nla_flag);
SignalTXA(index,:) is out-put , TX is input signal , BKOFF describes back-off input –in here is
variable between 0-12 – and also nla_flag describe kind of NLA . this figure shows us behavior
of NLA :
% AM [vector] amplitude assigned to X
% PM [vector] extra phase rotation assigned to X

Now we can follow effect of NLA on signal via eye diagram. Here I selected a high back-off
number for useful channel signal for showing non-linear effect of amplifier :
Time
−10 −5 0 5 10
−1
−0.5
0
0.5
1
−10 −5 0 5 10
−1
−0.5
0
0.5
1
Time

But best approach for understanding effect of NLA is obvious of PSD figures because as you see
in the introduction part, PSD describes how the power of a signal or time series is distributed
over the different frequencies. In this simulation It is performed by an average sliding window
estimator Welch’s periodogram.
Power Spectrum density (All)
−80 −60 −40 −20 0 20 40 60 80
−120
−100
−80
−60
−40
−20
0
Useful
Left mode
right mode
Co−Ch
Total

−80 −60 −40 −20 0 20 40 60 80
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
main channel
−80 −60 −40 −20 0 20 40 60 80
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Right Channel

−80 −60 −40 −20 0 20 40 60 80
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Left Channel
−80 −60 −40 −20 0 20 40 60 80
−110
−100
−90
−80
−70
−60
−50
−40
−30
−20
Co−Channel

Just for summaries, we know Raised Cosine filter and NLA effects can be manage and control by
three essential factors, Roll-off factor (𝜌), splitting factor(𝛼), 𝛽𝑖𝑛 back-off factor and also we
know by managing different values of above parameters, the system can work in best
performance. If we want to reach 𝐸𝑅𝑡𝑎𝑟𝑔𝑒𝑡. we have to evaluate Cost Function and set
parameters to achieve minimum cost.
In this simulation 𝐵𝐸𝑅𝑡𝑎𝑟𝑔𝑒𝑡 = 5𝑒 − 3.
As you can follow in Introduction part of this report, semi-analytic estimator we use for the
computation of the error probability depends on the vector of the received samples and on the
𝜂 = (𝐸 𝑏/𝑁0)values in this approach:
𝑃𝑏(𝑒|𝑆𝑖, 𝜎 𝑛
2) =
1
4
𝑒𝑟𝑓𝑐
(
|𝑆𝐼,𝑖|
√2𝜎𝐼,𝑛
2
)
+
1
4
𝑒𝑟𝑓𝑐
(
|𝑆 𝑄,𝑖|
√2𝜎 𝑄,𝑛
2
)
Received signal samples are affected by complex noise samples distributed as 𝑁𝑐(0, 𝜎 𝑛
2
) and
also
the variance associated to the additive noise that can be expressed as a function of 𝜂 =
(𝐸 𝑏/𝑁0)
𝜎𝐼,𝑛
2
= 𝜎 𝑄,𝑛
2
=
𝑃𝑥. 𝑇𝑏. 𝜖(ℎ 𝑟)
2. (
𝐸 𝑏
𝑁0
)
At the same time, we know how the performance of our system depends on a trade-off
between the power efficiency (measured by output back-off) and the bit error probability
(measured by the signal to noise ratio 𝜂 = (𝐸 𝑏/𝑁0). If both these quantity decreases, the
quality of the system increases from a bit error probability point of view and from a power
efficiency standpoint.
For this reason has been introduced an important function around which is based the
procedure of optimization. This function, called Cost Function, is defined as follows
𝐹(𝛼, 𝜌, 𝛽𝑖𝑛) =
𝐸 𝑏
𝑁0
| 𝑑𝐵+𝛽 𝑜𝑢𝑡
And for optimization just enough
𝑃𝑜𝑝𝑚 = 𝑎𝑟𝑔𝑚𝑖𝑛𝐹𝑡𝑎𝑟𝑔𝑒𝑡(𝑝)
And we can easily write :
𝜎2
=
𝑃𝑜𝑢𝑡𝑠𝑎𝑡. 𝑇𝑏. 𝜖(ℎ 𝑟)
2. 𝐹
The optimization of the system will be performed that will be followed by a performance
analysis and losses.in this case prior the beginning of the starting the real optimization taking

into account all the parameters jointly, a first optimization necessary to obtain a raw
approximation of the optimal back-off 𝛽𝑖𝑛,𝑜𝑝𝑡 is performed with an initial Roll-off factor (𝜌 =
0.5), splitting factor(𝛼 = 0.5).
Cost function vs β @ BER
Input Backoff (β )
in
result for QPSK :
The minimum cost is : 7.158 dB and Raw optimum input back off around = 3.
Optimized Parameter Alfa Rho Back-off Cost
QPSK/Ideal Channel 0.5 0.5 3 7.158
Actually this result is at the presence of effect of Co-channel and adjacent interfering channel(marked
TOTAL).
0 2 4 6 8 10 12
7
7.5
8
8.5
9
9.5
10
10.5
11
11.5
12
in target

For better understanding of effect of Co-channel and adjacent interfering channel we
evaluated results for amplitude signal in absent of Co-Channel and Adjacent Channel
effect(marked NLA ) , amplitude signal in absent of Co-Channel but in presence of
Adjacent Channel effect(marked NLA+ACI or NLA+Adj) and finally for signal without use
of nonlinear amplifier and interference of Adjacent channel and Co-Channel (marked IDEAL).
Here there is table for introduce results for QPSK at the optimal point :
Ideal NLA NLA+Adj Total
QPSK 7.158 6.445 6.816 5.045
Also for OQPSK we can compute the result first one we have to find back-off and optimal point
and after that we can find the results for different situation:
OQPSK/Ideal Channel 0.5 0.5 2 6.494
0 1 2 3 4 5 6 7 8 9
6
6.5
7
7.5
8
8.5
9
9.5
X: 2
Y: 6.494
Input Back-off
Cost function

As you can see in the figure back-off is 2 and cost function for total is 6.494
QPSK 5.045 6.445 6.816 7.158
OQPSK 5.042 6.141 6.279 6.494
Now we can calculate lost table according to previous table:
Loss /Ideal channel Ideal NLA NLA+Adj Total
QPSk 0 1.4 1.771 2.113
OQPSK 0 1.099 1.237 1.452
As you can see in all situation OQPSK results is better than QPSK and also back-off optimal for
OQPSK is less than QPSK than main advantage of OQPSK over QPSK is that the amplifiers can be
used in more power efficient manner.Another point is related to NLA , deference between ideal
case and NLA is too much than deference between NLA and NLA+Adj orTotal case, then for
improve this system in reality first one we have to improve behavior of NLA because more part
of loss is related to NLA.
BER , Ideal Channel//Optimized Paramete Alfa :0.5 Rho: 0.5 Back−off pt = 3
0
5 6 7 8 9 10 11 12 13 14 15

BER /Ideal Channel/OQPSK
Optimized Parameter rho: 0.5 Alfa: 0.5 BackOffpt :2
−3
Let me to improve my result, because in the estimation of back-off our interval is 1 and
we compute the back-off between 0-12, this point is important with small changes in
back-off we have changing in Roll-off factor, splitting factor than for better result I
decided to compute results according to this parameter :
For QPSK –Ideal Channel ,also I compute result for intervals around of optimal point.
rho_v = 0.6:0.05:0.7; % vector of roll off
alphat_v = 0.6:0.05:0.7; % vector of splitting ratios at the transmitter
BKOFF_v = BKOFF_opt-2:0.2:BKOFF_opt+2; % vector of input back off
You can see the result in this table :
QPSK/Ideal Channel 0.65 0.6 2.2 6.873

We discuss about this result in the next assignment but just I want to show you
optimized parameter are very sensitive to variation.
QPSK 5.136 dB 6.240 6.582 6.873
Also we can follow the results for OQPSK for this interval is
OQPSK/Ideal Channel 0.45 0.7 1.2 6.260
OQPSK 5.054 5.720 5.877 6.114
Final result can be summaries:
QPSK 5.136 6.240 6.582 6.873
OQPSK 5.054 5.720 5.877 6.114

loss analysis:
loss analysis Ideal NLA NLA+Adj Total
QPSK 0 1.04 1.446 1.77
OQPSK 0 0.666 0.823 1.06
As you can see – same as past loss analysis table – result of OQPSK is better than QPSK .
BER /QPSK/Ideal Channel
QPSK/Ideal Channel 0.65 0.6 2.2 6.873
-1
BER /QPSK/Ideal Channel
OQPSK/Ideal Channel 0.45 0.7 1.2 6.260

-1
We discussed about 2-rays channel in Channel part in Introduction section, now we want to
implement it with this properties as a FIR filter coefficient:
A_2rays =1;
B_2rays=[1 zeros(1,(3*Ns)-1) 0.15*exp(1i*pi/4)];
Same as ideal channel first we calculate back-off according to Roll-off factor (𝜌 = 0.5), splitting
factor(𝛼 = 0.5).

Result is easily visible in the figure, The minimum cost is: 7.811 dB and raw optimum input
back-off around= 3.00 dB
QPSK/2-ray Channel 0.5 0.5 3 7.811
0 2 4 6 8 10 12
7
8
9
10
11
12
13
X: 3
Y: 7.811
Input Backoff
Cost function

2-ray Channel Ideal NLA NLA+Adj Total
QPSK 5.678 7.051 7.369 7.811
BER/2−ray/QPSK
Optimized Parameter alfa:0.5 rho: 0.5 Backoffpt: 3
−2

Result for OQPSK is easily visible in the figure, The minimum cost is: 7.283 dB raw optimum
input back off around = 2.00 dB
OQPSK/2-ray Channel 0.5 0.5 2 7.283
0 2 4 6 8 10 12
7
8
9
10
11
12
13
Input Backoff
Cost function

OQPSK 5.625 6.822 7.063 7.283
−3
We can all results for 2-ray Channel in this table:
QPSK 5.678 7.051 7.369 7.811
OQPSK 5.625 6.822 7.063 7.283

Now we can prepare loss analysis table
QPSK 0 1.373 1.691 2.133
OQPSK 0 1.197 1.438 1.658
As you can see again OQPSK results is better than QPSK and also in back-off OQPSK score is
better than QPSK and this means OQPSK is better modulator for our systems according to
optimal point.
Now we can find better point for working around optimal point, just enough to compute results
in this intervals:
BKOFF_v = 1:0.2:3; % vector of input back off
Result will be :
QPSK/2-ray Channel 0.650 0.650 2 .40 7.401
And also result and figure for QPSK and OQPSK are shown in the tables and aslo you can see
back-off figures Result for OQPSK 2-ray
OQPSK/2-ray Channel 0.45 0.7 1.2 6.260

BER/2-ray Channel/QPSK
QPSK/2-ray Channel 0.650 0.650 2 .40 7.401
−1

BER/2-ray Channel/QPSK
OQPSK/2-ray Channel 0.45 0.7 1.2 6.260
−1

9.5 10 10.5 11 11.5 12 12.5 13 13.5 14

LAB7
In this case we face with three interval for optimized parameters Roll-off factor (𝜌), splitting
factor(𝛼), 𝛽𝑖𝑛 back-off factor.
We started with Ideal channel , Intervals are described as this way :
BKOFF_opt=3;
Idealchannel/QPSK Alfa Rho Back-off cost
0.65 0.6 2.2 6.873
0 2 4 6 8 10 12
6
7
8
9
10
11
12
Input Backoff
Cost function

And also you can see result for specific signal
Qpsk Ideal NLA NLA+Adj Total
Cost 5.136 6.240 6.582 6.901
−2
We repeated results for OQPSK and for Ideal channel :
Idealchannel/OQPSK Alfa Rho Back-off cost
0.45 0.7 1.2 6.260
And again we computed results for ideal signal, signal after amplification , signal after
amplification and after effect of adjacent channel and finally signal after amplification and after
effect of all interfere effects
OQPSK Ideal NLA NLA+Adj Total
Cost 5.054 5.720 5.877 6.114

0 2 4 6 8 10 12
6
7
8
9
10
11
12
Input Backoff
Cost function

−3
Now we can see result for ideal channel altogether in this table :
Cost OQPSK 5.054 5.720 5.877 6.114
Cost QPSK 5.136 6.240 6.582 6.901
Finally loss analysis table for ideal channel
Loss analysis Ideal NLA NLA+Adj Total
Cost OQPSK 0 0.666 0.823 1.06
Cost QPSK 0 1.104 1.446 1.765

As you can see again result describe not good behavior of NLA in OQPSK is less than QPSK and
also in in general total loss of case1 (OQPSK) is less than case2 (QPSK). Also Case 1 with less
back-off has efficient use from NLA.
2-ray channel/QPSK Alfa Rho Back-off cost
0.65 0.65 2.4 7.558
2-ray channel Ideal NLA NLA+Adj Total
Cost 5.765 6.837 7.156 7.558
0 2 4 6 8 10 12
7
8
9
10
11
12
13
Input Backoff
Cost function

−2
2-ray channel/OQPSK Alfa Rho Back-off cost
0.45 0.9 0.8 6.861
2-ray channel OQPSK Ideal NLA NLA+Adj Total
Cost 5.571 5.968 6.492 6.861

Finally here you can see all results together
Cost QPSK 5.765 6.837 7.156 7.558
Cost OQPSK 5.571 5.968 6.492 6.861
Loss analysis describe in this table
Loss QPSK 0 1.072 1.391 1.793
Loss OQPSK 0 0.397 0.921 1.29

Let me to introduce my simulation , in general we can see in all cases with implementing OQPSK
as modulator result was better than others. And also OQPSK in all situation had less back-off –
means NLA was in better condition for working in comparing with others .
For better understanding we can follow results in scattering diagram
(Note : left-up: Ideal Case , Right-Up: NLA Case , Left-Down : NLA+Adj , Right-Down:Total)

(Note : left-up: Ideal Case , Right-Up: NLA Case , Left-Down : NLA+Adj , Right-Down:Total)
QPSK/Ideal Channel 0.5 0.5 3 7.158
QPSK 7.158 6.445 6.816 5.045
QPSk 0 1.4 1.771 2.113

OQPSK/Ideal Channel 0.5 0.5 2 6.494
OQPSK 5.042 6.141 6.279 6.494
OQPSK 0 1.099 1.237 1.452

QPSK/2-ray Channel 0.5 0.5 3 7.811
QPSK 5.678 7.051 7.369 7.811
QPSK 0 1.373 1.691 2.133

OQPSK/2-ray Channel 0.5 0.5 2 7.283
OQPSK 5.625 6.822 7.063 7.283
OQPSK 0 1.197 1.438 1.658

Idealchannel/QPSK Alfa Rho Back-off cost
0.65 0.6 2.2 6.873
Cost QPSK 5.136 6.240 6.582 6.901
Cost QPSK 0 1.104 1.446 1.765

Idealchannel/OQPSK Alfa Rho Back-off cost
0.45 0.7 1.2 6.260
Cost OQPSK 5.054 5.720 5.877 6.114
Cost OQPSK 0 0.666 0.823 1.06

2-ray channel/QPSK Alfa Rho Back-off cost
0.65 0.65 2.4 7.558
Cost 5.765 6.837 7.156 7.558
Loss QPSK 0 1.072 1.391 1.793

2-ray channel/OQPSK Alfa Rho Back-off cost
0.45 0.9 0.8 6.861
2-ray channel OQPSK Ideal NLA NLA+Adj Total
Cost 5.571 5.968 6.492 6.861
Loss OQPSK 0 0.397 0.921 1.29

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