INDEX

S.No. Name of Experiment Date Sign

1 To Plot the power spectrum pattern of the Gaussian

Minimum Shift keying

2 Write a program to plot the attenuated signal pattern,

when the signal is propagated over a long distance (Km)

3 Write a program to Plot the different Pattern of Gaussian

Function by varying the standard deviation.

4 Write a program to plot different pattern of Rayleigh

function by varying the Rayleigh constant

5 Write a program to plot the Doppler fading power

spectrum pattern & compare the result by varying the

Doppler frequency shift.

6 To find the output of convolution encoder & decoder

7 To Plot the power spectrum pattern of the

Gaussian frequency shift keying.

8 To Plot the power spectrum pattern of the Minimum

Shift keying

It is hereby certified that …………………………………………, student of M.tech(ECE), Final

year, Roll No……………………………….has performed all the above experiments

successfully.

(Assoc. Prof. Alka Kalra)Teacher Incharge

Experiment No. – 1

Aim:- To Plot the power spectrum pattern of the Gaussian Minimum Shift keying.

Theory:

GMSK modulation is based on MSK, which is itself a form of phase shift keying. One of

the problems with standard forms of PSK is that sidebands extend out from the carrier.

To overcome this, MSK and its derivative GMSK can be used.

MSK and also GMSK modulation are known as a continuous phase scheme. It means

there are no phase discontinuities in MSK and GMSK. This arises as a result of the

unique factor of MSK that the frequency difference between the logical one and logical

zero states is always equal to half the data rate. This can be expressed in terms of the

modulation index, and it is always equal to 0.5.

Figure : Signal using MSK modulation

In GMSK ,the sidelobe levels of the spectrum are further reduced by passing the

modulating signal through a low pass filter prior to applying it to the carrier. The

requirements for the filter are that it should have a sharp cut-off, narrow bandwidth. The

ideal filter is known as a Gaussian filter which has a Gaussian shaped response to an

impulse.

Figure : Spectral density of MSK and GMSK signals

Generating GMSK modulation

One of the way in which GMSK modulation can be done is to filter the modulating

signal using a Gaussian filter and then apply this to a frequency modulator where the

modulation index is set to 0.5. This method is very simple and straightforward but it has

the drawback that the modulation index must exactly equal 0.5. In practice this method is

not suitable because component tolerances drift and cannot be set exactly.

Figure : Generating GMSK using a Gaussian filter and VCO

Block Diagram

GMSK filter may be completely defined from the 3 dB baseband bandwidth, B and

baseband symbol duration,T. It is therefore customary to define GMSK by its BT

product. As the BT product decreases, the sidelobe levels falls off very rapidly. However

reducing BT increases the error rate produced by filter due to intersymbol interference.

Experiment No. – 2

Aim :- Write a program to plot the attenuated signal pattern, when the signal is propagated

over a long distance (Km)

Theory:- Free-space path loss (FSPL) is the loss in signal strength of an electromagnetic wave

that would result from a line-of-sight path through free space, with no obstacles

nearby to cause reflection or diffraction. It does not include factors such as the gain of

the antennas used at the transmitter and receiver, nor any loss associated with

hardware imperfections.

To understand the reasons for the free space path loss, it is convenient to imagine a

signal spreading out from a transmitter. The signal will move away from the source

spreading out in the form of a sphere. As it does so, the surface area of the sphere

increases. According to law of conservation of energy, as the surface area of the

sphere increases, so the strength of the signal must decrease. As a result of this it is

found that the signal strength decreases in a way that is inversely proportional to the

square of the distance from the source of the radio signal.

Pr(d) =

Pt = Transmitted Power

Pr= Received Power

L= System Loss Factor not related to propagation (L1)

= wavelength in meters

Gain of an Antenna

Where Ae = effective Aperture.

Program

% To plot the Attenuated signal Pattern

pt=50; % transmitted power in watt

Gt=1; % gain of transmitter

Gr=1; % gain of receiver

λ =.6; % wavelength

l=1; % loss factor

d=10:10:100; % distance between transmitter and receiver

pr(d)=(pt*Gt*Gr*( λ )^2)./((4*pi)^2*(d).^2); % Received power in watt

subplot(2,1,1);

plot(d,pr(d));

xlabel(‘Distance between Transmitter and Receiver’);

ylabel(‘Received power in Watt’);

title(‘Attenuated signal Pattern’)

Experiment No. – 3

Aim :- Write a program to Plot the different Pattern of Gaussian Function by varying the

standard deviation.

Theory: A Gaussian function, or distribution, has the form:

where σ is the standard deviation, and μ is the x-offset of the Gaussian function from

zero.

Gaussian functions approximate the shapes of many observables in astronomy, such as

the profiles of seeing disks, the width of spectral lines, and the distribution of noise in

radio receivers. In error analysis, the Gaussian function is often used to determine the

significance of a measurement. Astronomers like to talk about whether a result is 2-σ, or

3-σ etc. This refers to the likelihood that an experimental result is a certain number of

standard deviations from the measured value. 67% of all results are within 1-σ of the

mean whereas 95.4% of within 2-σ and 99.7% with 3-σ. Thus a 1-σ result is 33% likely

to be due to random fluctuations, whereas a 3-σ result is very secure. The Gaussian

distribution is sometimes called the "normal distribution ".

Program

clear all

% To Plot the different Pattern of Gaussian Function by varying the standard deviation.

n=[1 2 3 4 5];

j=1;

for i=1:5

for x=-25:0.1:25;

X(j,1)=x;

Y(j,i)=(exp((-x.^2)./(2*(n(i)).^2)))./(sqrt(2*pi.*(n(i)).^2));

j=j+1;

end

end

plot(X,Y,'linewidth',2);

grid on

legend('\sigma= 1','\sigma = 2','\sigma = 3','\sigma = 4','\sigma = 5',5);

xlabel('x')

ylabel('F(x)')

AS the value of standard deviation increases, Gaussian PDF decreases. Gaussian PDF has

peak at the point where random variable has its value equal to its mean.

Experiment No. – 4

Aim :- Write a program to plot different pattern of Rayleigh function by varying the Rayleigh

constant

Theory:-

Rayleigh fading is a model that can be used to describe the form of fading that occurs

when multipath propagation exists. In built-up urban areas, fading occurs because the

height of the mobile antennas are well below the height of surrounding structures, so

there is no single line-of-sight path to the base station. Even when a line of sight path

exists, multipath still occur due to reflections from the ground and surrounding structures.

Also there is often movement of the transmitter or the receiver and it can cause the path

lengths to change and accordingly the signal level will vary.

The incoming radio waves arrive at receiver from different directions with different

propagation delays. The composite signal at the receiver is a combination of all the

signals that have reached the receiver via the multitude of different paths that are

available. The resultant signal will depend upon the phase of different individual

multipath signals. Dependent upon the way in which these multipath signals sum

together, the resultant signal will vary in strength. If the former are all in phase with each

other, they would all add together. However this is not normally the case, as some will be

in phase and others out of phase, depending upon the various path lengths, and therefore

some will tend to add to the overall signal, whereas others will subtract.

Rayleigh distribution

It is a continuous probability distribution. A Rayleigh distribution is often observed

when the overall magnitude of a vector is related to its directional components. One

example where the Rayleigh distribution naturally arises is when wind speed is analyzed

into its orthogonal 2-dimensional vector components. Assuming that the magnitude of

each component is uncorrelated and normally distributed with equal variance, and then

the overall wind speed (vector magnitude) will be characterized by a Rayleigh

distribution.

FORMULAE:

The Rayleigh probability density function is

and Rayleigh cumulative distribution function

for and parameter σ.

Program

%Rayleigh Function

close all

clear all

N = 100;

x = randn(1,N); % gaussian random variable, mean 0, variance 1

y = randn(1,N); % gaussian random variable, mean 0, variance 1

z = (x + j*y); % complex random variable

% probability density function of abs(z)

r = [0:0.01:7];

sigma2 = 1;

pzTheory = (r/sigma2).*exp(-(r.^2)/(2*sigma2)); % theory

plot(r,pzTheory,'b.-')

xlabel('z');

ylabel('probability density, p(z)');

title('Probability density function of Rayleigh distribution' )

axis([0 7 0 0.7]);

grid on

figure(2)

%%calculate power of rayleigh channel

h = 1/sqrt(2)*(randn(1,N) + j*randn(1,N)) % rayleigh random variable

hP = h.*conj(h); % finding the power

hist(hP) % plotting the histogram

xlabel('z'); ylabel('CDF(z)');

title(' Rayleigh cumulative distribution function ' )

0 1 2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

z

prob

abili

ty d

ensi

ty, p

(z)

Probability density function of Rayleigh function

Experiment No. – 5

Aim :- Write a program to plot the Doppler fading power spectrum pattern & compare the

result by varying the Doppler frequency shift.

Theory:-

To mitigate the effects of multipath interference, which prevail in mobile wireless

communication systems, directional antennas may be used at one or both ends of the

radio link . A good understanding of the channel behaviors is essential for evaluating and

analyzing the performance of the systems employing directional antennas. The Doppler

power spectral density is the key parameter in the description of mobile radio channel

The Doppler power spectral density of a fading channel describes how much spectral

broadening it causes. This shows how a pure frequency e.g. a pure sinusoid, which is an

impulse in the frequency domain is spread out across frequency when it passes through

the channel. It is the Fourier transform of the time-autocorrelation function. For Rayleigh

fading with a vertical receive antenna with equal sensitivity in all directions, power

spectral density has been shown as

Where v is the frequency shift relative to the carrier frequency. This spectrum is shown

in the figure for a maximum Doppler shift of 10 Hz. The 'bowl shape' or 'bathtub shape' is

the classic form of this Doppler spectrum.

Program

clc;

clear all;

%10Hz max doppler.

v=-9.9:0.1:9.9;

fd=10;

jakes_psd=1./(pi*fd*sqrt(1-(v./fd).^2));

plot(v,jakes_psd);

xlabel('Frequency -Hz');

ylabel('Power Spectral Density - watts/Hz');

title('Doppler Frequency spectrum');

Experiment No. – 6

Aim :- To find the output of convolution encoder & decoder

Theory

In telecommunication, a convolutional code is a type of error-correcting code in which

each m-bit information symbol (each m-bit string) to be encoded is transformed into

an n-bit symbol, where m/n is the code rate (n ≥ m) and

the transformation is a function of the last k information symbols, where k is the

constraint length of the code.

Convolutional codes are used extensively in numerous applications in order to achieve

reliable data transfer, including digital video, radio, mobile communication, and satellite

communication. These codes are often implemented in concatenation with a hard-

decision code, particularly Reed Solomon. Prior to turbo codes, such constructions were

the most efficient, coming closest to the Shannon limit.

convolutional encoding

To convolutionally encode data, start with k memory registers, each holding 1 input bit.

Unless otherwise specified, all memory registers start with a value of 0. The encoder

has n modulo-2 adders (a modulo 2 adder can be implemented with a

single Boolean XOR gate, where the logic is: 0+0 = 0, 0+1 = 1, 1+0 = 1, 1+1 = 0),

and n generator polynomials — one for each adder (see figure below). An input bit m1 is

fed into the leftmost register. Using the generator polynomials and the existing values in

the remaining registers, the encoder outputs n bits. Now bit shift all register values to the

right (m1 moves to m0, m0 moves to m-1) and wait for the next input bit. If there are no

remaining input bits, the encoder continues output until all registers have returned to the

zero state.

The figure below is a rate 1/3 (m/n) encoder with constraint length (k) of 3. Generator

polynomials are G1 = (1,1,1), G2 = (0,1,1), and G3 = (1,0,1). Therefore, output bits are

calculated (modulo 2) as follows:

n1 = m1 + m0 + m-1

n2 = m0 + m-1

n3 = m1 + m-1.

Figure: Convolution encoder

Trellis decoding of convolution codes

To implement the decoding of convolution codes is trellis decoding which involves the

use of a trellis, which is a time-indexed graph that represents a given linear code. More

formally, a trellis T = (V, E) of depth n is a finite, directed, edge-labeled graph with the

following properties:

1. Each vertex v V has an associated depth dv {0, 1, 2, 3,...n}.

2. Each edge e E connects a vertex at depth i to a vertex at depth i + 1, for some i.3. There is one vertex, called the root, at depth 0 and one vertex, called the toor, at

depth n.A trellis for a code establishes a one-to-one correspondence between code words and

paths from the root to the toor. One may consider a trellis to be a definite finite

automaton with one start state and one finish state and no loops, so that a code

represented by the trellis is precisely the language of the trellis.

Figure : A trellis for the linear code. Solid lines represent transition on 1 bits and dashed

lines represent transitions on 0 bits.

Given a received, possibly error-corrupted word, we may associate bit-error probabilities with

weights on each edge in the trellis so that the problem of maximum-likelihood decoding reduces to

the problem of finding the minimum-weight path from the root to the toor in the trellis.

Program

%% convolution encoder

close all;

clear all;

%input_bit_num is the number of bits entering

%the memory register of encoder at one time

input_bit_num=1;

% g is the generator matrix of the convolution code

%determine the output sequence of a binary convolution code

%with no rows at p*input_bit_num colomns,its row are

%g1,g2........gn

g=[1 0 1;1 1 1;];

%input the binary input sequence

input=[1 0 1 1 1 0 0];

n=length(input)/input_bit_num;

%determine p and no

p=size(g,2)/input_bit_num;

no=size(g,1);

%add extra zeros

encd_seq=[zeros(1,2),input,zeros(1,2)];

%generate uu,a matrix whose colomns are the contents of

%convolution encoder at various clock pulses

u1=encd_seq(p*input_bit_num:-1:1);

for i=1:n+p-2

u1=[u1,encd_seq((i+p)*input_bit_num:-1:i*input_bit_num+1)];

end

uu=reshape(u1,p*input_bit_num,n+p-1);

%dtermine the output

output=reshape(rem(g*uu,2),1,no*(p+n-1))

Input

1 0 1 1 1 0 0

output

1 1 0 1 0 0 1 0 0 1 1 0 1 1 0 0 0 0

%Convolution decoder

clear all;

close all;

u=[1 0 1 1 1 0 0 ];trellis=poly2trellis(3,[5 7]);

codeword=convenc(u,trellis)

msg=vitdec(codeword,trellis,5,'term','hard')

codeword =

1 1 0 1 0 0 1 0 0 1 1 0 1 1

msg =

1 0 1 1 1 0 0

Experiment No. – 7

Aim :- To Plot the power spectrum pattern of the Gaussian frequency shift keying.

Theory:-

Gaussian Frequency-Shift Keying (GFSK) is a type of Frequency Shift

Keying modulation that uses a Gaussian filter to smooth positive/negative frequency

deviations, which represent a binary 1 or 0.

In a GFSK modulator, everything is same as an FSK modulator except that before

the baseband pulses (-1, 1) go into the FSK modulator, they are passed through  Gaussian

filter to make the pulse smoother so as to limit its spectral width. Gaussian filtering is

one of the very standard ways for reducing the spectral width and it is known as "pulse

shaping".

If we use -1 for fc − fd and 1 for fc + fd, once when we jump from -1 to 1 or 1 to -1, the

modulated waveform changes rapidly, which introduces large out-of-band spectrum. If

we change the pulse going from -1 to 1 as -1, -.98, -.93 ..... .96, .99, 1, and we use this

smoother pulse to modulate the carrier, the out-of-band spectrum will be reduced

BT = 0.3

BT = 0.3

BT = 0.5

Thus increasing the BT product results into reduction of sidelobes but at the same time width of

main lobe increases reuting into poor spectral efficiency.

BT = 0.5

Experiment No. – 8

Aim :- To Plot the power spectrum pattern of the Minimum Shift keying.

Theory:-

One of the problems with standard forms of PSK is that sidebands extend out from the

carrier. To overcome this, MSK can be used. MSK is known as a continuous phase

scheme. It means there are no phase discontinuities in MSK.. This arises as a result of the

unique factor of MSK that the frequency difference between the logical one and logical

zero states is always equal to half the data rate. This can be expressed in terms of the

modulation index, and it is always equal to 0.5.

Figure : Signal using MSK modulation

MSK is a special type of continuous phase shift keying (CPFSK) where the peak

frequency deviation is equal to ¼ the bit rate. In the other words, MSK is CPFSK with a

modulation index of 0.5. The modulation index of an PSK signal is similar to FM

modulation index.

KFSK = 2Δf / Rb

Where Δf is peak frequency deviation and Rb is bit rate

A modulation index of 0.5 corresponds to the minimum frequency spacing that allows

two FSK signals to be coherently orthogonal or MSK implies the minimum frequency

spectrum (i.e. bandwidth) that allows orthogonal detection.

MSK is spectrally efficient modulation scheme and is particularly attractive for use in

mobile radio channel communication software. It posses properties such as constant

envelope, spectral efficiency, good BER performance.

Block Diagram

MSK spectrum has lower sidelobes than Quadrature Phase Shift Keying and Offset QPSK. Main

lobe of MSK is wider than that of QPSK and hence when compared in terms of first null

bandwidth, MSK is less spectrally efficient than QPSK and OQPSK. Continuous phase property

of MSK results into simple demodulation circuits.