Sinc Function, Ssb-Am, Fm

Topics: Amplitude modulation, Signal processing, Digital signal processing Pages: 8 (1024 words) Published: August 22, 2013
TELECOMMUNICATION 1

2012-2013 Fall

Project

SINC FUNCTION, SSB-AM, FM

Name: Onur Mustafa Erdoğan
ID Number: 10014044
Submission Date: 24.12.2012

Abstract:
In these project, I will analyze Fourier Transform of sinc function and it’s modulation.(SSB-AM,FM) I will explain SSB-AM and FM theoretically and solve their math model in steps. After all, I’ll use simulations and graphics to prove my solutions and In the and, I will write my conclusions down.

Introduction

SSB-AM

[1]In DSB-SC it is observed that there is symmetry in the band structure. So, even if one half is transmitted, the other half can be recovered at the received. By doing so, the bandwidth and power of transmission is reduced by half. Depending on which half of DSB-SC signal is transmitted, there are two types of SSB modulation; 1. Lower Side Band (LSB) Modulation

2. Upper Side Band (USB) Modulation

FM

[3]Frequency modulation (FM) is a method of impressing data onto an alternating-current (AC) wave by varying the instantaneous frequency of the wave. This scheme can be used with analog or digital data. Frequency modulation is similar in practice to phase modulation (PM). When the instantaneous frequency of a carrier is varied, the instantaneous phase changes as well. The converse also holds: When the instantaneous phase is varied, the instantaneous frequency changes. But FM and PM are not exactly equivalent, especially in analog applications. When an FM receiver is used to demodulate a PM signal, or when an FM signal is intercepted by a receiver designed for PM, the audio is distorted. This is because the relationship between frequency and phase variations is not linear; that is, frequency and phase do not vary in direct proportion.

System Model

[4]We are looking for Fourier Transform of sinc function, in first step, we put our function in Fourier Transform Formula.

if 

  

and if  and 
  

After these equations, we can say that Fourier Transform of sinc(t)=πrect(w/2)

Next step is modulating our signal for transmission. I will use SSB-AM and FM for this.

SSB-AM for x(t)=sinc(t);
General Formula for SSB-AM is;

Upper sideband

Lower sideband

H{x(t)} is Hilbert Transform, it’s defined as;

I put sinc(t) in the formula and used Hilbert Transform;

Upper Sideband

Lower Sideband

And in the frequency domain I reached;

Upper Sideband

Lower Sideband

FM for x(t)=sinc(t)
We can show FM signal as;

But we need to know phase;

Let’s put sinc(t) instead of x(t)

[5] We can use Dirichlet Condition to solve this integral.

This is the Frequency Modulation of sinc signal.

Simulations

Fourier Transform of Sinc

clc;
T = 1/10000; % samplefreq
t = -0.01:T:0.01-T; % time-axes
N = length(t);

% time-signal
y = sinc(10000 * t./pi);

% frequency-axes 1
freq1 = 1/T * (0:N-1) ./ N;

% frequency-axes 2
f = freq1(1:N/2)';
freq2 = [-1*flipud(f)' f(2:end)'];

% FFT
Y = abs(fft(y, N)); % fft
Z = fftshift(Y); % shiften

plot(freq2, Z(2:end));
title('Frequency Domain shifted');
xlabel('Freq in Hz');
ylabel('[Vs]');
grid on;

SSB-AM in Time Domain

clc;
clear all;
T = 1/10000; % sample frequency
t=-0.01:T:0.01
f=100; % carrier frequency

x=sinc(10000*t./pi); % message signal

xh=hilbert(x); % hilbert transform

yup=sqrt(2)/2*(x.*cos(2*pi*f*t)-xh.*sin(2*pi*f*t)); % upper sideband
ylow=sqrt(2)/2*(x.*cos(2*pi*f*t)+xh.*sin(2*pi*f*t)); % lower sideband

subplot(3,1,1)
plot(t,yup)
xlabel('time');
ylabel('upper side band modulation');
grid on

subplot(3,1,2)
plot(t,ylow)
xlabel('time');
ylabel('lower side band modulation');
grid on

SSB-AM in Frequency Domain

clc;
clear all;
T = 1/10000; % samplefreq
t=-0.01:T:0.01
N=length(t);
f=100;
% time-signal
x=sinc(10000*t./pi);
% frequency-axes 1
freq1 = 1/T * (-N/2:(N-1)/2) ./ N;

xh=hilbert(x);

yup=sqrt(2)/2*(x.*cos(2*pi*f.*t)-xh.*sin(2*pi*f.*t));...

References: [1]http://nptel.iitm.ac.in/courses/IIT-MADRAS/Principles_Of_Communication/pdf/Lecture21-22_SSB_VSB.pdf
[2] www.mathworks.com
[3]http://searchnetworking.techtarget.com/definition/frequency-modulation
[4]http://eagle.lamost.org/?p=38679
[5] http://en.wikipedia.org/wiki/Dirichlet_integral
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