**Chapter 1:**Introduction

**... 5-11**

**Chapter 2:**Tasks and their simulation results

(a)
Task1: Simulation results, Plots/graphs, figures,

**... 12**
tables
etc

(b)
Task2: Simulation results, Plots/graphs, figures,

**... 14**
tables
etc

(c)
Task3: Simulation results, Plots/graphs, figures,

**... 16****tables etc**

(d)
Task4:
Simulation results, Plots/graphs, figures,

**... 16-17****tables etc**

**Chapter 3:**Conclusions

**... 18**

**Appendix1: Source code ... 19-21**

**References ... 22**

**CHAPTER 1**

**Introduction:**

A discrete signal or discrete-time signal is a time series consisting of a sequence of quantities. In other words, it is a
time series that is a function over a domain of
integers. Unlike a continuous-time
signal, a discrete-time signal is not a function of a continuous
argument however, it may have been obtained by sampling from
a continuous-time signal, and then each value in the sequence is called a sample.
When a discrete-time signal obtained by sampling a sequence corresponds to
uniformly space times, it has an associated sampling rate;
the sampling rate is not apparent in the data sequence, and so needs to be
associated as a characteristic unit of the system.

A digital
signal is a
discrete-time signal for which not only the time but also the amplitude has
been made discrete; in other words, its samples take on only values from a discrete set (a countable set that can be mapped one-to-one to a subset of integers).
If that discrete set is finite, the discrete values can be represented with digital words of a finite width. Most commonly, these discrete
values are represented as fixed-point words or floating-point words. After sampling, the process of
converting a continuous-valued discrete-time signal to a digital signal is
known as analogue-to-digital conversion. It usually
proceeds by replacing each original sample value by an approximation selected
from a given discrete set a process known as quantization. This process
loses information, and so discrete-valued signals are only an approximation of
the continuous-valued discrete-time signal, itself only an approximation of the
original continuous-valued continuous-time signal. Amplitude modulation (AM) is a modulation technique
used in electronic communication, most commonly for transmitting information
via a radio carrier wave. AM works by varying the strength
(amplitude) of the carrier in proportion to the waveform being sent that
waveform may, for instance, correspond to the sounds to be reproduced by
a loudspeaker, or the light intensity of television pixels. This contrasts
with modulation, in the frequency which the frequency of
the carrier signal is varied, and phase modulation, in which
its phase is varied, by the modulating signal.

Discrete time views values of variables as occurring at distinct,
separate "points in time", or equivalently as being unchanged
throughout each non-zero region of time .Thus a variable jumps from one value
to another as time moves from time period to the next. In this framework, each
variable of interest is measured once at each time period. The number of
measurements between any two time periods is finite. Measurements are typically
made at sequential integer values
of the variable "time

**Project description:**

**Discrete Fourier transforms:**

In mathematics,
the discrete Fourier transform (DFT) converts a finite list of equally
spaced samples of
a function into
the list of coefficients of a finite combination of complex sinusoids,
ordered by their frequencies,
that has those same sample values. It can be said to convert the sampled function
from its original domain (often time or position along a line) to the frequency
domain. The input samples are complex
numbers (in practice,
usually real numbers),
and the output coefficients are complex as well. The frequencies of the output
sinusoids are integer multiples of a fundamental frequency, whose corresponding
period is the length of the sampling interval. The combination of sinusoids
obtained through the DFT is therefore periodic with that same period. The DFT differs
from the discrete-time Fourier transform (DTFT) in that it’s input and output sequences
are both finite; it is therefore said to be the Fourier analysis of
finite-domain (or periodic) discrete-time functions.

The DFT is the most
important discrete transform, used to perform Fourier
analysis in many
practical applications. In digital signal processing, the function is
any quantity or signal that
varies over time, such as the pressure of a sound wave,
a radio signal, or daily temperature readings, sampled over a finite time
interval (often defined by a window
function). In image
processing, the samples can be the values of pixels along a row or column of a raster image.
The DFT is also used to efficiently solve partial differential equations,
and to perform other operations such as convolutions or multiplying large integers. Since
it deals with a finite amount of data, it can be implemented in computers by numerical algorithms or even dedicated hardware.
These implementations usually employ efficient fast Fourier transform (FFT) algorithms; so much so that the terms
"FFT" and "DFT" are often used interchangeably. Prior to
its current usage, the "FFT" initialize may have also been used for the
ambiguous term "finite Fourier transform".

**Demodulation:**

Demodulation is the act of extracting the original
information-bearing signal from a modulated carrier wave.
A demodulator is an electronic circuit that is used to recover the
information content from the modulated carrier wave

^{ }there are many types of modulation so there are many types of demodulators. The signal output from a demodulator may represent sound images or binary data. These terms are traditionally used in connection with radio receivers, but many other systems use many kinds of demodulators. For example in a modem, which is a contraction of the terms modulator/demodulator. A demodulator is used to extract a serial digital data stream from a carrier signal which is used to carry it through a telephone line, coaxial cable, or optical fibre.
There are several
ways of demodulation depending on how parameters of the base-band signal are
transmitted in the carrier signal, such as amplitude, frequency or phase. For
example, for a signal modulated with a linear modulation, like AM (amplitude modulation), we can use a synchronous detector. On
the other hand, for a signal modulated with an angular modulation, we must use
an FM (frequency modulation) demodulator or a PM
(phase modulation) demodulator. Different kinds
of circuits perform these functions. Many techniques—such as carrier
recovery, clock
recovery, bit slip, frame synchronization, rake receiver, pulse
compression, Received Signal Strength Indication, error detection and correction,
etc., are only performed by demodulators, although any specific demodulator may
perform only some or none of these techniques. Many things can act as a
demodulator, if they pass the radio waves on nonlinearly:
for example, near a powerful radio station, it has been known for the metal
sides of a van to demodulate the radio signal as sound.

**Amplitude modulation:**

Amplitude modulation (AM) is
a modulation technique used in electronic communication, most
commonly for transmitting information via a radio carrier wave. AM
works by varying the strength (amplitude) of the carrier in proportion to the
waveform being sent. That waveform may, for instance, correspond to the sounds
to be reproduced by a loudspeaker, or the light intensity of television
pixels. This contrasts with modulation, in the frequency which the
frequency of the carrier signal is varied, and phase
modulation, in which its phase is varied, by the modulating signal. AM
was the earliest modulation method used to transmit voice by radio. It was
developed during the first two decades of the 20th century beginning
with Reginald Fessenden radio telephone experiments in 1900. It
remains in use today in many forms of communication; for example it is used in
portable two way radios, VHF aircraft radio and in computer
modems "AM" is often used to refer to medium wave AM
radio broadcasting. In some modulation
systems based on AM, a lower transmitter power is required through partial or
total elimination of the carrier component; however receivers for these signals
are more complex and costly. The receiver may regenerate a copy of the carrier
frequency (usually as shifted to the intermediate frequency from a greatly
reduced "pilot" carrier to use in the demodulation process. Even with
the carrier totally eliminated in double-sideband suppressed-carrier
transmission, carrier regeneration is possible using a Costas phase-locked
loop.

**Discrete time sequence:**

A discrete signal or discrete-time signal is a time
series consisting of a sequence of quantities. In other words, it is a time
series that is a function over a domain of integers.

**Discrete Fourier Transform (DFT):**

Frequency
analysis of DT sequences is usually and most conveniently performed on a
digital Signal Processor, which may be a general purpose computer or specially
designed digital hardware. To perform frequency analysis on DT sequences

*x*[*n*] a discrete time Fourier transform (**DTFT**), continuous function of frequency and therefore it is not a computationally convenient compute in a digital computer. The analysis and Synthesis equations for N point DFT are defined as follows.**Sampling theorem:**

In the field of
digital signal processing, the sampling
theorem is a fundamental bridge between continuous signals
(analogue domain) and discrete signals (digital domain). Reconstructing a
continuous function from samples is done by interpolation algorithms. The Whittaker–Shannon interpolation
formula is
mathematically equivalent to an ideallowpass
filter whose input is
a sequence of Dirac delta functions that are modulated (multiplied) by the
sample values. When the time interval between adjacent samples is a constant (T), the sequence of delta functions
is called a Dirac comb.
Mathematically, the modulated Dirac comb is equivalent to the product of the
comb function with s(t). That purely mathematical abstraction is sometimes referred
to as impulse sampling.

Most sampled signals
are not simply stored and reconstructed. But the fidelity of a theoretical
reconstruction is a customary measure of the effectiveness of sampling. That
fidelity is reduced when s(t) contains frequency components whose periodicity is smaller
than 2 samples; or equivalently the ratio of cycles to samples exceeds ½ (see Aliasing). The quantity

**½**cycles/sample ×**f**samples/sec =_{s}**f**/2 cycles/sec (hertz) is known as the Nyquist frequency of the sampler. Therefore s(t) is usually the output of a low pass filter, functionally known as an anti-aliasing filter. Without an anti-aliasing filter, frequencies higher than the Nyquist frequency will influence the samples in a way that is misinterpreted by the interpolation process._{s}**Uniform Sampling Theorem:**

The sampling theorem is significant in
communication systems because it provides the basis for transmitting analogue
signals by use of digital Techniques.

·

**Time domain Statement:**A band limited signal having no frequency components higher than*Fm*Hz may be completely recovered from the knowledge of its samples taken at the rate of at least 2*Fm*samples per second.
·

**Frequency domain Statement:**A band limited signal having no frequency components higher than*Fm*Hz is completely described by its sample values at uniform intervals less than or equal to 1 / 2*Fm*seconds apart.**MODUALTION:**

When low frequency signal of particular low
frequency is added to another high frequency signal then the resulted signal is
known as the modulated signal .This process is known as the modulation.There are three types of modulations. They
are 1.frequency modulation
2.amplitude modulation 3.phase modulation.

**Frequency Modulation (FM)**: It is a system where the amplitude of a carrier wave is held constant while the frequency is varied in sympathy with the voltage of the modulating Signal

**Amplitude Modulation (AM)**: It is a system where the frequency of a carrier wave is held constant while the amplitude is varied in sympathy with the voltage of the modulating signal

**Phase Modulation (PM): It**is a similar system where the phase of the carrier wave is varied in sympathy with the voltage of the modulating signal, and as in frequency modulation, the amplitude of a carrier is held constant.

In this project we
mainly use amplitude modulated D.T.sequences.Amplitude Modulation (AM) is a
system where the frequency of a carrier wave is held constant while the
amplitude is varied in sympathy with the voltage of the modulating signal. Amplitude modulation consists of multiplying a relatively
slowly varying signal by a relatively quickly varying periodic signal. The
frequency response of the combined signal is equivalent to the frequency
response of the slowly varying signal shifted by the frequency of the rapidly
changing signal in the positive and negative directions. Here we
are simply adding the carrier amplitude with message signal to obtain AM signal
then the instantaneous amplitude of carrier gets altered with respect to
modulating signal. Thus the carrier amplitude varies according to the base band
signal (message signal).Once this information is received, the low frequency
information must be removed from the high frequency carrier. This process is
known as “Demodulation.
Demodulation is the act of extracting the original information-bearing
signal from a modulated carrier
wave. A demodulator is an electronic circuit (or computer
program in a software-defined radio) that is used to recover the information content from the
modulated carrier
wave.

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