Frequency Analysis of Amplitude Modulated D.T. Sequences


Content:

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 × fs samples/sec = fs/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.
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.
Bhanu Namikaze

Bhanu Namikaze is an Ethical Hacker, Web Developer, Student and Mechanical Engineer. He Enjoys writing articles, Blogging, Solving Errors and Social Networking. Feel Free to let me know any of your concerns about hacking or let me know if you need any more methods on hacking anything. Enjoy Learning

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