Frequency Analysis of Amplitude Modulated D.T. Sequences

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.

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 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.

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_s samples/sec = f_s/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.