Natural sampling.

$ y(t) = p(t) \times y_\delta (t)\, ... \, ...(1) $To get the sampled spectrum, consider Fourier transform on both sides for equation 1 Here $P(\omega) = T Sa({\omega T \over 2}) = 2 \sin \omega T/ \omega$It is the minimum sampling rate at which signal can be converted into samples and can be recovered back without distortion.Nyquist interval = ${1 \over fN}$ = $ {1 \over 2fm}$ seconds.In case of band pass signals, the spectrum of band pass signal X[ω] = 0 for the frequencies outside the range fTo overcome this, the band pass theorem states that the input signal x(t) can be converted into its samples and can be recovered back without distortion when sampling frequency fthe spectrum of sampled signal is given by $Y[\omega] = {1 \over T} \Sigma_{n=-\infty}^{\infty}\,X[ \omega - 2nB]$$F_n= {1 \over T} \int_{-T \over 2}^{T \over 2} p(t) e^{-j n \omega_s t} dt$ The output sample signal is represented by the samples.

Here, the amplitude of impulse changes with respect to amplitude of input signal x(t). Hence, a rate was fixed for this, called as Nyquist rate.Suppose that a signal is band-limited with no frequency components higher than A theorem called, Sampling Theorem, was stated on the theory of this Nyquist rate.The sampling theorem states that, “a signal can be exactly reproduced if it is sampled at the rate To understand this sampling theorem, let us consider a band-limited signal, i.e., a signal whose value is We need a sampling frequency, a frequency at which there should be no loss of information, even after sampling.
Here, the amplitude of impulse changes with respect to amplitude of input signal x(t). A signal is defined as any physical or virtual quantity that varies with time or space or any other independent variable or variables.. Graphically, the independent variable is represented by horizontal axis or x-axis.

Impulse sampling can be performed by multiplying input signal x(t) with impulse train $\Sigma_{n=-\infty}^{\infty}\delta(t-nT)$ of period 'T'.


That gap can be termed as a $f_{s}$ is the sampling frequency or the sampling rateFor an analog signal to be reconstructed from the digitized signal, the sampling rate should be highly considered. You cannot use this practically because pulse width cannot be zero and the generation of impulse train is not possible practically.Natural sampling is similar to impulse sampling, except the impulse train is replaced by pulse train of period T. i.e. For an analog signal to be reconstructed from the digitized signal, the sampling rate … Hence, this is also a good sampling rate.The resultant pattern will look like the following figure.We can observe from the above pattern that the over-lapping of information is done, which leads to mixing up and loss of information. The output of sampler is given by$= x(t) × \Sigma_{n=-\infty}^{\infty} \delta(t-nT)$$ y(t) = y_{\delta} (t) = \Sigma_{n=-\infty}^{\infty}x(nt) \delta(t-nT)\,...\,... 1 $To get the spectrum of sampled signal, consider Fourier transform of equation 1 on both sides$Y(\omega) = {1 \over T} \Sigma_{n=-\infty}^{\infty} X(\omega - n \omega_s ) $This is called ideal sampling or impulse sampling. What is Signal, Classification of Signals and the Role of Signals in Digital Communication What Is Signal? Input signal frequency denoted by Fm and sampling signal frequency denoted by Fs. The information is replaced without any loss. In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. In comparison to natural sampling flat top sampling can be easily obtained. In this sampling techniques, the top of the samples remains constant and is equal to the instantaneous value of the message signal x(t) at the start of sampling process. The sampling rate denotes the number of samples taken per second, or for a finite set of values. A common example is the conversion of a sound wave (a continuous signal) to a sequence of samples (a discrete-time signal). There are three types of sampling techniques: Impulse sampling. When a source generates an analog signal and if that has to be digitized, having The following figure indicates a continuous-time signal To discretize the signals, the gap between the samples should be fixed. Impulse Sampling.

Here, the top of the samples are flat i.e.

Digital Communication Methods Figure 1. If either of these is digital then for our purposes it is considered to be a digital communications system. Impulse sampling can be performed by multiplying input signal x(t) with impulse train $\Sigma_{n=-\infty}^{\infty}\delta(t-nT)$ of period 'T'. Hence, it is called as flat top sampling or practical sampling. This unwanted phenomenon of over-lapping is called as Aliasing.Aliasing can be referred to as “the phenomenon of a high-frequency component in the spectrum of a signal, taking on the identity of a low-frequency component in the spectrum of its sampled version.”The corrective measures taken to reduce the effect of Aliasing are −The signal which is sampled after filtering, is sampled at a rate slightly higher than the Nyquist rate.This choice of having the sampling rate higher than Nyquist rate, also helps in the easier design of the It is generally observed that, we seek the help of Fourier series and Fourier transforms in analyzing the signals and also in proving theorems. A communications system may be digital either by the nature of the information (also known as data) which is passed or in the nature of the signals which are transmitted. These samples are maintained with a gap, these gaps are termed as sample period or sampling interval (Ts… The sampling theorem can be defined as the conversion of an analog signal into a discrete form by taking the sampling frequency as twice the input analog signal frequency. Where, 1.


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