Nyquist Rate of SamplingThe theoretical minimum sampling rate at which a signal can be sampled and still can be reconstructed from its samples without any distortion is called the Nyquist rate of sampling.Mathematically, $$\mathrm{Nyquist\: Rate, \mathit{f_{N}}\mathrm{=}2\mathit{f_{m}}}$$Where, $\mathit{f_{m}}$is the maximum frequency component present in the signal.If the signal is sampled at ... Read More

Laplace TransformThe Laplace transform is a mathematical tool which is used to convert the differential equation in time domain into the algebraic equations in the frequency domain or s-domain.Mathematically, if $\mathrm{\mathit{x\left(t\right)}}$ is a time domain function, then its Laplace transform is defined as −$$\mathrm{\mathit{L\left[\mathit{x}\mathrm{\left(\mathit{t} \right )}\right ]\mathrm{=}X\mathrm{\left( \mathit{s}\right)}\mathrm{=}\int_{-\infty }^{\infty}x\mathrm{\left (\mathit{t} ... Read More

ConvolutionThe convolution of two signals $\mathit{x\left ( t \right )}$ and $\mathit{h\left ( t \right )}$ is defined as, $$\mathrm{\mathit{y\left(t\right)\mathrm{=}x\left(t\right)*h\left(t\right)\mathrm{=}\int_{-\infty }^{\infty}x\left(\tau\right)\:h\left(t-\tau\right)\:d\tau}}$$This integral is also called the convolution integral.Frequency Convolution TheoremStatement - The frequency convolution theorem states that the multiplication of two signals in time domain is equivalent to the convolution ... Read More

The Fourier series can be used to analyse only the periodic signals, while the Fourier transform can be used to analyse both periodic as well as non-periodic functions. Therefore, the Fourier transform can be used as a universal mathematical tool in the analysis of both periodic and aperiodic signals over ... Read More

Linear Time-Invariant SystemA system for which the principle of superposition and the principle of homogeneity are valid and the input/output characteristics do not change with time is called the linear time-invariant (LTI) system.Impulse Response of LTI SystemWhen the impulse signal is applied to a linear system, then the response of ... Read More

Fourier TransformFor a continuous-time function $\mathrm{\mathit{x\left ( t \right )}}$ , the Fourier transform of $\mathrm{\mathit{x\left ( t \right )}}$ can be defined as, $$\mathrm{\mathit{X\left ( \omega \right )\mathrm{=}\int_{-\infty }^{\infty }x\left ( t \right )e^{-j\omega t}dt }}$$And the inverse Fourier transform is defined as, $$\mathrm{\mathit{x\left ( t \right )\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\infty ... Read More

The transfer function of a continuous-time LTI system may be defined using Laplace transform or Fourier transform. Also, the transfer function of the LTI system can only be defined under zero initial conditions. The block diagram of a continuous-time LTI system is shown in the following figure.Transfer Function of LTI ... Read More

Condition of CausalityA causal system is the one which does not produce an output before the input is applied. Therefore, for an LTI (Linear Time-Invariant) system to be causal, the impulse response of the system must be zero for t less than zero, i.e., $$\mathrm{\mathit{h\left ( t \right )\mathrm{=}\mathrm{0};\; \; ... Read More

Distortion-less TransmissionWhen a signal is transmitted through a system and there is a change in the shape of the signal, it called the distortion. If the output of the system is an exact replica of the input signal, then the transmission of the signal through the system is called distortionless ... Read More

For a continuous-time function $\mathit{x(t)}$, the Fourier transform of $\mathit{x(t)}$ can be defined as$$\mathrm{\mathit{X\left ( \omega \right )\mathrm{\mathrm{=}}\int_{-\infty }^{\infty }x\left ( t \right )e^{-j\omega t}dt}}$$And the inverse Fourier transform is defined as, $$\mathrm{\mathit{F^{\mathrm{-1}}\left [ X\left ( \omega \right ) \right ]\mathrm{\mathrm{=}}x\left ( t \right )\mathrm{\mathrm{=}}\frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\infty }^{\infty }X\left ( \omega ... Read More