Laplace TransformThe linear time invariant (LTI) system is described by differential equations. The Laplace transform is a mathematical tool which converts the differential equations in time domain into algebraic equations in the frequency domain (or s-domain).If $\mathrm{\mathit{x\left ( t \right )}}$ is a time function, then the Laplace transform of ... Read More

Z-TransformThe Z-transform (ZT) is a mathematical tool which is used to convert the difference equations in time domain into the algebraic equations in z-domain.Mathematically, if $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a discrete-time signal or sequence, then its bilateral or two-sided Z-transform is defined as −$$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z^{-\mathit{n}}}\:\:\:\:\:\:...(1)}$$Where, z is a complex variable.Also, the unilateral or ... Read More

Discrete-Time Fourier TransformThe Fourier transform of the discrete-time signals is known as the discrete-time Fourier transform (DTFT). The DTFT converts a time domain sequence into frequency domain signal. The DTFT of a discrete time sequence $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is given by, $$\mathrm{\mathit{F}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{\omega}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{e^{-j\omega n}}\:\:\:\:\:\:...(1)}$$Z-TransformThe Z-transform is a mathematical which is used to convert ... Read More

Energy of a SignalThe energy of a signal $\mathit{x}\mathrm{\left(\mathit{t}\right)}$ is defined as the area under the curve of square of magnitude of that signal, i.e., $$\mathrm{\mathit{E}\:\mathrm{=}\:\int_{-\infty}^{\infty}\left|\mathit{x}\mathrm{\left(\mathit{t}\right)} \right|^{\mathrm{2}}\:\mathit{dt}}$$The energy signal exists only of the energy (E) of the signal is finite, i.e., only if 0 < E < $\infty$.Rayleigh’s Energy TheoremStatement ... Read More

Z-TransformThe Z-transform (ZT) is a mathematical tool which is used to convert the difference equations in time domain into the algebraic equations in z-domain.Mathematically, if $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a discrete-time signal or sequence, then its bilateral or two-sided Z-transform is defined as −$$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty }}^{\infty }\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z}^{-\mathit{n}}}$$Where, z is a complex variable.Region of ... 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 $\mathit{x}\mathrm{\left(\mathit{t}\right)}$ is a time domain function, then its Laplace transform is defined as −$$\mathrm{\mathit{L}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{t}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{s}\right)}\:\mathrm{=}\:\int_{-\infty}^{\infty}\mathit{x}\mathrm{\left(\mathit{t}\right)}\mathit{e^{-st}}\:\mathit{dt}}$$Inverse Laplace TransformThe inverse Laplace transform is ... Read More

Data ReconstructionThe data reconstruction is defined as the process of obtaining the analog signal $\mathrm{\mathit{x\left ( t \right )}}$ from the sampled signal $\mathrm{\mathit{x_{s}\left ( t \right )}}$. The data reconstruction is also known as interpolation.The sampled signal is given by, $$\mathrm{\mathit{x_{s}\left ( t \right )\mathrm{=}x\left ( t \right )\sum_{n\mathrm{=}-\infty ... 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 [ x\left ... 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 [ x\left ... Read More

An electric circuit consisting of a resistance (R) and a capacitor (C), connected in series, is shown in Figure-1. Consider the switch (S) is closed at $\mathrm{\mathit{t=\mathrm{0}}}$.Step Response of Series RC Circuit Using Laplace TransformTo obtain the step response of the series RC circuit, the applied input is given by, ... Read More