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

An electric circuit consisting of a resistance (R) and an inductor (L), connected in series, is shown in Figure-1. Consider the switch (S) is closed at time $\mathrm{\mathit{ t=\mathrm{0}}}$.Step Response of Series RL CircuitTo obtain the step response of the series RL circuit, the input $\mathrm{\mathit{x\left ( t \right )}}$ ... Read More

What is Region of Convergence?Region of Convergence (ROC) is defined as the set of points in s-plane for which the Laplace transform of a function $\mathit{x}\mathrm{\left(\mathit{t}\right)}$ converges. In other words, the range of $\mathit{Re}\mathrm{\left(\mathit{s} \right)}$ (i.e., σ) for which the function $\mathit{X}\mathrm{\left(\mathit{s}\right)}$ converges is called the region of convergence.ROC 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 −$$\mathrm{\mathit{L}\mathrm{\left[ \mathit{x}\mathrm{\left(\mathit{t}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{s}\right)}\:\mathrm{=}\:\int_{-\infty}^{\infty}\mathit{x}\mathrm{\left(\mathit{t}\right)}\mathit{e^{-st}}\:\mathit{dt}\:\:\:\:\:\:...(1)}$$Equation (1) gives the bilateral Laplace transform ... Read More