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

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 ... Read More

Power Spectral DensityThe distribution of average power of a signal $x\mathrm{\left(\mathit{t}\right)}$ in the frequency domain is called the power spectral density (PSD) or power density (PD) or power density spectrum. The PSD function is denoted by $\mathit{S\mathrm{\left({\mathit{\omega }}\right)}}$ and is given by, $$\mathrm{\mathit{S}\mathrm{\left(\mathit{\omega}\right)}\mathrm{=}\displaystyle\lim_{\tau \to \infty }\frac{\left| \mathit{X\mathrm{\left ( \mathit{\omega}\right)}}\right|^{2}}{\tau}\:\:\:\:\:\:...(1)}$$ExplanationIn order ... Read More

What is Data Reconstruction?Data reconstruction is defined as the process of obtaining the analog signal $x\mathrm{\left(\mathit{t}\right)}$ from the sampled signal $x_{\mathit{s}}\mathrm{\left ( \mathit{t}\right)}$. The data reconstruction is also known as interpolation.The sampled signal is given by, $$\mathrm{\mathit{x}_{\mathit{s}}\mathrm{\left ( \mathit{t}\right)}\:\mathrm{=}\:\mathit{x}\mathrm{\left(\mathit{t}\right)}\sum_{\mathit{n}=-\infty}^{\infty}\:\delta \mathrm{\left ( \mathit{t-nT} \right )}}$$$$\mathrm{\Rightarrow \mathit{x}_{\mathit{s}}\mathrm{\left ( \mathit{t}\right)}\:\mathrm{=}\sum_{\mathit{n}=-\infty}^{\infty}\:\mathit{x}\mathrm{\left(\mathit{nT}\right )}\delta\mathrm{\left(\mathit{t-nT}\right)}}$$Where, $\mathit{\delta}\mathrm{\left(\mathit{t-nT} \right)}$ ... 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

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 $\mathrm{\mathit{x\left ( t \right )}}$ converges. In other words, the range of 𝑅𝑒(𝑠) (i.e., 𝜎) for which the function 𝑋(𝑠) converges is called the region ... 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 $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$ is a time domain function, then its Laplace transform is defined as −$$\mathrm{\mathit{L\left [ x\left ... Read More