Laplace TransformThe Laplace transform is a mathematical tool which is used to convert the differential equations 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

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 $\mathit{x}\mathrm{(\mathit{t})}$ is a time domain function, then its Laplace transform is defined as −$$\mathit{L}\mathrm{[\mathit{x}\mathrm{(\mathit{t})}]}\:\mathrm{=}\:\mathit{X}\mathrm{(\mathit{s})}\:\mathrm{=}\:\int_{-\infty}^{\infty}\mathit{x}\mathrm{(\mathit{t})\mathit{e^{-st}}}\mathit{dt} \:\:\:...(1)$$Equation (1) gives the bilateral Laplace ... 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 $\mathit{x}\mathrm{(\mathit{t})}$is a time-domain function, then its Laplace transform is defined as −$$\mathit{L}\mathrm{[\mathit{x}\mathrm{(\mathit{t})}]}\:\mathrm{=}\:\mathit{X}\mathrm{(\mathit{s})}\:\mathrm{=}\:\int_{-\infty}^{\infty}\mathit{x}\mathrm{(t)}\mathit{e^{-st}}\mathit{dt} \:\:...(1)$$Equation (1) gives the bilateral Laplace transform of ... Read More

Detection of Periodic Signals in the Presence of NoiseThe noise signal is an unwanted signal which has random amplitude variation. The noise signals are uncorrelated with any periodic signal.Detection of the periodic signals masked by noise signals is of great importance in signal processing. It is mainly used in the ... Read More

Detection of Periodic Signals in the Presence of NoiseThe noise signal is an unwanted signal which has random amplitude variation. The noise signals are uncorrelated with any periodic signal.Detection of the periodic signals masked by noise signals is of great importance in signal processing. It is mainly used in the ... Read More

Cross Correlation FunctionThe cross correlation function between two different signals is defined as the measure of similarity or coherence between one signal and the time delayed version of another signal.The cross correlation function is defined separately for energy (or aperiodic) signals and power or periodic signals.Cross Correlation of Energy SignalsConsider ... Read More

Autocorrelation FunctionThe autocorrelation function defines the measure of similarity or coherence between a signal and its time delayed version. The autocorrelation function of a real energy signal $\mathit{x}\mathrm{(\mathit{t})}$ is given by, $$\mathit{R}\mathrm{(\mathit{\tau})} \:\mathrm{=}\: \int_{-\infty}^{\infty}\mathit{x\mathrm(\mathit{t})}\:\mathit{x}\mathrm{(\mathit{t-\tau})}\:\mathit{dt}$$Energy Spectral Density (ESD) FunctionThe distribution of the energy of a signal in the frequency domain is ... Read More

Energy Spectral DensityThe distribution of the energy of a signal in the frequency domain is known as energy spectral density (ESD) or energy density (ED) or energy density spectrum. The ESD function is denoted by $\mathrm{\mathit{\psi \left ( \omega \right )}}$ and is given by, $$\mathrm{\mathit{\psi \left ( \omega \right ... Read More

What is Autocorrelation?The autocorrelation function of a signal is defined as the measure of similarity or coherence between a signal and its time delayed version. Thus, the autocorrelation is the correlation of a signal with itself.The autocorrelation function is defined separately for energy or aperiodic signals and power or periodic ... 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