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Found 1024 Articles for Digital Electronics
3K+ Views
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 transform of the function $\mathit{x}\mathrm{(\mathit{t})}$. But for the causal signals, the unilateral Laplace transform is applied, which is defined as −$$\mathit{L}\mathrm{[\mathit{x}\mathrm{(\mathit{t})}]}\:\mathrm{=}\:\mathit{X}\mathrm{(\mathit{s})}\:\mathrm{=}\:\int_{\mathrm{0}}^{\infty}\mathit{x}\mathrm{(\mathit{t})\mathit{e^{-st}}}\mathit{dt}\:\:\:..(2)$$Linearity Property of Laplace TransformStatement − The Linearity property of Laplace transform states that the Laplace transform of a weighted sum of two signals is equal to the weighted sum of ... Read More
23K+ Views
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 transform of the function $\mathit{x}\mathrm{(\mathit{t})}$. But for the causal signals, the unilateral Laplace transform is applied, which is defined as −$$\mathit{L}\mathrm{[\mathit{x}\mathrm{(\mathit{t})}]}\:\mathrm{=}\:\mathit{X}\mathrm{(\mathit{s})}\:\mathrm{=}\:\int_{\mathrm{0}}^{\infty}\mathit{x}\mathrm{(\mathit{t})\mathit{e^{-st}}}\mathit{dt}\:\:\:..(2)$$Final Value TheoremThe final value theorem of Laplace transform enables us to find the final value of a function$\mathit{x}\mathrm{(\mathit{t})}$[i.e., $\mathit{x}\mathrm{(\infty)}$] directly from its Laplace transform X(s) without the need for finding the ... Read More
24K+ Views
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 transform of the function $\mathit{x}\mathrm{(\mathit{t})}$. But for the causal signals, the unilateral Laplace transform is applied, which is defined as −$$\mathit{L}\mathrm{[\mathit{x}\mathrm{(\mathit{t})}]}\:\mathrm{=}\:\mathit{X}\mathrm{(\mathit{s})}\:\mathrm{=}\:\int_{\mathrm{0}}^{\infty}\mathit{x}\mathrm{\mathrm{(\mathit{t})}\mathit{e^{-st}}}\mathit{dt} \:\:\:...(2)$$Initial Value TheoremThe initial value theorem of Laplace transform enables us to calculate the initial value of a function $\mathit{x}\mathrm{(\mathit{t})}$[i.e., $\:\:\mathit{x}\mathrm{(0)}$] directly from its Laplace transform X(s) without the ... Read More
4K+ Views
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 the function $\mathit{x}\mathrm{(\mathit{t})}$. But for the causal signals, the unilateral Laplace transform is applied, which is defined as −$$\mathit{L}\mathrm{[\mathit{x}\mathrm{(\mathit{t})}]}\:\mathrm{=}\:\mathit{X}\mathrm{(\mathit{s})}\:\mathrm{=}\:\int_{\mathrm{0}}^{\infty}\mathit{x}\mathrm{(t)}\mathit{e^{-st}}\mathit{dt} \:\: ...(2)$$Frequency Derivative Property of Laplace TransformStatement − The differentiation in frequency domain or s-domain property of Laplace transform states that the multiplication of the function by $\mathit{'t'}$ in time domain ... Read More
2K+ Views
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 detection of radar and sonar signals, the detection of periodic components in brain signals, in the detection of periodic components in sea wave analysis and in many other areas of geophysics etc. The solution of these problems can be easily provided by thecorrelation techniques. The cross-correlation function, therefore can be ... Read More
1K+ Views
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 detection of radar and sonar signals, the detection of periodic components in brain signals, in the detection of periodic components in sea wave analysis and in many other areas of geophysics etc. The solution of these problems can be easily provided by the correlation techniques. The autocorrelation function, therefore can ... Read More
11K+ Views
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 two energy signals $\mathit{x_{\mathrm{1}}}\mathrm{(\mathit{t})}$ and $\mathit{x_{\mathrm{2}}}\mathrm{(\mathit{t})}$. The cross correlation of these two energy signals is defined as −$$\mathit{R_{\mathrm{12}}}\mathrm{(\tau)}\:\mathrm{=}\:\int_{-\infty}^{\infty}\mathit{x_{\mathrm{1}}}\mathrm{(\mathit{t})}x_{\mathrm{2}}^{*}\mathrm{(\mathit{t-\tau})}\mathit{dt} \:\mathrm{=}\: \int_{-\infty}^{\infty}\mathit{x_{\mathrm{1}}}\mathrm{(\mathit{t+\tau})}\mathit{x_\mathrm{2}^*}\mathrm{(\mathit{t})}\mathit{dt}$$Where, the variable $\tau$ is called the delay parameter or scanning parameter or searching parameter.The cross correlation of two energy signals is defined in another form as −$$\mathit{R_{\mathrm{12}}}\mathrm{(\mathit{\tau})} \:\mathrm{=}\: \int_{-\infty}^{\infty}\mathit{x_\mathrm{2}}\mathrm{(t)}\mathit{x_\mathrm{1}^*}\mathrm{(t-\tau)}\:\mathit{dt}$$Properties of ... Read More
1K+ Views
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 called the energy spectral density.The ESD function of a signal is given by, $$\mathit{\psi}\mathrm{(\mathit{\omega})}\: \mathrm{=}\: \mathrm{|\mathit{X}\mathrm{(\mathit{\omega})}|}^\mathrm{2} \:\mathrm{=}\: \mathit{X}\mathrm{(\mathit{\omega})} \mathit{X}\mathrm{(\mathit{-\omega})}$$Autocorrelation TheoremStatement − The autocorrelation theorem states that the autocorrelation function $\mathit{R}\mathrm{(\mathrm{\tau})}$ and the ESD (Energy Spectral Density) function $\mathit{\psi}\mathrm{(\mathit{\omega})}$ of an energy signal $\mathit{x}\mathrm{(\mathit{t})}$ form a Fourier transform pair, i.e., $$\mathit{R}\mathrm{(\mathit{\tau})} ... Read More
7K+ Views
Energy Spectral DensityThe distribution of the energy of a signal in the frequency domain is known as energy spectral density (ESD) or energy density (ED) or energy density spectrum. The ESD function is denoted by $\mathrm{\mathit{\psi \left ( \omega \right )}}$ and is given by, $$\mathrm{\mathit{\psi \left ( \omega \right )\mathrm{=}\left|X\left ( \omega \right ) \right|^{\mathrm{2}}}}$$For an energy signal, the total area under the energy spectral density curve plotted as the function of frequency is equal to the total energy of the signal.ExplanationConsider a linear system having $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$ and $\mathrm{\mathit{y\left ( \mathit{t} \right )}}$ as input ... Read More
17K+ Views
What is Autocorrelation?The autocorrelation function of a signal is defined as the measure of similarity or coherence between a signal and its time delayed version. Thus, the autocorrelation is the correlation of a signal with itself.The autocorrelation function is defined separately for energy or aperiodic signals and power or periodic signals.Autocorrelation Function for Energy SignalsThe autocorrelation function of an energy signal $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$ is defined as −$$\mathrm{\mathit{R_{\mathrm{11}}\left ( \tau \right )\mathrm{=}R\left ( \tau \right )\mathrm{=}\int_{-\infty }^{\infty }x\left ( t \right )x^{\ast }\left ( t-\tau \right )dt\mathrm{=}\int_{-\infty }^{\infty }x\left ( t\mathrm{+ }\tau \right )x^{\ast }\left ( t \right ... Read More