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 to drive the power spectral density (PSD) function, consider a power signal as a limiting case of an energy signal, i.e., the signal $\mathit{Z\mathrm{\left({\mathit{t }}\right)}}$ is zero outside the interval $\left|\tau /2 \right|$ as shown in the figure.The signal $\mathit{Z\mathrm{\left({\mathit{t }}\right)}}$ is given by, $$\mathrm{\mathit{Z\mathrm{\left({\mathit{t }}\right)}}\mathrm{=}\begin{cases} x\mathrm{\left(\mathit{t}\right)}\:\left|t \right|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, $\mathit{\delta}\mathrm{\left(\mathit{t-nT} \right)}$ is zero except at the instants t = nT. A reconstruction filter which is assumed to be linear and time invariant has unit impulse response $\mathit{h\mathrm{\left({\mathit{t}}\right)}}$. The output of the reconstruction filter is given by the convolution as, $$\mathrm{\mathit{y\mathrm{\left({\mathit{t}}\right)}}\:\mathrm{=}\:\int_{-\infty}^{\infty}\sum_{\mathit{n}=-\infty}^{\infty}\:\mathit{x}\mathrm{\left(\mathit{nT} \right )}\delta\mathrm{\left(\mathit{k-nT} \right)}\mathit{h}\mathrm{\left ( \mathit{t-k} \right )}\mathit{dk}}$$By rearranging the order of ... 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 of convergence.ROC of Two-Sided SignalsA signal $\mathrm{\mathit{x\left ( t \right )}}$ is said to be a two sided signal if it extends from -∞ to +∞. The two sided signal can be represented as the sum of two non-overlapping signals, one of which is right-sided signal and the other is ... 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 −$$\mathrm{\mathit{L\left [ x\left ( t \right ) \right ]\mathrm{=}X\left ( s \right )\mathrm{=}\int_{-\infty }^{\infty }x\left ( t \right )e^{-st}\; dt\; \; \; \cdot \cdot \cdot \left ( \mathrm{1} \right )}}$$Equation (1) gives the bilateral Laplace transform of the function $\mathrm{\mathit{x\left ( t \right )}}$. But for the causal signals, the unilateral 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 $\mathrm{\mathit{x\left ( t \right )}}$ is a time domain function, then its Laplace transform is defined as −$$\mathrm{\mathit{L\left [ x\left ( t \right ) \right ]\mathrm{=}X\left ( s \right )\mathrm{=}\int_{-\infty }^{\infty }x\left ( t \right )e^{-st}\; dt\; \; \; \cdot \cdot \cdot \left ( \mathrm{1} \right )}}$$Equation (1) gives the bilateral Laplace transform of the function $\mathrm{\mathit{x\left ( t \right )}}$ . But for the causal signals, the unilateral 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 $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$ is a time domain function, then its Laplace transform is defined as −$$\mathrm{\mathit{L\left [ x\left ( \mathrm{t} \right ) \right ]}= \mathit{X\left ( s \right )}=\int_{-\infty }^{\infty}\mathit{x\left ( \mathrm{t} \right )e^{-st}\; dt}\; \; ...\left ( 1 \right )}$$Equation (1) gives the bilateral Laplace transform of the function $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$. But for the causal signals, the unilateral Laplace transform is applied, which is ... Read More

In engineering analysis, a complex mathematically modelled physical system is converted into a simpler, solvable model by employing an integral transform. Once the model is solved, the inverse integral transform is used to provide the solution in the original form. There are two most commonly used integral transforms namely – Laplace Transform and Fourier Transform. In both these transforms, a physical system represented in differential equations is converted into algebraic equations or in easily solvable differential equations of lower degree. Thus, Laplace Transform and Fourier Transform make the problem easier to solve. In this article, we will learn ... 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 $x\mathrm{\left ( \mathit{t}\right)}$ is a time domain function, then its Laplace transform is defined as −$$\mathrm{\mathit{L\mathrm{\left[\mathit{x\mathrm{\left(\mathit{t} \right )}}\right ]}}\mathrm{=}\mathit{X\mathrm{\left(\mathit{s} \right )}}\mathrm{=}\int_{-\infty }^{\infty}\mathit{x\mathrm{\left(\mathit{t} \right )}e^{-st}}\:\mathit{dt}\:\:\:\:\:\:...(1)}$$Equation (1) gives the bilateral Laplace transform of the function $x\mathrm{\left ( \mathit{t}\right)}$. But for the causal signals, the unilateral Laplace transform is applied, which is defined as, $$\mathrm{\mathit{L\mathrm{\left[\mathit{x\mathrm{\left(\mathit{t} \right )}}\right ]}}\mathrm{=}\mathit{X\mathrm{\left(\mathit{s} \right )}}\mathrm{=}\int_{\mathrm{0} }^{\infty}\mathit{x\mathrm{\left(\mathit{t} \right )}e^{-st}}\:\mathit{dt}\:\:\:\:\:\:...(2)}$$Laplace Transform of Damped Hyperbolic Sine FunctionThe damped hyperbolic sine function ... Read More

What is Correlation?The correlation of two functions or signals or waveforms is defined as the measure of similarity between those signals. There are two types of correlations −Cross-correlationAutocorrelationCross-correlationThe cross-correlation between two different signals or functions or waveforms is defined as the measure of similarity or coherence between one signal and the time-delayed version of another signal. The cross-correlation between two different signals indicates the degree of relatedness between one signal and the time-delayed version of another signal.The cross-correlation of energy (or aperiodic) signals and power (or periodic) signals is defined separately.Cross-correlation of Energy SignalsConsider two complex signals $\mathit{x_{\mathrm{1}}\mathrm{\left ( \mathit{t} ... Read More

What is Sampling?The process of converting a continuous-time signal into a discrete-time signal is called sampling. Once the sampling is done, the signal is defined at discrete instants of time and the time interval between two successive sampling instants is called the sampling period.Nyquist Rate of SamplingThe Nyquist rate of sampling is the theoretical minimum sampling rate at which a signal can be sampled and still be reconstructed from its samples without any distortion.Effects of Under Sampling (Aliasing)If a signal is sampled at less than its Nyquist rate, then it is called undersampled.The spectrum of the sampled signal is given ... Read More