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Found 1024 Articles for Digital Electronics
![Manish Kumar Saini](https://www.tutorialspoint.com/assets/profiles/334420/profile/60_45466-1624275142.png)
13K+ 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 $\mathrm{\mathit{x\left ( t \right )}}$ 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
![Manish Kumar Saini](https://www.tutorialspoint.com/assets/profiles/334420/profile/60_45466-1624275142.png)
8K+ 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 $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$ 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
![Kiran Kumar Panigrahi](https://www.tutorialspoint.com/assets/profiles/390736/profile/60_2598-1635242143.png)
34K+ Views
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
![Manish Kumar Saini](https://www.tutorialspoint.com/assets/profiles/334420/profile/60_45466-1624275142.png)
758 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 $x\mathrm{\left ( \mathit{t}\right)}$ 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
![Manish Kumar Saini](https://www.tutorialspoint.com/assets/profiles/334420/profile/60_45466-1624275142.png)
13K+ Views
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
![Manish Kumar Saini](https://www.tutorialspoint.com/assets/profiles/334420/profile/60_45466-1624275142.png)
9K+ Views
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
![Manish Kumar Saini](https://www.tutorialspoint.com/assets/profiles/334420/profile/60_45466-1624275142.png)
15K+ Views
Nyquist Rate of SamplingThe theoretical minimum sampling rate at which a signal can be sampled and still can be reconstructed from its samples without any distortion is called the Nyquist rate of sampling.Mathematically, $$\mathrm{Nyquist\: Rate, \mathit{f_{N}}\mathrm{=}2\mathit{f_{m}}}$$Where, $\mathit{f_{m}}$is the maximum frequency component present in the signal.If the signal is sampled at the rate greater than the Nyquist rate, then the signal is called over sampled.If the signal is sampled at the rate less than its Nyquist rate, then it is said to be under sampled.Nyquist IntervalWhen the rate of sampling is equal to the Nyquist rate, then the time interval between ... Read More
![Manish Kumar Saini](https://www.tutorialspoint.com/assets/profiles/334420/profile/60_45466-1624275142.png)
18K+ 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 $\mathrm{\mathit{x\left(t\right)}}$ is a time domain function, then its Laplace transform is defined as −$$\mathrm{\mathit{L\left[\mathit{x}\mathrm{\left(\mathit{t} \right )}\right ]\mathrm{=}X\mathrm{\left( \mathit{s}\right)}\mathrm{=}\int_{-\infty }^{\infty}x\mathrm{\left (\mathit{t} \right )}e^{-st} \:dt}}$$Circuit Analysis Using Laplace TransformThe Laplace transform can be used to solve the different circuit problems. In order to solve the circuit problems, first the differential equations of the circuits are to be written and then these differential equations are solved by using the Laplace transform. Also, the ... Read More
![Manish Kumar Saini](https://www.tutorialspoint.com/assets/profiles/334420/profile/60_45466-1624275142.png)
8K+ Views
ConvolutionThe convolution of two signals $\mathit{x\left ( t \right )}$ and $\mathit{h\left ( t \right )}$ is defined as, $$\mathrm{\mathit{y\left(t\right)\mathrm{=}x\left(t\right)*h\left(t\right)\mathrm{=}\int_{-\infty }^{\infty}x\left(\tau\right)\:h\left(t-\tau\right)\:d\tau}}$$This integral is also called the convolution integral.Frequency Convolution TheoremStatement - The frequency convolution theorem states that the multiplication of two signals in time domain is equivalent to the convolution of their spectra in the frequency domain.Therefore, if the Fourier transform of two signals $\mathit{x_{\mathrm{1}}\left ( t \right )}$ and $\mathit{x_{\mathrm{2}}\left ( t \right )}$ is defined as$$\mathrm{\mathit{x_{\mathrm{1}}\left(t\right)\overset{FT}{\leftrightarrow} X_{\mathrm{1}}\left(\omega\right)} }$$And$$\mathrm{\mathit{x_{\mathrm{2}}\left(t\right)\overset{FT}{\leftrightarrow} X_{\mathrm{2}}\left(\omega\right)}}$$Then, according to the frequency convolution theorem, $$\mathrm{\mathit{x_{\mathrm{1}}\left(t\right).x_{\mathrm{2}}\left(t\right)\overset{FT}{\leftrightarrow}\frac{\mathrm{1}}{\mathrm{2}\pi }\left [ X_{\mathrm{1}}\left(\omega\right)* X_{\mathrm{2}}\left(\omega\right)\right ]}}$$ProofFrom the definition of Fourier transform, we have, ... Read More
![Manish Kumar Saini](https://www.tutorialspoint.com/assets/profiles/334420/profile/60_45466-1624275142.png)
11K+ Views
The Fourier series can be used to analyse only the periodic signals, while the Fourier transform can be used to analyse both periodic as well as non-periodic functions. Therefore, the Fourier transform can be used as a universal mathematical tool in the analysis of both periodic and aperiodic signals over the entire interval. The Fourier transform of periodic signals can be found using the concept of impulse function.Now, consider a periodic signal $\mathit{x\left(t\right )}$ with period $\mathit{T}$. Then, the expression of $\mathit{x\left(t\right )}$ in terms of exponential Fourier series is given by, $$\mathrm{\mathit{x\left(t\right)=\sum_{n=-\infty }^{\infty } C_{n}\:e^{jn\omega _{\mathrm{0}}t}}}$$Where $\mathit{C_{n}}$ be the ... Read More