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Manish Kumar Saini has Published 1379 Articles
Manish Kumar Saini
5K+ Views
Fourier TransformFor a continuous-time function 𝑥(𝑡), the Fourier transform of 𝑥(𝑡) can be defined as, $$\mathrm{X\left ( \omega \right )=\int_{-\infty }^{\infty}x\left ( t \right )e^{-j\omega t}dt}$$And the inverse Fourier transform is defined as, $$\mathrm{x\left ( t \right )=\frac{1}{2\pi }\int_{-\infty }^{\infty}X\left ( \omega \right )e^{j\omega t}d\omega}$$Fourier Transform of Complex FunctionsConsider a ... Read More
Manish Kumar Saini
3K+ Views
ConvolutionThe convolution of two signals 𝑥(𝑡) and ℎ(𝑡) is defined as, $$\mathrm{y\left ( t \right )=x\left( t \right )\ast h\left ( t \right )=\int_{-\infty }^{\infty}x\left ( \tau \right )h\left ( t-\tau \right )d\tau}$$This integral is also called the convolution integral.Time Convolution TheoremStatement – The time convolution theorem states that the ... Read More
Manish Kumar Saini
3K+ Views
Linear System – A system for which the principle of superposition and the principle of homogeneity is valid is called a linear system.Filter Characteristics of Linear SystemFor a given linear system, an input signal 𝑥(𝑡) produces a response signal 𝑦(𝑡). Therefore, the system processes the input signal 𝑥(𝑡) according to ... Read More
Manish Kumar Saini
8K+ Views
What is a Linear System?System − An entity which acts on an input signal and transforms it into an output signal is called the system.Linear System − A linear system is defined as a system for which the principle of superposition and the principle of homogeneity are valid.Superposition PrincipleThe principle ... Read More
Manish Kumar Saini
6K+ Views
Hilbert TransformWhen the phase angles of all the positive frequency spectral components of a signal are shifted by (-90°) and the phase angles of all the negative frequency spectral components are shifted by (+90°), then the resulting function of time is known as Hilbert transform of the given signal.In case ... Read More
Manish Kumar Saini
3K+ Views
Energy Spectral DensityThe distribution of energy of a signal in the frequency domain is called the energy spectral density (ESD) or energy density (ED) or energy density spectrum. It is denoted by $\psi (\omega )$ and is given by, $$\mathrm{\psi (\omega )=\left | X(\omega ) \right |^{2}}$$AutocorrelationThe autocorrelation function gives ... Read More
Manish Kumar Saini
6K+ Views
Fourier TransformFor a continuous-time function x(t), the Fourier transform of x(t) can be defined as, $$\mathrm{X(\omega)=\int_{-\infty }^{\infty}x(t)\:e^{-jwt}\:dt}$$And the inverse Fourier transform is defined as, $$\mathrm{x(t)=\frac{1}{2\pi}\int_{-\infty }^{\infty}X(\omega)\:e^{jwt}\:d\omega}$$Time Integration Property of Fourier TransformStatementThe time integration property of continuous-time Fourier transform states that the integration of a function x(t) in time domain is ... Read More
Manish Kumar Saini
21K+ Views
Linear Time Invariant SystemA system for which the principle of superposition and the principle of homogeneity are valid and the input/output characteristics do not with time is called the linear time invariant (LTI) system.Properties of LTI SystemA continuous-time LTI system can be represented in terms of its unit impulse response. ... Read More
Manish Kumar Saini
16K+ Views
For a continuous-time function 𝑥(𝑡), the Fourier transform of 𝑥(𝑡) can be defined as, $$\mathrm{X\left ( \omega \right )=\int_{-\infty }^{\infty }x\left ( t \right )e^{-j\omega t}\: dt}$$Time-Shifting Property of Fourier TransformStatement – The time shifting property of Fourier transform states that if a signal 𝑥(𝑡) is shifted by 𝑡0 in ... Read More
Manish Kumar Saini
7K+ Views
For a continuous-time function 𝑥(𝑡), the Fourier transform of 𝑥(𝑡) can be defined as, $$\mathrm{X\left ( \omega \right )=\int_{-\infty }^{\infty }x\left ( t \right )e^{-j\omega t}\: dt}$$Time Reversal Property of Fourier TransformStatement – The time reversal property of Fourier transform states that if a function 𝑥(𝑡) is reversed in time ... Read More