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

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

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

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

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

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

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

Signal BandwidthThe spectral components of a signal extends from (−∞) to ∞ and any practical signal has finite energy. Consequently, the spectral components approach zero when the frequency 𝜔 tends to ∞. Therefore, those spectral components can be neglected which have negligible energy and hence only a band of frequency ... Read More

ConvolutionThe convolution is a mathematical operation for combining two signals to form a third signal. In other words, the convolution is a mathematical way which is used to express the relation between the input and output characteristics of an LTI system.Mathematically, the convolution of two signals is given by, $$\mathrm{x_{1}\left ... Read More

Fourier TransformThe Fourier transform of a continuous-time function $x(t)$ can be defined as, $$\mathrm{X(\omega)=\int_{−\infty}^{\infty}x(t)e^{-j\omega t}dt}$$Fourier Transform of Unit Impulse FunctionThe unit impulse function is defined as, $$\mathrm{\delta(t)=\begin{cases}1 & for\:t=0 \0 & for\:t ≠ 0 \end{cases}}$$If it is given that$$\mathrm{x(t)=\delta(t)}$$Then, from the definition of Fourier transform, we have, $$\mathrm{X(\omega)=\int_{−\infty}^{\infty}x(t)e^{-j\omega t}dt=\int_{−\infty}^{\infty}\delta(t)e^{-j\omega t}dt}$$As ... Read More