For a continuous-time function $\mathit{x(t)}$, the Fourier transform of $\mathit{x(t)}$ can be defined as, $$\mathrm{\mathit{X\left(\omega\right )\mathrm{=}\int_{-\infty }^{\infty} x\left(t\right)\:e^{-j\omega t}\:dt} }$$Fourier Transform of Gaussian SignalGaussian Function - The Gaussian function is defined as, $$\mathrm{\mathit{g_{a}\left(t\right)\mathrm{=} e^{-at^{\mathrm{2}}} ;\:\:\mathrm{for\:all} \:t} }$$Therefore, from the definition of Fourier transform, we have, $$\mathrm{\mathit{X\left(\omega\right)\mathrm{=} F\left [e^{-at^\mathrm{2}} \right ]=\int_{-\infty ... Read More

Power Spectral DensityThe distribution of average power of a signal in the frequency domain is called the power spectral density (PSD) or power density (PD) or power density spectrum. The power spectral density is denoted by $\mathit{S\left (\omega \right )}$ and is given by, $$\mathrm{\mathit{S\left (\omega \right )\mathrm{=}\lim_{\tau \rightarrow \infty ... Read More

What is a Filter?A filter is a frequency selective network, i.e., it allows the transmission of signals of certain frequencies with no attenuation or with very little attenuation and it rejects all other frequency components.What is an Ideal Filter?An ideal filter is a frequency selective network that has very sharp ... 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 called the Hilbert transform of the signal.The Hilbert transform ... Read More

A distortion is defined as the change of the shape of the signal when it is transmitted through the system. Therefore, the transmission of a signal through a system is said to be distortion-less when the output of the system is an exact replica of the input signal. This replica, ... Read More

For a continuous-time function 𝑥(𝑡), the Fourier transform of 𝑥(𝑡) can be defined as, $$\mathrm{X\left ( \omega \right )=\int_{-\infty }^{\infty }x\left ( t \right )e^{-j\omega t}dt}$$System Analysis with Fourier TransformConsider an LTI (Linear Time-Invariant) system, which is described by the differential equation as, $$\mathrm{\sum_{k=0}^{N}a_{k}\frac{\mathrm{d}^{k}y\left ( t \right ) }{\mathrm{d} t^{k}}=\sum_{k=0}^{M}b_{k}\frac{\mathrm{d}^{k}x\left ... 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 Scaling Property of Fourier TransformStatement – The time-scaling property of Fourier transform states that if a signal is expended in time by a quantity (a), then ... Read More

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

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

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