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, $$\mathrm{\mathit{X\left(\omega\right)\mathrm{=} F\left [e^{-at^\mathrm{2}} \right ]=\int_{-\infty }^{\infty}e^{-at^\mathrm{2}} \:e^{-j\omega t} \:dt}}$$$$\mathrm{\Rightarrow \mathit{X\left(\omega\right) \mathrm{=}\int_{-\infty}^{\infty} e^{-\left(at^\mathrm{2}+j\omega t\right) }\:dt \mathrm{=}e^{-\left(\omega^\mathrm{2}/\mathrm{4}a\right)}\int_{-\infty}^{\infty}e^{\left [{-t\sqrt{a}+(j\omega/\mathrm{2}\sqrt{a})}\right]^{2}}}dt }$$Let, $$\mathrm{\mathit{\left [t\sqrt{a}+(j\omega /\mathrm{2}\sqrt{a})\right ]\mathrm{=} u}}$$Then, $$\mathrm{\mathit{du\mathrm{=} \sqrt{a} \:dt\:\mathrm{and}\: \:dt\mathrm{=} \frac{du}{\sqrt{a}}}}$$$$\mathrm{\mathit{\therefore X\left(\omega\right)\mathrm{=}e^{-\left(\omega^\mathrm{2}/\mathrm{4}a\right)}\int_{-\infty }^{\infty} \frac{e^{-u^{\mathrm{2}}}}{\sqrt{a}}\:du\:\mathrm{=} \frac{e^{-\left(\omega^\mathrm{2}/\mathrm{4}a\right)}}{\sqrt{a}}\int_{-\infty }^{\infty}e^{-u^{\mathrm{2}}} \:du}}$$$$\mathrm{\mathit{\because\int_{-\infty }^{\infty}e^{-u^{\mathrm{2}}} \:du\mathrm{=} \sqrt{\pi}}}$$$$\mathrm{\mathit{\therefore X\left(\omega\right)\mathrm{=}\frac{e^{-\left(\omega^\mathrm{2}/\mathrm{4}a\right)}}{\sqrt{a}}\cdot \sqrt{\pi}\mathrm{=} \sqrt{\frac{\pi}{a}} \cdot e^{-\left(\omega^\mathrm{2}/\mathrm{4}a\right)} } }$$Therefore, the Fourier transform of the Gaussian function is, $$\mathrm{\mathit{F\left [e^{-at^{\mathrm{2}}}\right ] \mathrm{=}\sqrt{\frac{\pi}{a}} ... 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, $$\mathrm{\mathit{S\left (\omega \right )\mathrm{=}\lim_{\tau \rightarrow \infty }\frac{\left | X\left (\omega \right ) \right |^{\mathrm{2}}}{\tau }}}$$AutocorrelationThe autocorrelation function gives the measure of similarity between a signal and its time-delayed version. The autocorrelation function of power (or periodic) signal $\mathit{x\left ( t \right ) }$ with any time period T is given by, $$\mathrm{\mathit{R\left(\tau \right)=\lim_{T\rightarrow \infty }\mathrm{\frac{1}{\mathit{T}}}\int_{-\left(T/\mathrm{2}\right)}^{T/\mathrm{2}}x\left(t\right)\:x^{*}\left(t-\tau \right)\:dt}}$$Where, ... 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 cut-off characteristics, i.e., it transmits the signals of certain specified band of frequencies exactly and totally rejects the signals of frequencies outside this band. Therefore, the phase spectrum of an ideal filter is linear.Ideal Filter CharacteristicsBased on the frequency response characteristics, the ideal filters can be of following types −Ideal ... 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 of a signal$\mathit{x\left(t\right)}$ is obtained by the convolution of $\mathit{x\left(t\right)}$ and (1/πt), i.e., , $$\mathrm{\mathit{\hat{x}\left(t\right)=x\left(t\right)*\left ( \frac{\mathrm{1}}{\mathit{\pi t}} \right )}}$$Properties of Hilbert TransformThe statement and proofs of the properties of the Hilbert transform are given as follows −Property 1The Hilbert transform does not change the domain of a signal.ProofLet a ... Read More

ConvolutionConvolution is a mathematical tool for combining two signals to produce a third signal. In other words, the convolution can be defined as a mathematical operation that is used to express the relation between input and output an LTI system.Consider two signals $\mathit{x_{\mathrm{1}}\left( t\right )}$ and $\mathit{x_{\mathrm{2}}\left( t\right )}$. Then, the convolution of these two signals is defined as$$\mathrm{ \mathit{\mathit{y\left(t\right)=x_{\mathrm{1}}\left({t}\right)*x_{\mathrm{2}}\left({t}\right)\mathrm{=}\int_{-\infty }^{\infty }x_{\mathrm{1}}\left(\tau\right)x_{\mathrm{2}}\left(t-\tau\right)\:d\tau=\int_{-\infty }^{\infty }x_{\mathrm{2}}\left(\tau \right)x_{\mathrm{1}}\left(t-\tau\right)\:d\tau }}}$$Properties of ConvolutionContinuous-time convolution has basic and important properties, which are as follows −Commutative Property of Convolution − The commutative property of convolution states that the order in which we convolve two signals does not ... 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, i.e., the output of the system may have different magnitude and also it may have different time delay.A constant change in the magnitude and a constant time delay in the output signal is not considered as distortion. Only the change in the shape of the signal is considered the distortion.Mathematically, ... 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, $$\mathrm{\sum_{k=0}^{N}a_{k}\frac{\mathrm{d}^{k}y\left ( t \right ) }{\mathrm{d} t^{k}}=\sum_{k=0}^{M}b_{k}\frac{\mathrm{d}^{k}x\left ( t \right ) }{\mathrm{d} t^{k}}}$$Taking Fourier transform on both sides of the above equation, we get, $$\mathrm{F\left [ \sum_{k=0}^{N}a_{k}\frac{\mathrm{d}^{k}y\left ( t \right ) }{\mathrm{d} t^{k}} \right ]=F\left [ \sum_{k=0}^{M}b_{k}\frac{\mathrm{d}^{k}x\left ( t \right ) }{\mathrm{d} t^{k}} \right ]}$$By using linearity property $\mathrm{\left [ i.e., \: ax_{1}\left ( t \right )+bx_{2}\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 its Fourier transform is compressed in frequency by the same amount. Therefore, if$$\mathrm{x\left ( t \right )\overset{FT}{\leftrightarrow} X\left ( \omega \right )}$$Then, according to the time-scaling property of Fourier transform$$\mathrm{x\left ( at \right )\overset{FT}{\leftrightarrow}\frac{1}{\left | a \right |} X\left ( \frac{\omega }{a} \right )}$$When 𝑎 > 1, then 𝑥(𝑎𝑡) is ... 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 complex function 𝑥(𝑡) that is represented as −$$\mathrm{x\left ( t \right )=x_{r}\left ( t \right )+jx_{i}\left ( t \right )}$$Where, 𝑥𝑟 (𝑡) and 𝑥𝑖 (𝑡) are the real and imaginary parts of the function respectively.Now, the Fourier transform of function 𝑥(𝑡) is given by, $$\mathrm{F\left [ x\left ( t \right ... 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 convolution in time domain is equivalent to the multiplication of their spectrum in frequency domain. Therefore, if the Fourier transform of two time signals is given as, $$\mathrm{x_{1}\left ( t \right )\overset{FT}{\leftrightarrow}X_{1} \left ( \omega \right )}$$And$$\mathrm{x_{2}\left ( t \right )\overset{FT}{\leftrightarrow}X_{2} \left ( \omega \right )}$$Then, according to the time ... Read More