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SVG stands for Scalable Vector Graphics and is a language for describing 2D-graphics and graphical applications in XML and the XML is then rendered by an SVG viewer. Most of the web browsers can display SVG just like they can display PNG, GIF, and JPG.You can draw shapes like circle, rectangle, line, etc using SVG in HTML5 easily. Let’s see an example to draw a rectangle using SVG.ExampleYou can try to run the following code to draw a rectangle in HTML5. The element will be used #svgelem { ... 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 )}$$Then, according to the time ... Read More
Linear System – A system for which the principle of superposition and the principle of homogeneity is valid is called a linear system.Filter Characteristics of Linear SystemFor a given linear system, an input signal 𝑥(𝑡) produces a response signal 𝑦(𝑡). Therefore, the system processes the input signal 𝑥(𝑡) according to the characteristics of system. The spectral density function of the input signal 𝑥(𝑡) is given by 𝑋(𝑠) in s-domain or 𝑋(𝜔) in frequency domain. Also, the spectral density function of the response signal 𝑦(𝑡) is given by 𝑌(𝑠) in s-domain and 𝑌(𝜔) in frequency domain. Therefore, $$\mathrm{Y\left ( s \right ... Read More
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 of superposition states that the response of the system to a weighted sum of input signals is equal to the corresponding weighted sum of the outputs of the system to each of the input signals.Therefore, if an input signal x1(t) produces an output signal y1(t) and another input signal x2(t) ... 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 of Hilbert transformation of a signal, the magnitude spectrum of the signal does not change, only phase spectrum of the signal is changed. Also, Hilbert transform of a signal does not change the domain of the signal.Let a signal x(t) with Fourier transform X(ω). The Hilbert transform of x(t) is ... 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 the measure of similarity between a signal and its time delayed version. The autocorrelation function of an energy signal x(t) is given by, $$\mathrm{R(\tau )=\int_{-\infty }^{\infty}x(t)\:x^{*}(t-\tau )\:dt}$$Where, the parameter $\tau$ is called the delayed parameter.Relation between ESD and Autocorrelation FunctionThe autocorrelation function $R(\tau$) and the energy spectral density (ESD) function ... Read More