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Found 1024 Articles for Digital Electronics
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Fourier TransformThe Fourier transform of a continuous-time function π₯(π‘) can be defined as, $$\mathrm{X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt}$$Convolution Property of Fourier TransformStatement β The convolution of two signals in time domain is equivalent to the multiplication of their spectra in frequency domain. Therefore, if$$\mathrm{x_1(t)\overset{FT}{\leftrightarrow}X_1(\omega)\:and\:x_2(t)\overset{FT}{\leftrightarrow}X_2(\omega)}$$Then, according to time convolution property of Fourier transform, $$\mathrm{x_1(t)*x_2(t)\overset{FT}{\leftrightarrow}X_1(\omega)*X_2(\omega)}$$ProofThe convolution of two continuous time signals π₯1(π‘) and π₯2(π‘) is defined as, $$\mathrm{x_1(t)*x_2(t)=\int_{-\infty}^{\infty}x_1(\tau)x_2(t-\tau)d\tau}$$Now, from the definition of Fourier transform, we have, $$\mathrm{X(\omega)=F[x_1(t)*x_2(t)]=\int_{-\infty}^{\infty}[x_1(t)*x_2(t)]e^{-j \omega t}dt}$$$$\mathrm{\Rightarrow F[x_1(t)*x_2(t)]=\int_{-\infty}^{\infty}[\int_{-\infty}^{\infty}x_1(\tau)x_2(t-\tau)d\tau]e^{-j \omega t}dt }$$By interchanging the order of integration, we get, $$\mathrm{\Rightarrow F[x_1(t)*x_2(t)]=\int_{-\infty}^{\infty}x_1(\tau)[\int_{-\infty}^{\infty}x_{2}(t-\tau)e^{-j \omega t}dt]d\tau }$$By replacing (π‘ β π) = π’ in the second integration, ... Read More
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Fourier SeriesIf π₯(π‘) is a periodic function with period T, then the continuous-time Fourier series of the function is defined as, $$\mathrm{x(t)=\sum_{n=-\infty}^{\infty}C_ne^{jn\omega_{0}t}\:\:\:\:\:.....(1)}$$Where, πΆπ is the exponential Fourier series coefficient, that is given by$$\mathrm{C_n=\frac{1}{T}\int_{t_0}^{t_0+T}x(t)e^{-jn\omega_0t}dt\:\:\:\:\:.....(2)}$$Convolution Property of Fourier SeriesAccording to the convolution property, the Fourier series of the convolution of two functions π₯1(π‘) and π₯2(π‘) in time domain is equal to the multiplication of their Fourier series coefficients in frequency domain.If π₯1(π‘) and π₯2(π‘) are two periodic functions with time period T and with Fourier series coefficients πΆπ and π·π. Then, if$$\mathrm{x_1(t)\overset{FS}{\leftrightarrow}C_n}$$$$\mathrm{x_2(t)\overset{FS}{\leftrightarrow}D_n}$$Then, the convolution property of continuous time Fourier series states that$$\mathrm{x_1(t)*x_2(t)\overset{FS}{\leftrightarrow}TC_nD_n}$$ProofBy ... Read More
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Fourier TransformFor a continuous-time function x(t), the Fourier transform of x(t) can be defined as, $$\mathrm{X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt}$$Conjugation Property of Fourier TransformStatement β The conjugation property of Fourier transform states that the conjugate of function x(t) in time domain results in conjugation of its Fourier transform in the frequency domain and Ο is replaced by (βΟ), i.e., if$$\mathrm{x(t)\overset{FT}{\leftrightarrow}X(\omega)}$$Then, according to conjugation property of Fourier transform, $$\mathrm{x^*(t)\overset{FT}{\leftrightarrow}X^*(-\omega)}$$ProofFrom the definition of Fourier transform, we have$$\mathrm{X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt}$$Taking conjugate on both sides, we get$$\mathrm{X^*(\omega)=[\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt]^*}$$$$\mathrm{\Rightarrow X^*(\omega)=\int_{-\infty}^{\infty}x^*(t)e^{j\omega t}dt}$$Now, by replacing (Ο) by (βΟ), we obtain, $$\mathrm{X^*(-\omega)=\int_{-\infty}^{\infty}x^*(t)e^{-j\omega t}dt=F[x^*(t)]}$$$$\mathrm{\therefore F[x^*(t)]=X^*(-\omega)}$$Or, it can also be represented as, $$\mathrm{x^*(t)\overset{FT}{\leftrightarrow}X^*(-\omega)}$$Autocorrelation Property ... Read More
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Exponential Fourier SeriesPeriodic signals are represented over a certain interval of time in terms of the linear combination of orthogonal functions. If these orthogonal functions are the exponential functions, then the Fourier series representation of the function is called the exponential Fourier series.The exponential Fourier series is the most widely used form of the Fourier series. In this representation, the periodic function x(t) is expressed as a weighted sum of the complex exponential functions. The complex exponential Fourier series is the convenient and compact form of the Fourier series, hence, its findsextensive application in communication theory.ExplanationLet a set of complex ... Read More
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When a voltage of V volts is applied across a resistance of R Ξ©, then a current I flows through it. The power dissipated in the resistance is given by, $$\mathrm{P=I^2R=\frac{V^2}{R}\:\:\:\:\:\:....(1)}$$But when the voltage and current signals are not constant, then the power varies at every instant, and the equation for the instantaneous power is given by, $$\mathrm{P=i^2(t)R=\frac{V^2(t)}{R}\:\:\:\:\:\:....(2)}$$Where, π(π‘) and π£(π‘) are the corresponding instantaneous values of current and voltage respectivelyNow, if the value of the resistance (R) is 1 Ξ©, then the instantaneous power can be represented as, $$\mathrm{p=i^2(t)=v^2(t)\:\:\:\:\:\:....(3)}$$Therefore, the instantaneous power of a signal x(t) can be given ... Read More
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Fourier TransformFor a continuous-time function x(t), the Fourier transform of x(t) can be defined as$$\mathrm{X(\omega)= \int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt}$$Duality Property of Continuous-Time Fourier TransformStatement β If a function x(t) has a Fourier transform X(Ο) and we form a new function in time domain with the functional form of the Fourier transform as X(t), then it will have a Fourier transform X(Ο) with the functional form of the original time function, but it is a function of frequency.Mathematically, the duality property of CTFT states that, if$$\mathrm{x(t)\overset{FT}{\leftrightarrow}X(\omega)}$$Then, according to duality property, $$\mathrm{X(t)\overset{FT}{\leftrightarrow}2\pi x(-\omega)}$$ProofFrom the definition of inverse Fourier transform, we have$$\mathrm{x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega)e^{j\omega t}d\omega }$$$$\mathrm{\Rightarrow 2\pi.x(t)=\int_{-\infty}^{\infty}X(\omega)e^{j\omega ... Read More
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The Fourier series representation of a periodic signal has various important properties which are useful for various purposes during the transformation of signals from one form to another.Consider two periodic signals π₯1(π‘) and π₯2(π‘) which are periodic with time period T and with Fourier series coefficients πΆπ and π·π respectively. With this assumption, let us proceed and check the various properties of a continuoustime Fourier series.Linearity PropertyThe linearity property of continuous-time Fourier series states that, if$$\mathrm{x_{1}(t)\overset{FS}{\leftrightarrow}C_{n}\: and\:x_{2}(t)\overset{FS}{\leftrightarrow}D_{n}}$$Then$$\mathrm{Ax_{1}(t)+Bx_{2}(t)\overset{FS}{\leftrightarrow}AC_{n}+BD_{n}}$$Time Shifting PropertyThe time scaling property of Fourier series states that, if$$\mathrm{x(t)\overset{FS}{\leftrightarrow}C_{n}}$$Then$$\mathrm{x(t-t_{0})\overset{FS}{\leftrightarrow}e^{-jn\omega_{0}t_{0}}C_{n}}$$Time Scaling PropertyThe time scaling property of Fourier series states that, if$$\mathrm{x(t)\overset{FS}{\leftrightarrow}C_{n}}$$Then$$\mathrm{x(at)\overset{FS}{\leftrightarrow}C_{n}\:with\:\omega_{0}\rightarrow a\omega_{0}}$$Time ... Read More
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Importance of Wave SymmetryIf a periodic signal π₯(π‘) has some type of symmetry, then some of the trigonometric Fourier series coefficients may become zero and hence the calculation of the coefficients becomes simple.Even or Mirror SymmetryWhen a periodic function is symmetrical about the vertical axis, it is said to have even symmetry or mirror symmetry. The even symmetry is also called the reflection symmetry. Mathematically, a periodic function x(t) is said to have even symmetry, if$$\mathrm{π₯(π‘) = π₯(βπ‘)\:\:\:\:\:\: ...(1)}$$Some examples of functions having even symmetry are shown in the figure. The even functions are always symmetrical about the vertical axis.ExplanationAs ... Read More
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Fourier TransformFourier transform is a transformation technique that transforms signals from the continuous-time domain to the corresponding frequency domain and vice-versa.The Fourier transform of a continuous-time function $x(t)$ is defined as, $$\mathrm{X(\omega)=\int_{-\infty}^{\infty} x(t)e^{-j\omega t}dtβ¦ (1)}$$Inverse Fourier TransformThe inverse Fourier transform of a continuous-time function is defined as, $$\mathrm{x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega)\:e^{j\omega t}d\omegaβ¦ (2)}$$Equations (1) and (2) for $X(\omega)$ and $x(t)$ are known as Fourier transform pair and can be represented as β$$\mathrm{X(\omega)=F[x(t)]}$$And$$\mathrm{x(t)=F^{-1}[X(\omega)]}$$Table of Fourier Transform PairsFunction, x(t)Fourier Transform, X(Ο)$\delta(t)$1$\delta(t-t_{0})$$e^{-j \omega t_{0}}$1$2\pi \delta(\omega)$u(t)$\pi\delta(\omega)+\frac{1}{j\omega}$$\sum_{n=β\infty}^{\infty}\delta(t-nT)$$\omega_{0}\sum_{n=β\infty}^{\infty}\delta(\omega-n\omega_{0});\:\:\left(\omega_{0}=\frac{2\pi}{T} \right)$sgn(t)$\frac{2}{j\omega}$$ e^{j\omega_{0}t}$$ 2\pi\delta(\omega-\omega_{0})$$ cos\:\omega_{0}t$$\pi[\delta(\omega-\omega_{0})+\delta(\omega+\omega_{0})]$$sin\:\omega_{0}t$$-j\pi[\delta(\omega-\omega_{0})-\delta(\omega+\omega_{0})]$$e^{-at}u(t);\:\:\:a >0$$\frac{1}{a+j\omega}$$t\:e^{at}u(t);\:\:\:a >0$$\frac{1}{(a+j\omega)^{2}}$$e^{-|at|};\:\:a >0$$\frac{2a}{a^{2}+\omega^{2}}$$e^{-|t|}$$\frac{2}{1+\omega^{2}}$$\frac{1}{\pi t}$$-j\:sgn(\omega)$$\frac{1}{a^{2}+t^{2}}$$\frac{\pi}{a}e^{-a|\omega|}$$\Pi (\frac{t}{Ο})$$Ο\:sin c(\frac{\omega Ο}{2})$$\Delta(\frac{t}{Ο})$$\frac{Ο}{2}sin C^{2}(\frac{\omega Ο}{4})$$\frac{sin\:at}{\pi t}$$P_{a}(\omega)=\begin{cases}1 & for\:|\omega|\:< a\0 & for\:|\omega|\: > a ... Read More
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Trigonometric Fourier SeriesA periodic function can be represented over a certain interval of time in terms of the linear combination of orthogonal functions. If these orthogonal functions are the trigonometric functions, then it is known as trigonometric Fourier series.Mathematically, the standard trigonometric Fourier series expansion of a periodic signal is, $$\mathrm{x(t)=a_{0}+ \sum_{n=1}^{\infty}a_{n}\:cos\:\omega_{0}nt+b_{n}\:sin\:\omega_{0}nt\:\:β¦ (1)}$$Exponential Fourier SeriesA periodic function can be represented over a certain interval of time in terms of the linear combination of orthogonal functions, if these orthogonal functions are the exponential functions, then it is known as exponential Fourier series.Mathematically, the standard exponential Fourier series expansion for a periodic ... Read More