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Manish Kumar Saini has Published 1379 Articles
Manish Kumar Saini
9K+ Views
Fourier SeriesConsider a periodic signal π(π‘) be periodic with period T, then the Fourier series of the function π(π‘) is defined as, $$\mathrm{g(t)=\sum_{n=-\infty}^{\infty}C_{n}e^{jn\omega_{0}t}\:\:\:\:....(1)}$$Where, πΆπ is the Fourier series coefficient and is given by, $$\mathrm{C_{n}=\frac{1}{T}\int_{\frac{-T}{2}}^{\frac{T}{2}}g(t)e^{-jn\omega_{0}t}dt\:\:\:\:....(2)}$$Derivation of Fourier Transform from Fourier SeriesLet π₯(π‘) be a non-periodic signal and let the relation between ... Read More
Manish Kumar Saini
9K+ Views
The graph plotted between the Fourier coefficients of a periodic function $x(t)$ and the frequency (Ο) is known as the Fourier spectrum of a periodic signal.The Fourier spectrum of a periodic function has two parts βAmplitude Spectrum β The amplitude spectrum of the periodic signal is defined as the plot ... Read More
Manish Kumar Saini
12K+ Views
A periodic signal can be represented over a certain interval of time in terms of the linear combination of orthogonal functions, if these orthogonal functions are trigonometric functions, then the Fourier series representation is known as trigonometric Fourier series.ExplanationConsider a sinusoidal signal $x(t)=A\:sin\:\omega_{0}t$ which is periodic with time period $T$ ... Read More
Manish Kumar Saini
1K+ Views
Fourier SeriesIf $x(t)$ is a periodic function with period $T$, then the continuous-time exponential Fourier series of the function is defined as, $$\mathrm{x(t)=\sum_{n=β\infty}^{\infty}C_{n}e^{jn\omega_{0} t}β¦ (1)}$$Where, $C_{n}$ is the exponential Fourier series coefficient, which is given by, $$\mathrm{C_{n}=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}x(t)e^{-jn\omega_{0} t}dtβ¦ (2)}$$Time Shifting Property of Fourier SeriesLet $x(t)$ is a periodic function with ... Read More
Manish Kumar Saini
10K+ Views
Fourier TransformThe Fourier transform of a continuous-time function $x(t)$ can be defined as, $$\mathrm{X(\omega)=\int_{β\infty}^{\infty}x(t)e^{-j\omega t}dt}$$And the inverse Fourier transform is defined as, $$\mathrm{x(t)=\frac{1}{2\pi}\int_{β\infty}^{\infty}X(\omega)e^{j\omega t}d \omega}$$Time Differentiation Property of Fourier TransformStatement β The time differentiation property of Fourier transform states that the differentiation of a function in time domain is equivalent ... Read More
Manish Kumar Saini
1K+ Views
Fourier SeriesIf $x(t)$ is a periodic function with period $T$, then the continuous-time exponential Fourier series of the function is defined as, $$\mathrm{x(t)=\sum_{n=β\infty}^{\infty}C_{n}\:e^{jn\omega_{0} t}β¦ (1)}$$Where, $C_{n}$ is the exponential Fourier series coefficient, which is given by, $$\mathrm{C_{n}=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}x(t)e^{-jn\omega_{0} t}dtβ¦ (2)}$$Time Differentiation Property of Fourier SeriesIf $x(t)$ is a periodic function with ... Read More
Manish Kumar Saini
2K+ Views
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 ... Read More
Manish Kumar Saini
4K+ Views
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 ... Read More
Manish Kumar Saini
4K+ Views
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 ... Read More
Manish Kumar Saini
690 Views
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 ... Read More