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Published on 07-Dec-2021 07:49:56
Quarter Wave SymmetryA periodic function $x(t)$ which has either odd symmetry or even symmetry along with the half wave symmetry is said to have quarter wave symmetry.Mathematically, a periodic function $x(t)$ is said to have quarter wave symmetry, if it satisfies the following condition −$$\mathrm{x(t)=x(-t)\:or\:x(t)=-x(-t)\:and\:x(t)=-x\left (t ± \frac{T}{2}\right )}$$Some examples ... Read More
Published on 07-Dec-2021 07:45:25
Fourier SeriesIf $x(t)$ is a periodic function with period $T$, then the continuous-time exponential Fourier series of the function is defined as, $$\mathrm{x(t)=\sum_{n=−\infty}^{\infty}C_{n}\:e^{jn\omega_{0} t}… (1)}$$Where, $C_{n}$ is the exponential Fourier series coefficient, which is given by, $$\mathrm{C_{n}=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}x(t)\:e^{-jn\omega_{0} t}\:dt… (2)}$$Parseval’s Theorem and Parseval’s IdentityLet $x_{1}(t)$ and $x_{2}(t)$ two complex periodic functions ... Read More
Published on 07-Dec-2021 07:43:04
Importance of Wave symmetryIf a periodic signal $x(t)$ 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.Odd or Rotation SymmetryWhen a periodic function $x(t)$ is antisymmetric about the vertical axis, then the function is ... Read More
Published on 07-Dec-2021 07:40:30
Fourier TransformThe Fourier transform is defined as a transformation technique which transforms signals from the continuous-time domain to the corresponding frequency domain and vice-versa. In other words, the Fourier transform is a mathematical technique that transforms a function of time $x(t)$ to a function of frequency X(ω) and vice-versa.For a ... Read More
Published on 07-Dec-2021 07:37:32
Importance of Wave SymmetryIf a periodic signal $x(t)$ 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.Half Wave SymmetryA periodic function $x(t)$ is said to have half wave symmetry, if it satisfies the following ... Read More
Published on 06-Dec-2021 13:19:37
Exponential Fourier SeriesA 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 exponential functions, then it is called the exponential Fourier seriesFor any periodic signal 𝑥(𝑡), the exponential form of Fourier series is given ... Read More
Published on 06-Dec-2021 12:51:20
The graph plotted between the Fourier coefficients of a periodic function $x(t)$ and the frequency (ω) is known as the Fourier spectrum of a periodic signal.The Fourier spectrum of a periodic function has two parts −Amplitude Spectrum − The amplitude spectrum of the periodic signal is defined as the plot ... Read More
Published on 06-Dec-2021 12:46:07
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
Published on 06-Dec-2021 12:41:06
Fourier SeriesIf $x(t)$ is a periodic function with period $T$, then the continuous-time exponential Fourier series of the function is defined as, $$\mathrm{x(t)=\sum_{n=−\infty}^{\infty}C_{n}e^{jn\omega_{0} t}… (1)}$$Where, $C_{n}$ is the exponential Fourier series coefficient, which is given by, $$\mathrm{C_{n}=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}x(t)e^{-jn\omega_{0} t}dt… (2)}$$Time Shifting Property of Fourier SeriesLet $x(t)$ is a periodic function with ... Read More
Published on 06-Dec-2021 12:36:38
Fourier TransformThe Fourier transform of a continuous-time function $x(t)$ can be defined as, $$\mathrm{X(\omega)=\int_{−\infty}^{\infty}x(t)e^{-j\omega t}dt}$$And the inverse Fourier transform is defined as, $$\mathrm{x(t)=\frac{1}{2\pi}\int_{−\infty}^{\infty}X(\omega)e^{j\omega t}d \omega}$$Time Differentiation Property of Fourier TransformStatement – The time differentiation property of Fourier transform states that the differentiation of a function in time domain is equivalent ... Read More
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