What is Fourier Series?In the domain of engineering, most of the phenomena are periodic in nature such as the alternating current and voltage. These periodic functions could be analysed by resolving into their constituent components by a process called the Fourier series.Therefore, the Fourier series can be defined as under ... Read More

The cosine form of Fourier series is the alternate form of the trigonometric Fourier series. The cosine form Fourier series is also known as polar form Fourier series or harmonic form Fourier series.The trigonometric Fourier series of a function x(t) contains sine and cosine terms of the same frequency. That ... Read More

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

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

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

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

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

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

Fourier TransformThe Fourier transform of a continuous-time function 𝑥(𝑡) can be defined as, $$\mathrm{X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt}$$Convolution Property of Fourier TransformStatement – 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 ... Read More

Fourier SeriesIf 𝑥(𝑡) 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, 𝐶𝑛 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 ... Read More