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

Fourier SeriesConsider a periodic signal 𝑔(𝑡) be periodic with period T, then the Fourier series of the function 𝑔(𝑡) is defined as, $$\mathrm{g(t)=\sum_{n=-\infty}^{\infty}C_{n}e^{jn\omega_{0}t}\:\:\:\:....(1)}$$Where, 𝐶𝑛 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 𝑥(𝑡) be a non-periodic signal and let the relation between ... Read More

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

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