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 −“The representation of periodic signals over a certain interval of time in terms of linear combination of orthogonal functions (i.e., sine and cosine functions) is known as Fourier series.”The Fourier series is applicable only to the periodic signals i.e. the signals which repeat itself periodically over an interval from $(-\infty\:to\:\infty)$and ... 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 is, $$\mathrm{x(t)=a_{0}+\sum_{n=1}^{\infty}a_{n}\:cos\:n\omega_{0} t+b_{n}\:sin\:n\omega_{0} t… (1)}$$Where, $$\mathrm{a_{0}=\frac{1}{T}\int_{t_{0}}^{(t_{0}+T)}x(t)\:dt}$$$$\mathrm{a_{n}=\frac{2}{T}\int_{t_{0}}^{(t_{0}+T)}x(t)\:cos\:n\omega_{0} t\:dt}$$$$\mathrm{b_{n}=\frac{2}{T}\int_{t_{0}}^{(t_{0}+T)}x(t)\:sin\:n\omega_{0} t\:dt}$$In equation (1), by multiplying the numerator and denominator of the sine and cosine terms with ($\sqrt{a_{n}^{2}+b_{n}^{2}}$), we get, $$\mathrm{x(t)=a_{0}+\sum_{n=1}^{\infty}\left( \sqrt{a_{n}^{2}+b_{n}^{2}}\right)\left( \frac{a_{n}}{\sqrt{a_{n}^{2}+b_{n}^{2}}}cos\:n\omega_{0} t+\frac{b_{n}}{\sqrt{a_{n}^{2}+b_{n}^{2}}}sin\:n\omega_{0} t\right)… (2)}$$Putting the values in the equation (2) as, $$\mathrm{a_{0}=A_{0}}$$$$\mathrm{\sqrt{a_{n}^{2}+b_{n}^{2}}=A_{n}… (3)}$$$$\mathrm{\frac{a_{n}}{\sqrt{a_{n}^{2}+b_{n}^{2}}}=cos\:\theta_{n}\:\:and\:\:\frac{b_{n}}{\sqrt{a_{n}^{2}+b_{n}^{2}}}=-sin\:\theta_{n}}$$We obtain, $$\mathrm{x(t)=A_{0}+\sum_{n=1}^{\infty}A_{n}(cos\:\theta_{n}\:cos\:n\omega_{0} t-sin\:\theta_{n}\:sin\:n\omega_{0} t)}$$$$\mathrm{\Rightarrow\:x(t)=A_{0}+\sum_{n=1}^{\infty}A_{n}\:cos(n\omega_{0} t+\theta_{n})… (4)}$$Where, $$\mathrm{\theta_{n}=-tan^{-1} \left(\frac{b_{n}}{a_{n}}\right)… (5)}$$The ... 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 of periodic functions having quarter wave symmetry are shown in Figure-1.The Fourier series coefficients for the function having quarter wave symmetry are evaluated as follows −Case I – When n is odd$$\mathrm{x(t)=-x(-t)\:and\:x(t)=-x\left (t ± \frac{T}{2}\right )}$$For this case, $$\mathrm{a_{0}=0\:\:and\:\:a_{n}=0}$$And, $$\mathrm{b_{n}=\frac{8}{T} \int_{0}^{T/4}x(t)\:sin\:n\omega_{0}\:t\:dt}$$Case II – When n is even$$\mathrm{x(t)=x(-t)\:and\:x(t)=-x\left (t ± \frac{T}{2}\right ... 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 with period T and with Fourier series coefficients $C_{n}$ and $D_{n}$.If, $$\mathrm{x_{1}(t)\overset{FT}{\leftrightarrow}C_{n}}$$$$\mathrm{x_{2}(t)\overset{FT}{\leftrightarrow}D_{n}}$$Then, the Parseval’s theorem of continuous time Fourier series states that$$\mathrm{\frac{1}{T} \int_{t_{0}}^{t_{0}+T} x_{1}(t)\:x_{2}^{*}(t)\:dt =\sum_{n=−\infty}^{\infty} C_{n}\:D_{n}^{*}\:[for\:complex\: x_{1}(t)\: \& \: x_{2}(t)] … (3)}$$And the parseval’s identity of Fourier series states that, if$$\mathrm{x_{1}(t)=x_{1}(t)=x(t)}$$Then, $$\mathrm{\frac{1}{T}\int_{t_{0}}^{t_{0}+T}|x(t)|^{2}\:dt=\sum_{n=−\infty}^{\infty}|C_{n}|^{2}… (4)}$$Proof – Parseval’s theorem or Parseval’s relation or ... 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 said to have the odd symmetry or rotation symmetry.Mathematically, a function $x(t)$ is said to have odd symmetry, if$$\mathrm{x(t)=-x(-t)… (1)}$$Some functions having odd symmetry are shown in the figure. It is clear that the odd symmetric functions are always antisymmetrical about the vertical axis.ExplanationAs we know that any periodic signal ... 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 continuous-time function $x(t)$, the Fourier transform of $x(t)$ can be defined as$$\mathrm{X(ω)=\int_{−\infty}^{\infty}x(t)\:e^{-j\omega t}dt}$$Points about Fourier TransformThe Fourier transform can be applied for both periodic as well as aperiodic signals.The Fourier transform is extensively used in the analysis of LTI (linear time invariant) systems, cryptography, signal processing, signal analysis, etc.Fourier transform ... 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 condition −$$\mathrm{x(t)=-x\left ( t ± \frac{T}{2}\right )… (1)}$$Where, $T$ is the time period of the function.A periodic function $x(t)$ having half wave symmetry is shown in the figure. It can be seen that this function is neither purely even nor purely odd. For such type of functions, the DC component ... 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 of amplitude of Fourier coefficients versus frequency.Phase Spectrum − – The plot of the phase of Fourier coefficients versus frequency is called the phase spectrum of the signal.The amplitude spectrum and phase spectrum together are known as Fourier frequency spectra of the periodic signal $x(t)$. This type of representation of ... 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 by, $$\mathrm{X(t)=\sum_{n=-\infty}^{\infty}C_n e^{jn\omega_0t}\:\:\:...(1)}$$Where, 𝜔0 = 2𝜋⁄𝑇 is the angular frequency of the periodic function.Coefficients of Exponential Fourier SeriesIn order to evaluate the coefficients of the exponential series, we multiply both sides of the equation (1) by 𝑒−𝑗𝑚𝜔0𝑡 and integrate over one period, so we have, $$\mathrm{\int_{t_0}^{t_0+T}x(t)e^{-jm\omega_0t}dt=\int_{t_0}^{t_0+T}(\sum_{n=-\infty}^{\infty}C_ne^{jn\omega_0t})e^{-jm\omega_{0}t}dt}$$$$\mathrm{\Rightarrow\int_{t_0}^{t_0+T}x(t)e^{-jm\omega_0t}dt=\sum_{n=-\infty}^{\infty}C_n\int_{t_0}^{t_0+T}e^{jn\omega_0t}e^{-jm\omega_0t}dt}$$$$\mathrm{\because \int_{t_0}^{t_0+T}e^{jn\omega_0t}e^{-jm\omega_0t}dt=\begin{cases}0 & for\: m ... 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 $T$ such that $T=2\pi/ \omega_{0}$. If the frequencies of two sinusoids are integral multiples of a fundamental frequency $(\omega_{0})$, then the sum of these two sinusoids is also periodic.We can prove that a signal $x(t)$ that is a sum of sine and cosine functions whose frequencies are integral multiples of the ... Read More