Calculus: Fundamental Theorem of Calculus -1 , & \ \mbox{ if } \ell < x < 2\ell ; \end{cases} \quad = \quad 2 \left[ H(x/\ell ) - H(x/\ell -1) \right] -1 = 2\,H(x) -1 = Generation of a Square Wave. \\ Calculus: Fundamental Theorem of Calculus example. It is known that restoring a function from its Fourier coefficients is an ill-posed problem. Loading. A square wave function, also called a pulse wave, is a periodic waveform consisting of instantaneous transitions between two levels. \end{eqnarray*}. For practical applications, its regularization, can be obtained with Cesro's summation: Finally, we demonstrate animation for Cesro's approximation, Return to the main page (APMA0340) f_m (x) = \frac{4}{\pi} \sum_{n= 0}^m \left( 1 - \frac{n}{m+1} \right) \frac{1}{2n+1} \,\sin \left( \frac{(2n+1) \pi x}{\ell} \right) = \frac{4}{\pi} \sum_{k= 1}^m \left( 1 - \frac{k}{m} \right) \frac{1}{2k-1} \,\sin \left( \frac{(2k-1) \pi x}{\ell} \right) . a_0 &= \frac{1}{\ell} \,\int \delta (x)\,{\text d}x = \frac{1}{\ell} , \], Plot[Sign[x], {x, -2.5, 2.5}, PlotStyle -> {Thickness[0.01], Purple}]. 0, & \ \mbox{otherwise} . A square wave function, also called a pulse wave, is a periodic waveform consisting of instantaneous transitions between two levels. Return to the Part 2 Linear Systems of Ordinary Differential Equations b_n &= \frac{1}{\ell} \int \delta (x)\,\sin \frac{n\pi x}{\ell} \,{\text d}x = \frac{1}{\ell} \,\sin 0 =0 . \tag{1.2} H[t_] = 1/2 + 2/Pi*Sum[1/(2*k+1)*Sin[(2*k+1)*Pi*t/2], {k,0,25}]; \begin{eqnarray*} \mbox{oddsign}(x) = {\bf j}\sum_{n=-\infty}^{\infty} \frac{(-1)^n -1}{n\pi}\, e^{n{\bf j}\pi x/\ell} \], \[ \\ \]. \\ 1/2 , & \ \mbox{ if } t=\ell , \\ \\ example. \mbox{sign}(x) = \begin{cases} \phantom{-}1 , & \ \mbox{ if } 0 < x < +\infty , H(t-\ell ) = \begin{cases} 0 , & \ \mbox{ if } 0 < t < \ell , \], \[ \alpha_n = \frac{1}{2\ell} \int_{-\ell}^{\ell} \mbox{sign}(x)\,e^{-n{\bf j} \pi x/\ell} {\text d}x = {\bf j}\,\frac{\cos n\pi -1}{n\pi} , \qquad n=1,2,\ldots . We start with the Heaviside function on finite interval [-2,2], which we extend in periodic manner with period 4: We can also expand the Heaviside function into Fourier sine and Fourier cosine series on the interval [0,2]. Return to the Part 7 Special Functions, \begin{eqnarray*} \end{eqnarray*}. \tag{1.4} \delta_N (x) = \frac{1}{2\ell} + \frac{1}{\ell}\,\sum_{n= 1}^N \cos \frac{n\pi x}{\ell} = \frac{1}{2\ell} \left[ 1 + 2\,\cos \frac{\pi x}{\ell} + 2 \,\cos \frac{2\pi x}{\ell} + \cdots + 2\,\cos \frac{N\pi x}{\ell} \right] . example. \\ They are also used in driving sweep oscillators in. The square wave is sometimes also called the Rademacher function. \mbox{sign}(x) = \sum_{n\ge 1} \frac{2}{n\pi} \left[ (-1)^{n+1} + 1 \right] \sin (n\pi x) = \frac{4}{\pi} \sum_{k\ge 0} \frac{1}{2k+1} \,\sin \left( \frac{(2k+1) \pi x}{\ell} \right) . example. \], \[ Other common levels for the square wave includes - and . \], \begin{align*} Other common levels for square waves include (-1,1) and (0,1) (digital signals). H(t) = \begin{cases} 1 , & \ \mbox{ if } 0 < t < \ell , \], \( f(0) = \frac{a_0}{2} + a_1 + a_2 + \cdots . Return to the Part 1 Matrix Algebra \mbox{sinc} (x) = \frac{\sin \pi x}{\pi x} = \frac{1}{\Gamma (1+x)\,\Gamma (1-x)} = \prod_{n\ge 1} \left( 1 - \frac{x^2}{n^2} \right) . 2. powered by. \\ 1/2 , & \ \mbox{ if } t=0 , \\ \], \[ We start with the Heaviside function on . To achieve this, we expand the Heaviside function evenly and oddly from the given interval: How we define the shifted 2 periodic Heaviside function: Example 1: The triangular- wave and sawtooth wave outputs of function generators are commonly used for those applications which need a signal that increases (or reduces) at a specific linear rate. Loading. \], \[ hs[t_] = 2/Pi*Sum[1/n*(Cos[n*Pi/2] -(-1)^n)*Sin[n*Pi*t/4], {n,1,50}]; \[ 2\,\cos \theta = e^{{\bf j}\theta} + e^{-{\bf j}\theta} , Transformations: Scaling a Function. hs(t) = \begin{cases} 0 , & \ \mbox{ if } 0 < t < \ell , Conic Sections: Parabola and Focus. \frac{4}{\pi} \,\sum_{k\ge 0} \frac{1}{2k+1} \,\sin \frac{(2k+1)\pi x}{\ell} . Untitled Graph. \tag{1.1} We consider two cases of square waves that include the digital signal (0,1) and oscillation between (-1,1). a_n = \frac{2}{n\pi h} \,\sin \frac{nh\pi}{2\ell} = \frac{h}{\ell} \,\frac{\frac{\pi nh}{2\ell}}{\pi nh/(2\ell )} = \frac{h}{\ell} \, \mbox{sinc} \frac{n\pi h}{2\ell} , example \end{align*}, \[ 2 < x < 3}, {-1, 3 < x < 4}, {1, 4 < x < 5}}]; \[ \\ \\ 1/2 , & \ \mbox{ if } t=\ell , \\ 1 , & \ \mbox{ if } \ell < t < 2\ell ; \end{cases} &=& \frac{1}{2} - \frac{2}{\pi} \,\sum_{n\ge 1} \frac{1}{n}\,\sin \left( \frac{n\pi}{2} \right) \cos \frac{n\pi x}{2\ell} . We can also expand the signum function into complex Fourier series: We plot partial sums with n = 10, 100, and 1000 terms. Return to the Part 3 Non-linear Systems of Ordinary Differential Equations \delta_N (x) = \frac{1}{2\ell} \left[ 1 + e^{{\bf j}\theta} + e^{-{\bf j}\theta} + e^{2{\bf j}\theta} + e^{-2{\bf j}\theta} + \cdots + e^{N{\bf j}\theta} + e^{-N{\bf j}\theta} \right] , \qquad \theta = \pi x/\ell . \end{cases}. The square wave illustrated above has period 2 and levels -1/2 and 1/2. Calculus: Integral with adjustable bounds. Statistics: Linear . 0 , & \ \mbox{ if } -\ell < t < 0 ; \end{cases} &=& \frac{1}{2} + \], \[ Sum[(1 - k/10)*Sin[(2*k - 1)*x]/(2*k - 1), {k, 1, 10}]; sign[t_] = (4/Pi) Sum[(1/n) Sin[2 Pi n t], {n, 1, \[Infinity], 2}]; f[t_] = (4/Pi) Sum[(1/n) Sin[2 Pi n t], {n, 1, \[Infinity], 2}]; f[t_] = (4/Pi) Sum[(1/n) Sin[2 Pi n t], {n, 1, , 2}], (2 I (ArcTanh[E^(-2 I t)] - ArcTanh[E^(2 I t)]))/, Plot[f[t], {t, -1, 3}, PlotStyle -> Thick], f[M_, x_] := 4/Pi *Sum[Sin[(2*k+1)*x]/(2*k+1) , {k,0,M}], Plot[Table[f[M, x] ,{M, 1,4}], {x, -5,5}], \begin{align*} \\ \,(-1)^k \sin^2 \frac{k\pi}{\ell} = \frac{4}{k\pi} \times \begin{cases} \), Linear Systems of Ordinary Differential Equations, Non-linear Systems of Ordinary Differential Equations, Boundary Value Problems for heat equation, Laplace equation in spherical coordinates, Plot of the signum function on interval [-2, 2] using. Square wave function constitute a very important class of functions used in electrical engineering and computer science; in particular, in music synthesizors. \phantom{-}0 , & \ \mbox{ if } x=0 , \\ A wealth of information hidden in the Fourier series can be recovered upon utilization of a summability method. Thia section is aout Fourier series exoansions of square like functions. 0 , & \ \mbox{ if } \ell < t < 2\ell ; \end{cases} &=& \frac{1}{2} + \frac{2}{\pi} \,\sum_{n\ge 1} \frac{1}{n}\,\sin \left( \frac{n\pi}{2} \right) \cos \frac{n\pi x}{2\ell} . \phantom{-}0 , & \ \mbox{ when } x=0 , \\ Translating a Function. a_n &= \frac{1}{\ell} \int \delta (x)\,\cos \frac{n\pi x}{\ell} \,{\text d}x = \frac{1}{\ell} \,\cos 0 = \frac{1}{\ell} , 1 , & \ \mbox{ if } \ell < t < 2\ell ; \end{cases} &=& \frac{2}{\pi}\, \sum_{n\ge 1} \frac{1}{n}\left[ \cos \frac{n\pi}{2} - (-1)^n \right] \sin \frac{n\pi x}{2\ell} , The Gibbs phenomenon clearly indicates that it is impossible to recover the function in a neighbohood of the point of discontinuity. \\ This section is a collection of Fourier series expansions for different step functions. \frac{2}{\pi} \,\sum_{k\ge 0} \frac{1}{2k+1} \,\sin \frac{(2k+1)\pi t}{\ell} . 4*Sum[(1 - k/(m + 1))*Sin[(2*k + 1)*Pi*x]/(2*k + 1), {k, 0, m}]/Pi; C10[x_] = (4/Pi)* We consider two cases of square waves that include the digital signal (0,1) and oscillation between (-1,1). Generation of a Square Wave. Calculus: Integral with adjustable bounds. \delta (x) = \frac{1}{2\ell} + \frac{1}{\ell} \,\sum_{n\ge 1} \cos \frac{n\pi x}{\ell} . The uses of sinusoidal outputs and square-wave outputs have already been described in the earlier Arts. s[x_, m_] = \]. \]. g[x_] = Piecewise[{{1, 0 < x < 1}, {-1, b_n &= \frac{1}{\ell} \,\int \delta (x)\,\sin \frac{n\pi x}{\ell} \,{\text d}x = \frac{1}{h\ell} \,\int_{-h/2}^{h/2} \,\sin \frac{n\pi x}{\ell} \,{\text d}x = 0 . 1 < x < 2}, {-1, -1 < x < 0}, {1, -2 < x < -1}, {1, Square wave from Sine wave harmonics. \], FourierSinCoefficient[Sign[x], x, n, FourierParameters -> {1, 1}], \[ -1 , & \ \mbox{ if } -\infty < x < 0 ; \end{cases} \tag{1.3} a_n &= \frac{1}{\ell} \,\int \delta_h (x)\,\cos \frac{n\pi x}{\ell} \,{\text d}x = \frac{1}{h\ell} \,\int_{-h/2}^{h/2} \,\cos \frac{n\pi x}{\ell} \,{\text d}x = \frac{2}{n\pi h} \,\sin \frac{nh\pi}{2\ell} , hc(t) = \begin{cases} 0 , & \ \mbox{ if } 0 < t < \ell , 1 , & \ \mbox{if $k$ is odd}, \\ 0, & \ \mbox{if $k$ is even}. under the terms of the GNU General Public License (GPL). \\ f(x) \sim \frac{a_0}{2} + \sum_{n\ge 1} \left[ a_n \cos \frac{n\pi x}{\ell} + b_n \sin \frac{n\pi x}{\ell} \right] , Transformations: Inverse of a Function. \], Simplify[1/2/L *Integrate[Sign[x]*Exp[-n*I*Pi*x/L], {x, -L, L}]], (I (-1 + Cos[n \[Pi]]) Sign[L])/(n \[Pi]), \[ example. 1 , & \ \mbox{ if } \ell < t < 2\ell ; \end{cases} &=& \frac{1}{2} - \\ 0 , & \ \mbox{ if } \ell < t < 2\ell , \\ -1 , & \ \mbox{ if } -\ell < t < 0 ; \end{cases} &=& \frac{4}{\pi}\, \sum_{n\ge 1} \frac{1}{n}\,\sin^2 \left( \frac{n\pi}{4} \right) \sin \frac{n\pi x}{2\ell} , Calculus: Integral with adjustable bounds. \]. \], \[ \end{eqnarray*}. \int \delta_N (x)\,f(x)\,{\text d} x = \frac{a_0}{2} + a_1 + \cdots + a_N \,\to\, f(0), \qquad\mbox{as } N\to\infty . example. Log InorSign Up. The Square Function. We consider another related to the Heaviside function that is known as the signum function. We expect that the Fourier series will give a periodic expansion shown in the figure. \\ 1/2 , & \ \mbox{ if } t=\ell , \\ \mbox{oddsign}(x) = \begin{cases} \phantom{-}1 , & \ \mbox{ if } 0 < x < \ell , Hc(t) = \begin{cases} 1 , & \ \mbox{ if } -\ell < t < \ell , \tag{1.5} sign[x_,L_]=HeavisideTheta[x/L] - HeavisideTheta[-1 + x/L]; \[ \\ 1. 1, & \ \mbox{ when } -\frac{h}{2} < x < \frac{h}{2} , Return to the Part 6 Partial Differential Equations \], \[ \frac{2}{\pi} \,\sum_{k\ge 0} \frac{1}{2k+1} \,\sin \frac{(2k+1)\pi t}{\ell} . Return to the Part 5 Fourier Series \end{eqnarray*}. \delta_h (x) = \frac{1}{h} \times \begin{cases} \end{align*}, \[ Other common levels for the square wave includes - and . \\ 1/2 , & \ \mbox{ if } t=\ell , \\ As a result, we obtain the periodic version of the signum function that we denote as oddsign; its sine Fourier series is. The square wave, also called a pulse train, or pulse wave, is a periodic waveform consisting of instantaneous transitions between two levels. h[t_] = 1/2 - 2/Pi*Sum[1/(2*k+1)*Sin[(2*k+1)*Pi*t/2], {k,0,25}]; (2 (Cos[(n \[Pi])/2] - Cos[n \[Pi]]))/(n \[Pi]), (2 (-Sin[(n \[Pi])/2] + Sin[n \[Pi]]))/(n \[Pi]), \begin{eqnarray*} \delta_N (x) = \frac{1}{2\ell} \, \frac{e^{{\bf j} (N+ 1/2) \pi x /\ell} - e^{-{\bf j} (N+ 1/2) \pi x /\ell}}{e^{{\bf j}\pi x /\ell} - e^{-{\bf j}\pi x /\ell}} = \frac{1}{2\ell} \, \frac{\sin (N+1/2) x/\ell}{\sin (x/2/\ell)} Calculus: Fundamental Theorem of Calculus a_0 &= \frac{1}{\ell} \int \delta_h (x)\,{\text d}x = \frac{1}{\ell} \,\frac{1}{h} \times h = \frac{1}{\ell} , Hs[t_] = 4/Pi*Sum[1/n*(Sin[n*Pi/4])^2*Sin[n*Pi*t/4], {n,1,50}]; (Cos[n \[Pi]] - Cos[2 n \[Pi]])/(n \[Pi]), (-Sin[n \[Pi]] + Sin[2 n \[Pi]])/(n \[Pi]), \begin{eqnarray*} deltan[x_,M_] = (1/4)*Sin[(M+1/2)*x/2] /Sin[x/4]; \[ S10[x_] = (4/Pi)*Sum[Sin[(2*k - 1)*x]/(2*k - 1), {k, 1, 10}]; \[ Hs(t) = \begin{cases} 1 , & \ \mbox{ if } 0 < t < \ell , Conic Sections: Parabola and Focus Return to the Part 4 Numerical Methods \end{cases} b_k = \frac{2}{\ell} \int_0^{\ell} \mbox{sign}(x) \,\sin \left( \frac{k\pi x}{\ell} \right) {\text d}x = - \frac{4}{k\pi} powered by "x" x "y" y "a" squared a 2 "a" Superscript, "b . However, if you want remove the point valuse on the horizontal axis, you can include plot option Exclusions None.
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