\tag{7.42} \end{equation}, Such a series is known as a cosine Fourier series, and in this case the coefficients $a_n$ can be obtained directly. Fourier Series and Waves Text will be coming soon! Plot the function over a few periods, as well as a few truncations of the Fourier series. endobj The material covered in this chapter is also presented in Boas Chapter 7, Sections 3, 4, 5, 7, 8, and 9. /Filter /FlateDecode But the question arises: why only sine functions for the square and sawtooth waves, and why only cosine functions for the triangle and rectified sine waves? The initial settings of the Sin Wave blocks are set to generate the Fourier series expansion x (t) = 0+ X13 k=1 kodd 4 k sin (2kt) . CUPE 3913 And waves of different forms are easily distinguished by the ear. \tag{7.17} \end{equation}, But $e^{-in\pi} = (e^{-i\pi})^n = (-1)^n$, so that the factor within brackets is $(-1)^n - 1$, which is equal to $0$ when $n$ is even, and to $-2$ when $n$ is odd. It is analogous to a Taylor series, which represents functions as possibly infinite sums of monomial terms. NOTE: Math will not display properly in Safari - please use another browser. Making the substitution in Eq. This series is called the trigonometric Fourier series, or simply the Fourier series, of f (t). 6 0 obj << It is easy to see that the variables are related by $x = 2\pi t$. Workshop Requisition Form Here we see that the coefficients are proportional to $n^{-2}$, and therefore decrease much faster than those of the square wave considered previously. ; \tag{7.15} \end{equation}. Their "Example 1" shows how to derive the Fourier series of a sawtooth wave. Change the script so that it computes and plots the trig. A square wave. A triangle wave. The factor within brackets is $0$ when $n$ is even, and $2$ when $n$ is odd. A matlab function that accept number of harmonics, n as input and produce the Fourier series waveform up to and included n harmonics can be implemented as follow: function fs_tri(N) x . But what we're going to do in this case is we're going to add them. Connect and share knowledge within a single location that is structured and easy to search. /Type /XObject Graduate Schedule of Dates We need more feedback from him. . period of the square wave and triangular wave from Figure 10.2 is 2. The function is plotted in Fig. We look at a spike, a step function, and a rampand smoother functions too. This is why they are not practical: you need an awful lot of terms in the series to get a good approximation. Finally, verify that in the case of a sawtooth wave, $f(x)$ is an odd function. /Creator (LaTeX with hyperref package) What is this political cartoon by Bob Moran titled "Amnesty" about? /D [2 0 R /Fit] Why does a Triangular Fourier series converge faster than a Square Fourier series? 3 0 obj << As we shall see in a moment, Fourier series do better when $f(x)$ is continuous. Use of properties table including integration, linearity, and time shifting is explained, also how to take the coefficient of a dc off set is also analyzed. Plot the function over a few periods, as well as a few truncations of the Fourier series. The coefficients may be determined rather easily by the use of Table 1. f (t) f ( t) 2/ 0 f (t)dt, 0 0 2 / f ( t) d t, 0. For $c_0$ we have, \begin{equation} c_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x)\, dx = \frac{1}{2\pi} \int_{-\pi}^0 (-x)\, dx + \frac{1}{2\pi} \int_0^{\pi} (x)\, dx = \frac{\pi}{2}. This is not the same as computing Fourier Series. It can be advantageous to re-express the Fourier series of Eq. This infinite Fourier series converges to the triangle . The coefficients alternate in sign and decrease as $n^{-1}$ with increasing $n$. 4.12 { }{ } >> function can be approximated arbitrarily well by its Nth-order Fourier series approximation. This advances the waveform by T 0 / 4 s. In calculating the final plot, subtract T 0 / 4 from . 4 0 obj << This answers the important question: which mixture of sine and cosine waves forms the function $f(x)$? This implies that a smaller number of terms will be required of the Fourier series to give an adequate representation of the triangle wave. )^infty((-1)^((n-1)/2))/(n^2)sin((npix)/L), (4) which can be summed to yield the. Replace first 7 lines of one file with content of another file. (Boas Chapter 7, Section 8, Problem 15b) Find the Fourier series for the function $f(x)$ defined by $f = 1+2x$ for $-1 \leq x < 0$ and $f = 1-2x$ for $0 \leq x < 1$. Let the function be -periodic and suppose that it is presented by the Fourier series: Calculate the coefficients and Solution. /FormType 1 endobj Your response is private Was this worth your time? This means that f ( t) = f ( t) ODD f ( t) = f ( t) REAL therefore, c n = c n c n = c n IMAGINARY Fourier series approximation of a triangle wave Figure 6.3. RwN$e[5R4x? TriangleWave[x] gives a triangle wave that varies between -1 and +1 with unit period. Equation (7.14) allows us to obtain $a_n$ and $b_n$ from $c_n$ and $c_{-n} = +i/(\pi n)$. They are pretty simple series, though they are still infinite series. That's exactly what is given. Another Fourier series recipe for a triangle wave is also all of the odd harmonics. /Filter /FlateDecode Suppose that $f(x)$ is an even function of $x$. Mathematically, the triangle function can be written as: [Equation 1] We'll give two methods of determining the Fourier Transform of the triangle function. I just can't seem to figure out how to code the step function in a way that I can apply np.fft.fft() My latest (poor) attempt: By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. Trigonometric Fourier Series Any periodic waveform can be approximated by a DC component (which may be 0) and the sum of the fundamental and harmomic sinusoidal waveforms. 1, the Fourier series representation for the triangle wave is under the Fig.1 below. Video created by for the course " 1 ". endobj The function is periodic with period $2\pi$. The frequency of each wave in the sum, or harmonic, is an integer multiple of the periodic function's fundamental frequency. Waves of the same (dominant) frequency but of different forms are shown in Fig.7.1. Common periodic signals include the square wave, pulse train, and triangle wave. >> Does a beard adversely affect playing the violin or viola? endstream Thus, the (complex) Fourier coefficients are obtained by evaluating the integrals of Eq.(7.13). The function is plotted in Fig.7.3. This has important applications in many applications of electronics but is particularly crucial for signal processing and communications. (7.35), in which we set $x_0 = -L$. A Fourier series ( / frie, - ir / [1]) is a sum that represents a periodic function as a sum of sine and cosine waves. As a third example we examine the rectified sine wave described by, \begin{equation} f(x) = |\sin x|; \tag{7.27} \end{equation}. rev2022.11.7.43014. (7.14) is unaffected by the rescaling, so that we still have, \begin{equation} a_n = c_n + c_{-n}, \qquad b_n = i (c_n - c_{-n}). apply to documents without the need to be rewritten? 50 Stone Road E. Thanks for contributing an answer to Mathematics Stack Exchange! The rescaling implies that the Fourier coefficients are now given by, \begin{equation} c_n = \frac{1}{2L} \int_0^{2L} f(x) e^{-in\pi x/L}\, dx = \frac{1}{2L} \int_{x_0}^{x_0 + 2L} f(x) e^{-in\pi x/L}\, dx. Combining Eqs. I am generating a 100hz Triangle signal using the following code: Now how should i go about deriving the Fourier series of this signal, i am completely lost. Substituting black beans for ground beef in a meat pie. Each harmonic is going to have an amplitude that is 1 over n squared. As another example, let us calculate $c_1$. in the second expression we indicate that the integral can be evaluated for any interval of length $2\pi$. This time we multiply Eq. (clarification of a documentary). (7.14) and (7.30), we get that $b_n = 0$ and, \begin{equation} a_n = -\frac{4}{\pi(n^2-1)} \tag{7.31} \end{equation}, when $n$ is even. endobj Verify that the general result for $a_n$ applies to $a_0$. ds"^T x Lc0ue"S|^yp\(O%gD6q<5F9>7=OZ5#>ih/fM 18 0 obj << \tag{7.29} \end{align}, Both numerators are equal to $-2$ when $n$ is even, and to $0$ when $n$ is odd. You will have noticed that we changed our notation with respect to Sec.7.1. Apart from the constant $\pi/2$, the series involves cosine functions only. The series involves sine functions only. /ProcSet [ /PDF /ImageB ] a) Sketch the function on the interval $-3\pi \leq x \leq 3\pi$. Three different truncations are shown in Fig.7.5. Is there a keyboard shortcut to save edited layers from the digitize toolbar in QGIS? Typeset a chain of fiber bundles with a known largest total space. /BBox [0 0 612 449] Share. As Fig.7.4 shows, the Fourier series truncated after a small number of terms gives an excellent representation of the rectified sine wave. 2 Sawtooth Waveform For example, $c_0$ evaluated at $x + 2\pi$ is still $c_0$, while $\sin[n(x+2\pi)] = \sin(nx + 2n\pi) = \sin(nx) \cos(2n\pi) + \cos(nx) \sin(2n\pi) = \sin(nx)$, since $\cos(2n\pi) = 1$ and $\sin(2n\pi) = 0$. 2. Find the Fourier series for the odd function $f(x)$ defined by $f = \sinh x$ for $-\pi \leq x < \pi$. for the Fourier coefficients. The identity, in fact, applies to any interval of length $2\pi$; the integral could go from $x_0$ to $x_0 + 2\pi$, with $x_0$ completely arbitrary. (Boas Chapter 7, Section 5, Problem 3) Find the Fourier series for the function $f(x)$ defined by $f = 0$ for $-\pi \leq x < \pi/2$ and $f = 1$ for $\pi/2 \leq x < \pi$. Is it possible for a gas fired boiler to consume more energy when heating intermitently versus having heating at all times? Space - falling faster than light? Why are UK Prime Ministers educated at Oxford, not Cambridge? As stated, its Fourier series can admit cosine waves only, and it must take the form, \begin{equation} f(x) = c_0 + \sum_{n=1} a_n \cos \Bigl( \frac{n\pi x}{L} \Bigr). Your triangle wave is an odd function: it satisfies ##s(t) = -s(-t)## for all ##t##. Notice that $c_0$ cannot be obtained directly from this relation, because the manipulations required to evaluate the integral for $c_n$ do not apply when $n=0$. We therefore arrive at, \begin{equation} b_n = \frac{2}{L} \int_0^L f(x) \sin\Bigl( \frac{n\pi x}{L} \Bigr)\, dx \tag{7.37} \end{equation}. \[f(x) = x^2, \qquad -\pi \leq x \leq \pi,\]. The plot in black color shows how the reconstructed (Fourier Synthesis) signal will look like if the three terms are combined together. As we shall see in a moment, Fourier series do better when f(x) is continuous. The exact mixture of waves determines the precise shape of the total wave; different mixtures produce waves of different forms. Fourier Theory and Some Audio Signals. (Boas Chapter 7, Section 8, Problem 18) Find the Fourier seriesfor the function $f(x)$ defined by $f = x^2$ for $0 \leq x < 10$. /Keywords () (Boas Chapter 7, Section 5, Problem 11) Find the Fourier series for the function $f(x)$ defined by $f = 0$ for $-\pi \leq x < 0$ and $f = \sin x$ for $0 \leq x < \pi$. In the introduction we used $t$ as the independent variable, and took the wave $f(t)$ to be periodic with period $1$. 2 0 obj << /Resources 4 0 R As a test, draw the circle $e^{i\phi}$ in the complex plane, with $\phi$ the angle from the real axis. I don't understand the use of diodes in this diagram. Stack Exchange network consists of 182 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. /Length 295 In this video, Fourier series analysis and synthesis using coefficients of Periodic Triangle Wave, Periodic Square Wave, and Periodic Impulse train is derive. Plot the function over a few periods, as well as a few truncations of the Fourier series. the function is automatically periodic with period $2\pi$. when $n$ is odd. Overview. Fourier cosine series of a triangle wave function.Join me on Coursera: https://www.coursera.org/learn/differential-equations-engineersLecture notes at http:/. Would a bicycle pump work underwater, with its air-input being above water? Can FOSS software licenses (e.g. Overview of a simple guitar amplifier, power supply, volume . Guelph, Ontario, Canada Functions. \tag{7.20} \end{equation}. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. \tag{7.36} \end{equation}. This time the Fourier series can only admit sine waves, so that, \begin{equation} f(x) = \sum_{n=1} b_n \sin \Bigl( \frac{n\pi x}{L} \Bigr). which did not match with previous.My question is why they took the interval from - to + instead of 0 to 2. Consider the following function which is periodic but always linearly increasing (this is sometimes called the sawtooth wave): The equation describing this curve is \[ \begin{aligned} x(t) = 2A\frac{t}{\tau},\ -\frac{\tau}{2} \leq t \frac{\tau}{2} \end{aligned} \] Plot the function over a few periods, as well as a few truncations of the Fourier series. We will learn to become familiar with these steps in the examples that follow. A typical task of Fourier analysis is to obtain the coefficients $c_n$ (or $a_n$ and $b_n$) when $f(x)$ is known. 0. (Boas Chapter 7, Section 8, Problem 15c) Find the Fourier series for the function $f(x)$ defined by $f = x+x^2$ for $-1 \leq x < 0$ and $f = x-x^2$ for $0 \leq x < 1$. a triangle wave is continuous Quote from here: - The triangle wave has no discontinuous jumps, but the slope changes discontinuously twice per cycle Having the slope change discontinuously also means an infinite range of sinusoidal components. Also, you can integrate over any period of the function, so instead of integrating from 0 to 4, you can integrate from -2 to 2, which will allow you to exploit the oddness of the function. It only takes a minute to sign up. \tag{7.45} \end{equation}, Such a series is known as a sine Fourier series, and the coefficients $b_n$ can be calculated directly as follows. We have, \begin{align} a_n &= c_n + c_{-n} \nonumber \\ &= \frac{1}{2L} \int_{-L}^L f(x) \Bigl( e^{-in\pi x/L} + e^{in\pi x/L} \Bigr)\, dx \nonumber \\ &= \frac{1}{L} \int_{-L}^L f(x) \cos\Bigl( \frac{n\pi x}{L} \Bigr)\, dx, \tag{7.43} \end{align}, and notice that the integrand is an even function of $x$, since it is the product of two even functions. Conic Sections: Parabola and Focus. Exercise 7.6: Reproduce the steps leading to Eq. Plot the function over a few periods, as well as a few truncations of the Fourier series. Find the Fourier series for the square -periodic wave defined on the interval Example 1. /Length 1359 As a second example we examine a triangle wave described by, \begin{equation} f(x) = \left\{ \begin{array}{ll} -x & \quad -\pi \leq x < 0 \\ x & \quad 0 \leq x < \pi \end{array} \right. I am using matlab to study digital signalling and have come across a problem which i was wondering if anyone with more experience could help me with. Since the function is odd , Now consider the asymmetric triangle wave pinned an -distance which is ( )th of the distance . The triangular wave is the even 2-periodic function dened on x by the formula twave(x) = ( x 0 <x ; + x x 0: Theorem. axios authentication; uab neurosurgery fax number dmv select virginia dmv select virginia c) From your result in part (b), determine the exact value of the sum $\sum_{n=1}^\infty n^{-2}$. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. A simple rescaling of $x$ can accommodate any other period, and we shall explore this freedom systematically in Sec.7.7. Exercise 2.10: Fourier series of a triangle wave Consider the Fourier sine series approximation for the triangle wave depicted in Figure 2.32. f M (x)= n=1M ansin(nx) x [0,1] (a) Find the coefficients an,n= 1,2,. \tag{7.35} \end{equation}, The relation of Eq. When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. The function is plotted in Fig.7.2. (c) Plot the Fourier transform. xAo fLx`cfvVK;l;hm ,kM3@)m.LxQn .0-tL.M>lnUd93T-"C\p@}g jr_+w1Q_(46(1)JiDq b)uLM| +7;+)$#[0JyYm4YK+F f984"2*E9Y^YY k}mql96` ;MeL?K Why do a violin and a flute playing the same note sound so different? \tag{7.40} \end{equation}, The Fourier representation of the sawtooth wave is therefore, \begin{equation} f(x) = \frac{2L}{\pi} \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \sin\Bigl( \frac{n\pi x}{L} \Bigr). No examples provided. Understanding the Plots: In the first plot, the original square wave (red color) is decomposed into first three terms ( n=3) of the Fourier Series. What is the function of Intel's Total Memory Encryption (TME)? In the case of a triangle wave, continuity of the function allows the cosines to do a very good job of synthesizing the function. \tag{7.33} \end{equation}. stream Exercise 7.4: You should have done this before, but here's another chance: verify Eq. (7.9) by $e^{-ix}$ and integrate, to get, \begin{align} \int_0^{2\pi} f e^{-ix}\, dx &= c_0 \int_0^{2\pi} e^{-ix}\, dx + c_1 \int_0^{2\pi}\, dx + c_2 \int_0^{2\pi} e^{ix}\, dx + \cdots \nonumber \\ & \quad \text{} + c_{-1} \int_0^{2\pi} e^{-2ix}\, dx + c_{-2} \int_0^{2\pi} e^{-3ix}\, dx + \cdots. The factor of $2$ in front of $L$ is conventional and introduced for convenience. Consider the sawtooth wave f(x)=t, 0 < t < 0.5 f(x)= 1-t, 0.5 < t < 1 (a) Define this function using code. (7.30), and verify that $c_1 = c_{-1} = 0$. << This table shows the Fourier series analysis . MathJax reference. It is a periodic, piecewise linear, continuous real function. >> Here we let $x$ be the independent variable, and take $f(x)$ to be periodic with period $2\pi$. The choice $x_0 = -\pi$ is sometimes convenient. 1 0 obj The answer is that the sound waves generated by different instruments come with different forms, even when they come with the same (dominant) frequency. /PTEX.FileName (/usr/share/texmf-texlive/tex/latex/pdfscreen/overlay3.pdf) The function is periodic with period $2$. More precisely, we have the following result for the convergence of the Fourier series, stated without proof. We see that as in the case of the square wave in Sec.7.4, the Fourier series has difficulties reproducing the discontinuities of the sawtooth function. Exercise 7.12: Prove that $b_n = i(c_n - c_{-n}) = 0$ when $f(x)$ is an even function of $x$. So please do take the time to refresh your knowledge and understanding. The function is periodic with period $2\pi$. Why does sending via a UdpClient cause subsequent receiving to fail? >> endobj /PTEX.PageNumber 1 Shouldn't these waves make the same impression on our ears and be heard as identical sounds? Employee Portal @Phonon This is why I asked him if what he really wants is to compute the FFT of the signal, and not really 'deriving the fourier series'. A violin and a flute sound different because they produce different wave forms. The difference is that Fourier Series comes from continuous Fourier Transform, while FFT comes from discrete Fourier Transform. Typeset a chain of fiber bundles with a known largest total space. To learn more, see our tips on writing great answers. rev2022.11.7.43014. 1-519-824-4120 x 52261 Vote. (7.3), and reproduce the steps that lead to Eq.(7.7). % Fourier series representation, and plot its amplitude spectrum and phase. a) Find the Fourier series of this function. I was able to calculate the coefficient values and they are in vector form but I can't transform this into my output signal and plot it. Reviews (5) Discussions (3) % The user can design various sawtooth wave by determining its period, % time shift, dc value, etc. Did Twitter Charge $15,000 For Account Verification? (7.13) with $x_0 = -\pi$ to calculate the Fourier coefficients. Also included are a few examples that show, in a very basic way, a couple of applications of Fourier Theory, thought the number of applications and the ways that Fourier Theory is used are many. The other waves have frequencies $f = 2$, $f=3$, and $f = 4$, so that they oscillate twice, three times, and four times as fast as the dominant wave, respectively; these are the harmonics, or overtones, of the fundamental frequency. Any periodic function $f(t)$ can be thought of as being synthesized from a (potentially infinite) number of elementary sine and cosine waves. Graduate Calendar Let's investigate this question graphically. That's why I would be more careful. Fourier series for a non-periodic function on an Interval, Fourier Series Exponential Representation, Deriving Fourier series rigorously and other small questions. Course Outlines (7.2) into Eq. And it is also fun to use Spiral Artist and see how circles make waves. Poster Boards (7.7), which we rewrite as, \begin{equation} f = c_0 + c_1 e^{ix} + c_2 e^{2ix} + \cdots + c_{-1} e^{-ix} + c_{-2} e^{-2ix} + \cdots. Fourier Series--Triangle Wave Fourier Series--Triangle Wave Consider a triangle wave of length . The function is periodic with period $2\pi$. Method 1. In this example, you are asked to find the Fourier series for the given periodic voltage shown below . Therefore, When then When then Since the function is even, the Fourier coefficients are zero. (7.10) in the more general form \[ \int_{x_0}^{x_0 + 2\pi} e^{nix}\, dx = 2\pi \delta_{n0}. Recall from Sec.7.3 that only the length of the interval matters, and that the choice of starting point is arbitrary. But in a typical application of Fourier series the period may not be $2\pi$, and we should generalize our formulation to handle such cases. /Producer (dvips + Distiller) For $c_0$ we get, \begin{equation} c_0 = \frac{1}{2L} \int_{-L}^L f(x)\, dx = \frac{1}{2L} \int_{-L}^L x\, dx = \frac{x^2}{4L} \biggr|^L_{-L} = 0. Exercise 7.7: Verify that Eq. The following example explains how to use Equations 3-4 to calculate the Fourier coefficients for a specific periodic function. How Fourier transform is derived from Fourier series? The original coefficients $a_n$ and $b_n$ are given by, \begin{equation} a_0 = 2 c_0, \qquad a_n = c_n + c_{-n}, \qquad b_n = i ( c_n - c_{-n} ) \tag{7.8} \end{equation}. (4.38) --- and Sec.5.6 --- see Eq.(5.35). FOURIER SERIES AND INTEGRALS 4.1 FOURIER SERIES FOR PERIODIC FUNCTIONS This section explains three Fourier series: sines, cosines, and exponentials eikx. Plot the function over a few periods, as well as a few truncations of the Fourier series. Stack Overflow for Teams is moving to its own domain! the definition is extended to $x<0$ and $x \geq 2\pi$ according to the periodicity requirement $f(x+2\pi) = f(x)$. >> MathJax reference. (7.14) to obtain $a_n$ and $b_n$ as integrals involving $f(x)$ and ordinary trigonometric functions. (7.8), \begin{equation} a_n = c_n + c_{-n}, \qquad b_n = i ( c_n - c_{-n} ), \tag{7.14} \end{equation}. The reason is that when $f(x)$ is an even function of $x$, so that $f(-x) = f(x)$, its Fourier series can contain only cosine waves (which are even) and must exclude sine waves (which are odd). Space - falling faster than light? Visualization of Fourier series expansion of Triangular wave (MATLAB) . \tag{7.2} \end{equation}, \begin{equation} \cos\phi = \frac{1}{2}\bigl( e^{i\phi} + e^{-i\phi} \bigr), \qquad \sin\phi = \frac{1}{2i} \bigl( e^{i\phi} - e^{-i\phi} \bigr), \tag{7.3} \end{equation}, and substitution into Eq. Are witnesses allowed to give private testimonies? \tag{7.41} \end{equation}. We choose $(0,2\pi)$ as the reference interval, and note that while $f(x) = \sin x$ when $0 \leq x < \pi$, $f(x) = -\sin x$ when $\pi \leq x < 2\pi$. A sawtooth wave represented by a successively larger sum of trigonometric terms (7.7) brings us back to Eq.(7.1). Fourier Series Representation of Continuous Time Periodic Signals A signal is said to be periodic if it satisfies the condition x (t) = x (t + T) or x (n) = x (n + N). Is it possible to make a high-side PNP switch circuit active-low with less than 3 BJTs? can be exploited to find $a_n$ and $b_n$, and the real form of the Fourier transform can be constructed from Eq.(7.1). Making statements based on opinion; back them up with references or personal experience. This is the complex form of the Fourier series, which contains, in an attractive and economical package, the same information as Eq.(7.1). University of Guelph Example #1: triangle wave Here, we compute the Fourier series coefcients for the triangle wave plotted in Figure 1 below. Remembering the fact that we introduced a factor of i (and including a factor of 2 that just crops up . Where N is the total number of Fourier coefficients used for approximation. Cn=- (A To/ ( (n pi)^2)) ( (Sin ( (n pi)/2))^2) Hint: double differentiate your signal till you end up with dirac delta functions, they are easy to modify. The Fourier coefficients are obtained from Eq. Use MathJax to format equations. The best answers are voted up and rise to the top, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Without seeing your actual solution, the result should be the same. % 7.4, 7.5, 7.6, and 7.8 involved either sine waves or cosine waves, but not both. Exercise 7.3: Verify Eq. \tag{7.39} \end{align}, From this and Eq. Find the Fourier series for the triangle wave defined on the interval Solution. This choice of period is convenient, because the elementary waves making up the Fourier series are of the simple form $\sin(nx)$ and $\cos(nx)$. In each case the wave is synthesized from four simple waves of the form $\sin(2\pi f t)$, in which $f$ is the frequency. \tag{7.9} \end{equation}, To extract the coefficients $c_n$ we rely on the identity, \begin{equation} \int_0^{2\pi} e^{nix}\, dx = 2\pi \delta_{n0}, \tag{7.10} \end{equation}. (7.13) into Eq. An electrocardiogram (ECG) signal. c) On the same graph, plot $f(x)$ together with its Fourier series truncated after two terms. (Boas Chapter 7, Section 5, Problem 2) Find the Fourier series for the function $f(x)$ defined by $f = 0$ for $-\pi \leq x < 0$, $f = 1$ for $0 \leq x < \pi/2$, and $f = 0$ for $\pi/2 \leq x < \pi$. \begin{equation} c_0 = \frac{1}{2\pi} \int_0^{2\pi} f(x)\, dx = \frac{1}{2\pi} \int_0^\pi \sin x\, dx - \frac{1}{2\pi} \int_\pi^{2\pi} \sin x\, dx = \frac{2}{\pi} \tag{7.28} \end{equation}, \begin{align} c_n &= \frac{1}{2\pi} \int_0^{2\pi} f(x) e^{-inx}\, dx \nonumber \\ &= \frac{1}{2\pi} \int_0^\pi \sin x\, e^{-inx}\, dx - \frac{1}{2\pi} \int_\pi^{2\pi} \sin x\, e^{-inx}\, dx \nonumber \\ &= \frac{1}{4\pi i} \int_0^\pi (e^{ix} - e^{-ix}) e^{-inx}\, dx - \frac{1}{4\pi i} \int_\pi^{2\pi} (e^{ix} - e^{-ix}) e^{-inx}\, dx \nonumber \\ &= \frac{e^{-i(n-1)x}}{4\pi(n-1)} \biggr|^\pi_0 - \frac{e^{-i(n+1)x}}{4\pi(n+1)} \biggr|^\pi_0 - \frac{e^{-i(n-1)x}}{4\pi(n-1)} \biggr|^{2\pi}_\pi + \frac{e^{-i(n+1)x}}{4\pi(n+1)} \biggr|^{2\pi}_\pi \nonumber \\ &= \frac{(-1)^{n-1} - 1}{4\pi(n-1)} - \frac{(-1)^{n+1} - 1}{4\pi(n+1)} - \frac{1 - (-1)^{n-1}}{4\pi(n-1)} + \frac{1 - (-1)^{n+1}}{4\pi(n+1)} \nonumber \\ &= \frac{(-1)^{n-1} - 1}{2\pi(n-1)} - \frac{(-1)^{n+1} - 1}{2\pi(n+1)}.
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