Fourier Series Symmetry

11 Key excerpts on "Fourier Series Symmetry"

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  Linear Networks And Systems: Algorithms And Computer-aided Implementations (In 2 Volumes) (2nd Edition)

  (In 2 Volumes)

  • Wai-kai Chen(Author)
  • 1990(Publication Date)
  • World Scientific

    (Publisher)

  We remark that in plotting the ampli-tude and phase spectra, we convert (8.54) to the form of (8.30). In doing so we used the relation sin nco 0 t = cos (nco 0 t — 7i/2). Observe that in (8.54) all the cosine terms are missing. We next show that this is a direct consequence of certain symmetries in the signal. We will make use of this information in simplifying the computation of the Fourier coefficients, thus avoiding the needless work of evaluating integrals that result in zero values. 8-4 SYMMETRY PROPERTIES OF FOURIER SERIES In this section we show that if a signal possesses certain identifiable wave-form properties, these properties may be used to simplify the evaluation of the Fourier coefficients. A function /(t) is said to be even if it satisfies the condition fi-t) = f{t) (8.55) 338 Fourier Series and Signal Spectra Examples of the even functions are the familiar functions cos no) 0 t and the constant a 0 . Other examples are sin 2 t, t 2w , | sin t|, and e~'*'. Geometrically a function is even if its graph is symmetric in the vertical axis. For instance, the waveforms of Figures 8.1(a) and (c) are even. On the other hand, a function f{t) is said to be odd if it satisfies the condition / ( -« = -fit) (8.56) Examples of the odd functions are sin nco 0 t, t, and tan -1 t. Geometrically, a function is odd if its graph is symmetric in the origin. The rectangular-wave voltage of Figure 8.10 is odd because it is symmetric with respect to the origin. In the following we shall investigate in detail what effect the sym-metry of f(t) has on the evaluation of the Fourier coefficients. Before we do this, however, we recognize the following relationships: Even function x odd function = odd function Even function x even function = even function Odd function x odd function = even function For any even function f e (t), £ o / # (t)dt = 2 ^hWdt (8.57) For any odd function f 0 {t), P° / 0 (t) dt == 0 (8.58) J-to Equations (8.57) and (8.58) hold for any t 0 .

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  Dorf's Introduction to Electric Circuits

  • Richard C. Dorf, James A. Svoboda(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  Table 15.3-1 Fourier Series and Symmetry SYMMETRY FOURIER COEFFICIENTS 1. Odd function f t f t a n 0 for all n b n 4 T T 2 0 f t sin no 0 t dt 2. Even function f t f t b n 0 for all n a n 4 T T 2 0 f t cos no 0 t dt 3. Half-wave symmetry f t f t T 2 a 0 0 a n 0 for even n b n 0 for even n a n 4 T T 2 0 f t cos no 0 t dt for odd n b n 4 T T 2 0 f t sin no 0 t dt for odd n 4. Quarter-wave symmetry. (Half-wave symmetry and symmetry about the midpoints of the positive and negative half cycles) A. Odd function: a 0 0 a n 0 for all n b n 0 for even n b n 8 T T 4 0 f t sin no 0 t dt for odd n B. Even function: a 0 0 b n 0 for all n a n 0 for even n a n 8 T T 4 0 f t cos no 0 t dt for odd n Symmetry of the Function f ( t ) 735 E X A M P L E 1 5 . 3 - 1 Symmetry and the Fourier Series Determine the Fourier series for the triangular waveform v(t) shown in Figure 15.3-2. Solution Step 1: From Figure 15.3-2, we see that the period of v o (t) is T p 4 p 4 p 2 s The fundamental frequency is o 0 2p T 4 rad s Step 2: If we don’t take advantage of the symmetry of the triangle waveform, determining the Fourier coefficients a 0 , a n , and b n will require integration over a full period—either from 0 to T or from T 2 to T 2. Accordingly, we can represent v(t) from time T 2 to T, that is, from p 8 to p 2 seconds. By writing equations for the various straight-line segments that comprise the triangle waveform, we can represent v(t) as v t 32 p t 8 when 3p 8 t p 8 32 p t when p 8 t p 8 32 p t 8 when p 8 t 3p 8 32 p t 16 when 3p 8 t 5p 8 If we take advantage of symmetry, we will need to integrate only from 0 to T 2, that is, from 0 to p 8 seconds. If we need to represent v(t) only from 0 to p 8 seconds, we don’t have to write equations for so many straight-line segments. In this case, we need to write the equation only for one straight line to represent v(t) as v t 32 p t when p 8 t p 8 Step 3: Next, we will determine the Fourier coefficients a 0 , a n , and b n .

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  The Analysis and Design of Linear Circuits

  • Roland E. Thomas, Albert J. Rosa, Gregory J. Toussaint(Authors)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  It is helpful to recognize these symmetries, since they may simplify the calculation of the Fourier coefficients. The first expression in Eq. (13–3) shows that the amplitude of the dc compo- nent a 0 is the average value of the periodic waveform f t . If the waveform has equal area above and below the time axis, then the integral over one cycle vanishes, the average value is zero, and a 0 = 0. The square wave, triangular wave, and parabolic wave in Figure 13–4 are examples of periodic waveforms with zero average value. A waveform is said to have even symmetry if f − t = f t . The cosine wave, rec- tangular pulse, and triangular wave in Figure 13–4 are examples of waveforms with even symmetry. The Fourier series of an even waveform is made up entirely of cosine terms: that is, all of the b n coefficients are zero. To show this, we write the Fourier series for f t in the form f t = a 0 + ∞ n = 1 a n cos 2πnf 0 t + b n sin 2πnf 0 t (13 –10) Given the Fourier series for f t , we use the identities cos − x = cos x and sin − x = − sin x to write the Fourier series for f − t as follows: f − t = a 0 + ∞ n = 1 a n cos 2πnf 0 t − b n sin 2πnf 0 t (13 –11) For even symmetry f t = f − t and the right sides of Eqs. (13–10) and (13–11) must be equal. Comparing the Fourier coefficients term by term, we find that f t = f − t requires b n = − b n . The only way this can happen is for b n = 0 for all n. A waveform is said to have odd symmetry if − f − t = f t . The sine wave, square wave, and parabolic wave in Figure 13–4 are examples of waveforms with this type of symmetry. The Fourier series of odd waveforms are made up entirely of sine terms: that is, all of the a n coefficients are zero.

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  Signals and Systems Laboratory with MATLAB

  • Alex Palamides, Anastasia Veloni(Authors)
  • 2010(Publication Date)
  • CRC Press

    (Publisher)

  Regarding the coef fi cients a k of the complex exponential Fourier series, when the signal x ( t ) is even, the relationship a k ¼ a k stands, namely, the coef fi cients also have even symmetry. In order to demonstrate this property, the fi rst 11 coef fi cients a k of the signal that in one period is de fi ned as x ( t ) ¼ t 2 , 2 t 2 are computed and plotted. It is well known that x ( t ) ¼ t 2 is a signal with even symmetry. Commands Results t0 ¼ 2; T ¼ 4; w ¼ 2*pi = T; syms t x ¼ t ^ 2; k ¼ 5:5; a ¼ (1 = T)*int(x*exp( j*k*w*t),t,t0,t0 þ T); stem(k,eval(a)) legend( ' a_k ' ) −5 −4 −3 −2 −1 0 1 2 3 4 5 −1 −0.5 0 0.5 1 1.5 a k From the above graph, it is clear that the complex exponential Fourier series coef fi cients of an even signal have also even symmetry, namely, a k ¼ a k . 5.10.2 Odd Symmetry Recall that when a signal x ( t ) is an odd function of t , the relationship x ( t ) ¼ x ( t ) stands. In this case, the trigonometric Fourier series coef fi cients b k are zero. In order to demonstrate this property, the same procedure used in the previous section is followed. The coef fi cients b k of an odd signal are computed for one period time, i.e., T = 2 t T = 2 in two parts. 278 Signals and Systems Laboratory with MATLAB 1 First, the coef fi cients b k are computed for T = 2 t 0; that is, b k are computed for the signal x ( t ). However, because of the even symmetry x ( t ) ¼ x ( t ). Thus, Commands Results syms x t k T b1 ¼ sin(k*pi)*x = k = pi w ¼ 2*pi = T; b1 ¼ (2 = T)*int( x*cos(k*w*t),t, T = 2,0) Next, the coef fi cients b k are computed for 0 t T = 2, and in order to calculate the coef fi cients b k for the signal x ( t ), T = 2 t T = 2 the two parts are added. Commands Results b2 ¼ (2 = T)*int(x*cos(k*w*t),t,0,T = 2) b2 ¼ sin(k*pi)*x = k = pi b ¼ b1 þ b2 b ¼ 0 Indeed, the trigonometric Fourier series coef fi cients b k of a signal with odd symmetry are zero.

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  Introduction to Electric Circuits

  • Richard C. Dorf, James A. Svoboda(Authors)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  Answer: a 0 ¼ 0, a 1 ¼ A, a n ¼ 0 for n > 1, and b n ¼ 0 15.3 S y m m e t r y o f t h e F u n c t i o n f ( t ) Four types of symmetry can be readily recognized and then used to simplify the task of calculating the Fourier coefficients. They are the following: 1. Even-function symmetry. 2. Odd-function symmetry. 3. Half-wave symmetry. 4. Quarter-wave symmetry. A function is even when f t ðÞ¼ f t ð Þ, and a function is odd when f t ðÞ¼f t ð Þ. The function shown in Figure 15.2-2 is an even function. For even functions, all b n ¼ 0 and a n ¼ 4 T Z T =2 0 f t ðÞ cos no 0 t dt For odd functions, all a n ¼ 0 and b n ¼ 4 T Z T =2 0 f t ðÞ sin no 0 t dt An example of an odd function is sin o 0 t . Another odd function is shown in Figure 15.3-1. Half-wave symmetry for a function f (t) is obtained when f t ðÞ¼f t  T 2   ð15:3-1Þ In these half-wave symmetric waveforms, the second half of each period looks like the first half turned upside down. The function shown in Figure 15.3-2 has half-wave symmetry. If a function has half-wave 722 15. Fourier Series and Fourier Transform symmetry, then both a n and b n are zero for even values of n. We see that a 0 ¼ 0 for half-wave symmetry because the average value of the function over one period is zero. Quarter-wave symmetry describes a function that has half-wave symmetry and, in addition, has symmetry about the midpoint of the positive and negative half-cycles. An example of an odd function with quarter-wave symmetry is shown in Figure 15.3-1. If a function is odd and has quarter-wave symmetry, then a 0 ¼ 0; a n ¼ 0 for all n, b n ¼ 0 for even n. For odd n, b n is given by b n ¼ 8 T Z T =4 0 f t ðÞ sin no 0 t dt If a function is even and has quarter-wave symmetry, then a 0 ¼ 0; b n ¼ 0 for all n, and a n ¼ 0 for even n. For odd n, a n is given by a n ¼ 8 T Z T =4 0 f t ðÞ cos no 0 t dt The calculation of the Fourier coefficients and the associated effects of symmetry of the waveform f (t) are summarized in Table 15.3-1.

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  Introduction to Electric Circuits

  • James A. Svoboda, Richard C. Dorf(Authors)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  Answer: a 0 ¼ 0, a 1 ¼ A, a n ¼ 0 for n > 1, and b n ¼ 0 15.3 S y m m e t r y o f t h e F u n c t i o n f ( t ) Four types of symmetry can be readily recognized and then used to simplify the task of calculating the Fourier coefficients. They are the following: 1. Even-function symmetry. 2. Odd-function symmetry. 3. Half-wave symmetry. 4. Quarter-wave symmetry. A function is even when f t ð Þ ¼ f t ð Þ, and a function is odd when f t ð Þ ¼ f t ð Þ. The function shown in Figure 15.2-2 is an even function. For even functions, all b n ¼ 0 and a n ¼ 4 T Z T =2 0 f t ð Þ cos no 0 t dt For odd functions, all a n ¼ 0 and b n ¼ 4 T Z T =2 0 f t ð Þ sin no 0 t dt An example of an odd function is sin o 0 t . Another odd function is shown in Figure 15.3-1. Half-wave symmetry for a function f (t) is obtained when f t ð Þ ¼ f t  T 2   ð15:3-1Þ In these half-wave symmetric waveforms, the second half of each period looks like the first half turned upside down. The function shown in Figure 15.3-2 has half-wave symmetry. If a function has half-wave 750 15. Fourier Series and Fourier Transform symmetry, then both a n and b n are zero for even values of n. We see that a 0 ¼ 0 for half-wave symmetry because the average value of the function over one period is zero. Quarter-wave symmetry describes a function that has half-wave symmetry and, in addition, has symmetry about the midpoint of the positive and negative half-cycles. An example of an odd function with quarter-wave symmetry is shown in Figure 15.3-1. If a function is odd and has quarter-wave symmetry, then a 0 ¼ 0; a n ¼ 0 for all n, b n ¼ 0 for even n. For odd n, b n is given by b n ¼ 8 T Z T =4 0 f t ð Þ sin no 0 t dt If a function is even and has quarter-wave symmetry, then a 0 ¼ 0; b n ¼ 0 for all n, and a n ¼ 0 for even n.

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  Advanced Engineering Mathematics and Analysis

  • (Author)
  • 2014(Publication Date)
  • Orange Apple

    (Publisher)

  ________________________ WORLD TECHNOLOGIES ________________________ Chapter- 2 Fourier Series and Fourier Transform Fourier series The first four Fourier series approximations for a square wave In mathematics, a Fourier series decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or complex exponentials). The study of Fourier series is a branch of Fourier ________________________ WORLD TECHNOLOGIES ________________________ analysis. Fourier series were introduced by Joseph Fourier (1768–1830) for the purpose of solving the heat equation in a metal plate. The heat equation is a partial differential equation. Prior to Fourier's work, there was no known solution to the heat equation in a general situation, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a sine or cosine wave. These simple solutions are now sometimes called eigensolutions. Fourier's idea was to model a complicated heat source as a superposition (or linear combination) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions. This superposition or linear combination is called the Fourier series. Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems. The Fourier series has many applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, econometrics, thin-walled shell theory, etc. Fourier series is named in honour of Joseph Fourier (1768-1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli.

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  Advanced Mathematical Analysis and Engineering Mathematics

  • (Author)
  • 2014(Publication Date)
  • Library Press

    (Publisher)

  ________________________ WORLD TECHNOLOGIES ________________________ Chapter 7 Fourier Series and Fourier Transform Fourier series The first four Fourier series approximations for a square wave In mathematics, a Fourier series decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or complex exponentials). The study of Fourier series is a branch of Fourier ________________________ WORLD TECHNOLOGIES ________________________ analysis. Fourier series were introduced by Joseph Fourier (1768–1830) for the purpose of solving the heat equation in a metal plate. The heat equation is a partial differential equation. Prior to Fourier's work, there was no known solution to the heat equation in a general situation, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a sine or cosine wave. These simple solutions are now sometimes called eigen-solutions. Fourier's idea was to model a complicated heat source as a superposition (or linear combination) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions. This superposition or linear combination is called the Fourier series. Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems. The Fourier series has many applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, econometrics, thin-walled shell theory, etc. Fourier series is named in honour of Joseph Fourier (1768-1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli.

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  Fourier and Laplace Transforms

  • R. J. Beerends, H. G. ter Morsche, J. C. van den Berg, E. M. van de Vrie(Authors)
  • 2003(Publication Date)
  • Cambridge University Press

    (Publisher)

  The methods that were used turned out to be much more widely applicable. Thereafter, applying Fourier series would produce fruitful results in many different fields, even though 58 the mathematical theory had not yet fully crystallized. By now, the Fourier theory has become a very versatile mathematical tool. From the times of Fourier up to the present day, research has been carried out in this field, both concrete and abstract, and new applications are being developed. Contents of Chapter 3 Fourier series: definition and properties Introduction 60 3.1 Trigonometric polynomials and series 61 3.2 Definition of Fourier series 65 3.2.1 Fourier series 66 3.2.2 Complex Fourier series 68 3.3 The spectrum of periodic functions 71 3.4 Fourier series for some standard functions 72 3.4.1 The periodic block function 72 3.4.2 The periodic triangle function 74 3.4.3 The sawtooth function 75 3.5 Properties of Fourier series 76 3.5.1 Linearity 76 3.5.2 Conjugation 77 3.5.3 Shift in time 78 3.5.4 Time reversal 79 3.6 Fourier cosine and Fourier sine series 80 Summary 83 Selftest 83 CHAPTER 3 Fourier series: definition and properties I N T R O D U C T I O N Many phenomena in the applications of the natural and engineering sciences are pe-riodic in nature. Examples are the vibrations of strings, springs and other objects, rotating parts in machines, the movement of the planets around the sun, the tides of the sea, the movement of a pendulum in a clock, the voltages and currents in elec-trical networks, electromagnetic signals emitted by transmittters in satellites, light signals transmitted through glassfibers, etc. Seemingly, all these systems operate in complicated ways; the phenomena that can be observed often behave in an erratic way. In many cases, however, they do show some kind of repetition. In order to analyse these systems, one can make use of elementary periodic functions or signals from mathematics, the sine and cosine functions.

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  Bird's Higher Engineering Mathematics

  • John Bird(Author)
  • 2021(Publication Date)
  • Routledge

    (Publisher)

  The Fourier series for the periodic function defined by f (x) = { -3, when - π ⟨ x ⟨ 0 +3, when 0 ⟨ x ⟨ π and which has a period of 2π is: (a) an odd function and contains no cosine terms (b) an even function and contains no sine terms (c) an odd function and contains no sine terms (d) an even function and contains no cosine terms 6. The Fourier series for the periodic function defined by f (x) =    0, -2 ⟨ x ⟨ -1 2, -1 ⟨ x ⟨ 1 0, 1 ⟨ x ⟨ 2 and which has a period of 2π is: (a) an odd function and contains no cosine terms (b) an even function and contains no cosine terms (c) an odd function and contains no sine terms (d) an even function and contains no sine terms For fully worked solutions to each of the problems in Practice Exercises 267 and 268 in this chapter, go to the website: www.routledge.com/cw/bird Chapter 63 Fourier series over any range Why it is important to understand: Fourier series over any range As has been mentioned in preceding chapters, the Fourier series has many applications; in fact, any field of physical science that uses sinusoidal signals, such as engineering, applied mathematics and chemistry, will make use of the Fourier series. In communications, the Fourier series is essential to understanding how a signal behaves when it passes through filters, amplifiers and communications channels. In astron- omy, radar and digital signal processing Fourier analysis is used to map the planet. In geology, seismic research uses Fourier analysis, and in optics, Fourier analysis is used in light diffraction. This chapter explains how to determine the Fourier series of a periodic function over any range.

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  Music: A Mathematical Offering

  • Dave Benson(Author)
  • 2006(Publication Date)
  • Cambridge University Press

    (Publisher)

  This explains, for example, why a m = 0 in the example on page 41. The square wave is not quite an even function, because f (π ) = f (−π ), but changing the value of a function at a finite set of points in the interval of integration never affects the value of an integral, so we just replace f (π ) and f (−π ) by zero to obtain an even function with the same Fourier coefficients. There is a similar explanation for why b 2m = 0 in the same example, using a different symmetry. The discussion of even and odd functions depended on the symmetry θ → −θ of order two. For periodic functions of period 2π , there is another symmetry of order two, namely θ → θ + π . The functions f (θ ) sat- isfying f (θ + π ) = f (θ ) are half-period symmetric, while functions satisfying f (θ + π ) = − f (θ ) are half-period antisymmetric. Any function f (θ ) can be de- composed into half-period symmetric and antisymmetric parts: f (θ ) = f (θ ) + f (θ + π ) 2 + f (θ ) − f (θ + π ) 2 . Multiplying half-period symmetric and antisymmetric functions works in the same way as for even and odd functions. If f (θ ) is half-period antisymmetric, then 2π π f (θ ) dθ = − π 0 f (θ ) dθ and so 2π 0 f (θ ) dθ = 0. Now the functions sin(mθ ) and cos(mθ ) are both half-period symmetric if m is even, and half-period antisymmetric if m is odd. So we deduce that if f (θ ) is half- period symmetric, f (θ + π ) = f (θ ), then the Fourier coefficients with odd indices 46 Fourier theory (a 2m+1 and b 2m+1 ) are zero, while if f (θ ) is antisymmetric, f (θ + π ) = − f (θ ), then the Fourier coefficients with even indices a 2m and b 2m are zero (check that this holds for a 0 too!). This corresponds to the fact that half-period symmetry is really the same thing as being periodic with half the period, so that the frequency components have to be even multiples of the defining frequency; while half-period antisymmetric functions only have frequency components at odd multiples of the defining frequency.

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