Technology & Engineering
Fourier Coefficients
Fourier coefficients are the set of coefficients used to represent a periodic function as a sum of sine and cosine functions. They are obtained by projecting the original function onto the basis functions of the Fourier series. These coefficients provide a way to analyze and manipulate periodic signals in various engineering and technological applications, such as signal processing and communications.
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6 Key excerpts on "Fourier Coefficients"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
This has the advantage of simplifying many of the formulas involved and providing a formulation for Fourier series that more closely resembles the definition followed here. This passage from sines and cosines to complex exponentials makes it necessary for the Fourier Coefficients to be complex valued. The usual interpretation of this complex number is that it gives both the amplitude (or size) of the wave present in the function and the phase (or the initial angle) of the wave. This passage also introduces the need for negative frequencies. If θ were measured in seconds then the waves e 2 πiθ and e −2 πiθ would both complete one cycle per second, but they represent different frequencies in the Fourier transform. Hence, frequency no longer measures the number of cycles per unit time, but is closely related. There is an close connection between the definition of Fourier series and the Fourier transform for functions ƒ which are zero outside of an interval. For such a function we can calculate its Fourier series on any interval that includes the interval where ƒ is not identically zero. The Fourier transform is also defined for such a function. As we increase the length of the interval on which we calculate the Fourier series, then the Fourier series coefficients begin to look like the Fourier transform and the sum of the Fourier series of ƒ begins to look like the inverse Fourier transform. To explain this more precisely, suppose that T is large enough so that the interval [− T /2, T /2] contains the interval on which ƒ is not identically zero. Then the n -th series coefficient c n is given by : Comparing this to the definition of the Fourier transform it follows that since ƒ ( x ) is zero outside [− T /2, T /2]. Thus the Fourier Coefficients are just the values of the Fourier transform sampled on a grid of width 1/ T . As T increases the Fourier Coefficients more closely represent the Fourier transform of the function.
eBook - PDF- J. David Irwin, R. Mark Nelms(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
The techniques we will explore are based on the work of Jean Baptiste Joseph Fourier. Although our analyses will be confined to electric circuits, it is important to point out that the techniques are applicable to a wide range of engineering problems. In fact, it was Fourier’s work in heat flow that led to the techniques that will be presented here. In his work, Fourier demonstrated that a periodic function f (t ) could be expressed as a sum of sinusoidal functions. Therefore, given this fact and the fact that if a periodic function is expressed as a sum of linearly independent functions, each function in the sum must be periodic with the same period, and the function f (t ) can be expressed in the form f (t ) = a 0 + ∑ n = 1 ∞ D n cos (nω 0 t + θ n ) 15.1 560 CHAPTER 15 Fourier Analysis Techniques where ω 0 = 2π/T 0 and a 0 is the average value of the waveform. An examination of this ex- pression illustrates that all sinusoidal waveforms that are periodic with period T 0 have been included. For example, for n = 1, one cycle covers T 0 seconds, and D 1 cos (ω 0 t + θ 1 ) is called the fundamental. For n = 2, two cycles fall within T 0 seconds, and the term D 2 cos (2ω 0 t + θ 2 ) is called the second harmonic. In general, for n = k, k cycles fall within T 0 seconds, and D k cos (kω 0 t + θ k ) is the kth harmonic term. Since the function cos (nω 0 t + θ k ) can be written in exponential form using Euler’s iden- tity or as a sum of cosine and sine terms of the form cos nω 0 t and sin nω 0 t as demonstrated in Chapter 8, the series in Eq.
eBook - PDF- Lawrence Turyn(Author)
- 2013(Publication Date)
- CRC Press(Publisher)
9 Fourier Series 9.1 Orthogonality and Fourier Coefficients 9.1.1 Introduction A Fourier series is a way of decomposing a function into a possibly infinite sum of “harmonic components”. We call those pieces “harmonic” because they are sinusoidal functions whose frequencies are multiples of a common base frequency. Those func-tions are also solutions of harmonic oscillator problems. We call them components because decomposing a function in this way is like decomposing a vector into orthogonal components, as we did in Section 2.4, specifically Corollary 2.7 in Section 2.4. Suppose f ( x ) is a function defined only on the interval [ − L , L ] . Here, L is an unspecified positive number. The Fourier series “expansion,” that is, decomposition into components, is given by f ( x ) . = f s ( x ) = a 0 2 + ∞ n = 1 a n cos n π x L + b n sin n π x L , (9.1) where the real constants a 0 ; a 1 , a 2 , a 3 , ...; b 1 , b 2 , b 3 , ... are called the Fourier Coefficients . The notation f ( x ) . = f s ( x ) means that f is “represented” by f s , the Fourier series for f . We will see that f s ( x ) may be unequal to f ( x ) at some value(s) of x ; even worse, the Fourier series f s ( x ) may fail to converge at some value(s) of x . So, in principle, f s ( x ) and f ( x ) may be different functions. Note that even though the original function f ( x ) is defined only for − L ≤ x ≤ L , its Fourier series f s ( x ) may be defined for all x and is periodic with period 2 L , that is, f s ( x + k · 2 L ) = f s ( x ) for all integers k . One way to think of the relationship of f s to f is that it is like the relationship of your computer game’s “avatar” to you. In some circumstances your avatar may behave just like you, but perhaps not always. Another way to think of the relationship is between a movie’s “stunt double” and the actress she replaces: From certain viewpoints they may look and behave exactly alike.
eBook - PDF- Peter O'Neil(Author)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
AkeSak / Shutterstock.com Fourier series are the foundation for most modern imaging, signal processing, and analytical devices and algorithms. As such, they are a basic tool for the electrical engineers who design MRI and ultrasound machines for medical applications and NMR and IR spectrometers for chemical analysis. To translate the signal output from these machines into useful informa-tion for doctors and chemists, computer programmers write programs, based on the Fast Fourier Transform (FFT) algorithm, which combines the Fourier methods of this chapter with the matrix methods of previous chapters. Even broadcast and audio engineers use frequency filters based on Fourier analysis to amplify voices or cut out noise from their transmissions. Section 17.1 Problems In each of Problems 1–12, write the Fourier series for the function on the interval and determine the sum of the series. If software is available, graph some partial sums of the Fourier series. 1. f ( x ) = 4, − 3 ≤ x ≤ 3 2. f ( x ) = − x , − 1 ≤ x ≤ 1 3. f ( x ) = cosh (π x ) , − 1 ≤ x ≤ 1 4. f ( x ) = 1 − | x | , − 2 ≤ x ≤ 2 5. f ( x ) = − 4 for − π ≤ x ≤ 0, 4 for 0 < x ≤ π 6. f ( x ) = sin ( 2 x ) , − π ≤ x ≤ π 7. f ( x ) = x 2 − x + 3, − 2 ≤ x ≤ 2 8. f ( x ) = − x for − 5 ≤ x < 0, 1 + x 2 for 0 ≤ x ≤ 5 9. f ( x ) = 1 for − π ≤ x < 0, 2 for 0 ≤ x ≤ π 10. f ( x ) = cos ( x / 2 ) − sin ( x ) , − π ≤ x ≤ π 11. f ( x ) = cos ( x ) , − 3 ≤ x ≤ 3 12. f ( x ) = 1 − x for − 1 ≤ x ≤ 0, 0 for 0 < x ≤ 1. In each of Problems 13–19, determine the sum of the Fourier series on the interval. It is not necessary (or even helpful) to write this series. 13. f ( x ) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 2 x for − 3 ≤ x < − 2, 0 for − 2 ≤ x < 1 x 2 for 1 ≤ x ≤ 3 14. f ( x ) = 2 x − 2 for − π ≤ x < 1, 3 for 1 < x ≤ π Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 17.2 Sine and Cosine Series 597 15. f ( x ) = x 2 for − π ≤ x ≤ 0, 2 for 0 < x ≤ π 16.
- William Bober, Andrew Stevens(Authors)
- 2016(Publication Date)
- CRC Press(Publisher)
Fourier Transforms and Signal Processing ◾ 241 8.2 Mathematical Description of Periodic Signals: Fourier Series We begin with a signal x ( t ), which might represent audible sound that has been converted to a varying voltage using a transducer (e.g., a microphone). If x ( t ) is peri-odic as shown in Figure 8.1c as a sustained A note on a piano, then Fourier theory tells us that the waveform may be expressed as a weighted sum of pure sinusoids: … … ∑ ∑ = + π ⋅ ⋅ + π ⋅ ⋅ = = ( ) cos(2 ) sin(2 ) 0 1,2,3, 1,2,3, x t A A nf t B nf t n o n n o n (8.1) t x ( t ) 0.025 (a) t y ( t ) 0.025 (b) t z ( t ) 0.025 (c) Figure 8.1 Example of band-limited waveforms: (a) speech; (b) high-fidelity audio; (c) A above middle C on the piano. 242 ◾ Numerical and Analytical Methods with MATLAB The frequencies nf o in the second and third terms represent all possible sine and cosine components of x ( t ). The n = 1 component is the fundamental frequency f o , and components where n > 1 are called the harmonics . The factors A n and B n are called the Fourier Coefficients, and A 0 is the “DC” component of x ( t ). We can deter-mine A 0 , A n , and B n mathematically by multiplying x ( t ) by each possible frequency component and then integrating over one period of x ( t ).
eBook - PDF- Roland E. Thomas, Albert J. Rosa, Gregory J. Toussaint(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
This chapter covers Fourier Series. It discusses the devo- lution of periodic signals into an infinite series of sines and cosines. Fourier transforms do for aperiodic signals what Fourier series do for periodic ones. Fourier transforms let us understand the frequency content of aperiodic signals and apply that understanding to communication systems and other applications. An introduction to Fourier transforms is available in Chapter 18. Chapter Sections 13–1 Overview of Fourier Series 13–2 Fourier Coefficients 13–3 Waveform Symmetries 13–4 Circuit Analysis Using the Fourier Series 13–5 RMS Value and Average Power Chapter Learning Objectives 13-1 The Fourier Series (Sects. 13–1 to 13–3) (a) Given an equation or graph of a periodic waveform, derive expressions for the Fourier Coefficients. (b) Given a 0 , a n , and b n , calculate the Fourier coeffi- cients of a given periodic waveform. (c) Given a Fourier series of a periodic waveform, determine the properties of the waveform and plot its amplitude and phase spectra. 13-2 Fourier Series and Circuit Analysis (Sect. 13–4) (a) Given a linear circuit with a periodic input waveform, find the Fourier series of a steady-state response. (b) Given a network function with a periodic input, find the amplitude and phase spectra of the steady-state output. 13-3 RMS Value and Average Power (Sect. 13–5) (a) Given a periodic waveform, find the rms value of the waveform and the average power delivered to a spe- cified load. (b) Given the Fourier series of a periodic waveform, find the fraction of the average power carried by specified components and estimate the average power delivered to a specified load. 579 13–1 O V E R V I E W O F F O U R I E R S E R I E S In this chapter, we develop a method of finding the steady-state response of circuits to periodic signals. Periodic waveforms can be written as a Fourier series consisting of an infinite sum of harmonically related sinusoids.
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