MATH283_L14_Wk_05.pdf - Lecture 14 Introduction to Fourier series In this lecture we will introduce Fourier series Advanced Engineering Mathematics Week

MATH283_L14_Wk_05.pdf - Lecture 14 Introduction to Fourier...

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Lecture 14: Introduction to Fourier series In this lecture, we will . . . introduce Fourier series Advanced Engineering Mathematics Week 5, Thursday Lecture L14: 1 / 22
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Overview of Fourier Series In mathematics, representation of a function by infinite series is very important. It is used extensively in calculators and computers for evaluating values of many functions. The most familiar series expansions are power series, f ( x ) = X n =0 a n x n . There are other types of infinite series which are very useful. In particular, a Fourier series is an expansion in terms of sine and cosines functions. Fourier series is developed by Joseph Fourier, a French mathematician and physicist, in his investigation of problems of heat transfer. The Fourier transform and Fourier’s Law are also named in his honour. Fourier was taught by Lagrange and Laplace. Advanced Engineering Mathematics Week 5, Thursday Lecture L14: 2 / 22
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Why Fourier series? Fourier Series is really interesting, as it uses many of the mathematical techniques that you have learned before, like graphs, integration, differentiation, summation notation, trigonometry, etc. Fourier series and transforms are used in many other areas, as governing equations for other problems can be converted to heat equations. Fourier series, in particular, is used in the analysis of signals in electronics, and more. For example, pulse code modulation used for recording digital music. Fourier series are used in the analysis of periodic functions. An example of a periodic function Many of the phenomena studied in engineering and science are periodic in nature, for example, the current and voltage in an alternating current circuit. Advanced Engineering Mathematics Week 5, Thursday Lecture L14: 3 / 22
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Harmonics In presenting a non-sinusoidal function by a Fourier series, we will see that only certain sinusoids are required: 1 The terms a 1 cos t and b 1 sin t are known as the fundamental or first harmonic . Advanced Engineering Mathematics Week 5, Thursday Lecture L14: 4 / 22
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Harmonics In presenting a non-sinusoidal function by a Fourier series, we will see that only certain sinusoids are required: 1 The terms a 1 cos t and b 1 sin t are known as the fundamental or first harmonic . 2 The terms a 2 cos 2 t and b 2 sin 2 t are called the second harmonic . Advanced Engineering Mathematics Week 5, Thursday Lecture L14: 4 / 22
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Harmonics In presenting a non-sinusoidal function by a Fourier series, we will see that only certain sinusoids are required: 1 The terms a 1 cos t and b 1 sin t are known as the fundamental or first harmonic . 2 The terms a 2 cos 2 t and b 2 sin 2 t are called the second harmonic . 3 The terms a 3 cos 3 t and b 3 sin 3 t are called the third harmonic , etc. Advanced Engineering Mathematics Week 5, Thursday Lecture L14: 4 / 22
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Harmonics In presenting a non-sinusoidal function by a Fourier series, we will see that only certain sinusoids are required: 1 The terms a 1 cos t and b 1 sin t are known as the fundamental or first harmonic .
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