2014 • 106 Pages • 1.03 MB • English

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E1.10 Fourier Series and Transforms Mike Brookes E1.10 Fourier Series and Transforms (2014-5509) Sums and Averages: 1 – 1 / 14

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Syllabus • Syllabus Main fact: Complicated time waveforms can be • Optical Fourier Transform • Organization expressed as a sum of sine and cosine waves. 1: Sums and Averages Joseph Fourier 1768-1830 E1.10 Fourier Series and Transforms (2014-5509) Sums and Averages: 1 – 2 / 14

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Syllabus • Syllabus Main fact: Complicated time waveforms can be • Optical Fourier Transform • Organization expressed as a sum of sine and cosine waves. 1: Sums and Averages Why bother? Sine/cosine are the only bounded waves that stay the same when differentiated. Joseph Fourier 1768-1830 E1.10 Fourier Series and Transforms (2014-5509) Sums and Averages: 1 – 2 / 14

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Syllabus • Syllabus Main fact: Complicated time waveforms can be • Optical Fourier Transform • Organization expressed as a sum of sine and cosine waves. 1: Sums and Averages Why bother? Sine/cosine are the only bounded waves that stay the same when differentiated. Any electronic circuit: sine wave in ⇒ sine wave out (same frequency). Joseph Fourier 1768-1830 E1.10 Fourier Series and Transforms (2014-5509) Sums and Averages: 1 – 2 / 14

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Syllabus • Syllabus Main fact: Complicated time waveforms can be • Optical Fourier Transform • Organization expressed as a sum of sine and cosine waves. 1: Sums and Averages Why bother? Sine/cosine are the only bounded waves that stay the same when differentiated. Any electronic circuit: sine wave in ⇒ sine wave out (same frequency). Joseph Fourier 1768-1830 Hard problem: Complicated waveform → electronic circuit→ output = ? E1.10 Fourier Series and Transforms (2014-5509) Sums and Averages: 1 – 2 / 14

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Syllabus • Syllabus Main fact: Complicated time waveforms can be • Optical Fourier Transform • Organization expressed as a sum of sine and cosine waves. 1: Sums and Averages Why bother? Sine/cosine are the only bounded waves that stay the same when differentiated. Any electronic circuit: sine wave in ⇒ sine wave out (same frequency). Joseph Fourier 1768-1830 Hard problem: Complicated waveform → electronic circuit→ output = ? Easier problem: Complicated waveform → sum of sine waves → linear electronic circuit (⇒ obeys superposition) → add sine wave outputs → output = ? E1.10 Fourier Series and Transforms (2014-5509) Sums and Averages: 1 – 2 / 14

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Syllabus • Syllabus Main fact: Complicated time waveforms can be • Optical Fourier Transform • Organization expressed as a sum of sine and cosine waves. 1: Sums and Averages Why bother? Sine/cosine are the only bounded waves that stay the same when differentiated. Any electronic circuit: sine wave in ⇒ sine wave out (same frequency). Joseph Fourier 1768-1830 Hard problem: Complicated waveform → electronic circuit→ output = ? Easier problem: Complicated waveform → sum of sine waves → linear electronic circuit (⇒ obeys superposition) → add sine wave outputs → output = ? Syllabus: Preliminary maths (1 lecture) E1.10 Fourier Series and Transforms (2014-5509) Sums and Averages: 1 – 2 / 14

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Syllabus • Syllabus Main fact: Complicated time waveforms can be • Optical Fourier Transform • Organization expressed as a sum of sine and cosine waves. 1: Sums and Averages Why bother? Sine/cosine are the only bounded waves that stay the same when differentiated. Any electronic circuit: sine wave in ⇒ sine wave out (same frequency). Joseph Fourier 1768-1830 Hard problem: Complicated waveform → electronic circuit→ output = ? Easier problem: Complicated waveform → sum of sine waves → linear electronic circuit (⇒ obeys superposition) → add sine wave outputs → output = ? Syllabus: Preliminary maths (1 lecture) Fourier series for periodic waveforms (4 lectures) E1.10 Fourier Series and Transforms (2014-5509) Sums and Averages: 1 – 2 / 14

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Syllabus • Syllabus Main fact: Complicated time waveforms can be • Optical Fourier Transform • Organization expressed as a sum of sine and cosine waves. 1: Sums and Averages Why bother? Sine/cosine are the only bounded waves that stay the same when differentiated. Any electronic circuit: sine wave in ⇒ sine wave out (same frequency). Joseph Fourier 1768-1830 Hard problem: Complicated waveform → electronic circuit→ output = ? Easier problem: Complicated waveform → sum of sine waves → linear electronic circuit (⇒ obeys superposition) → add sine wave outputs → output = ? Syllabus: Preliminary maths (1 lecture) Fourier series for periodic waveforms (4 lectures) Fourier transform for aperiodic waveforms (3 lectures) E1.10 Fourier Series and Transforms (2014-5509) Sums and Averages: 1 – 2 / 14

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Optical Fourier Transform • Syllabus A pair of prisms can split light up into its component frequencies (colours). • Optical Fourier Transform • Organization 1: Sums and Averages E1.10 Fourier Series and Transforms (2014-5509) Sums and Averages: 1 – 3 / 14

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