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Signals & Systems

Signals & Systems. Lecture 13: Chapter 3 Spectrum Representation. Fourier Series Synthesis. Synthesis Example: Harmonic Signal (3 Frequencies). Spectrum Representation. Fourier Series (3-4) Any periodic signal can be synthesized with a sum of harmonically related sinusoids

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Signals & Systems

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  1. Signals & Systems Lecture 13: Chapter 3 Spectrum Representation

  2. Fourier Series Synthesis

  3. Synthesis Example: Harmonic Signal (3 Frequencies)

  4. Spectrum Representation • Fourier Series (3-4) • Any periodic signal can be synthesized with a sum of harmonically related sinusoids • Fourier series/ Fourier Synthesis Equation • Fourier series integral (to perform Fourier analysis) is known as Fourier Analysis Equation • Fourier series coefficients

  5. Strategy: x(t)---->ak

  6. Synthesis Vs. Analysis

  7. Integral Property of complex exponentials

  8. Integral Property of complex exponentials

  9. Product of complex exponentials: vl*(t) vk(t) Orthogonality Property

  10. Isolate one FS Coefficient Multiply both sides by vl*(t) and integrate over one period

  11. Isolate one FS Coefficient Multiply both sides by vl*(t) and integrate over one period

  12. General Waveforms • Waveforms can be synthesized by the equation x(t) = A0 + ∑Ak cos(2πfkt +k) • These waveforms maybe • constants • cosine signals ( periodic) • complicated-looking signals (not periodic) • So far we have dealt with signals whose amplitudes, phases and frequencies do not change with time

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