Signals & Systems Lecture 13: Chapter 3 Spectrum Representation

2 Today's lecture  Fourier series synthesis  Fourier series: spectrum representation  Synthesis versus Analysis  Integral property of complex exponentials  Product of complex exponentials  Computing Fourier coefficients  Analysis example

3 Fourier Series Synthesis

4 Synthesis Example: Harmonic Signal (3 Frequencies)

5 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

6 Strategy: x(t)---->a k

7 Synthesis Vs. Analysis

8 Integral Property of complex exponentials

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10 Product of complex exponentials: v l *(t) v k (t) Orthogonality Property

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

12 Isolate one FS Coefficient Multiply both sides by v l *(t) and integrate over one period

13 General Waveforms  Waveforms can be synthesized by the equation x(t) = A 0 + ∑A k cos(2πf k t +  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