Leo Lam © Signals and Systems EE235

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Exponential Fourier Series: formulas Leo Lam © Analysis: Breaking signal down to building blocks: Synthesis: Creating signals from building blocks

Harmonic Series (example) Leo Lam © Example with (t) (a “delta train”): Write it in an exponential series: Signal is periodic: only need to do one period The rest just repeats in time t T

Harmonic Series (example) Leo Lam © One period: Turn it to: Fundamental frequency: Coefficients: t T * All basis function equally weighted and real! No phase shift! Complex conj.

Harmonic Series (example) Leo Lam © From: To: Width between “spikes” is: t T Fourier spectra 0 1/T  Time domain Frequency domain

Example: Shifted delta-train Leo Lam © A shifted “delta-train” In this form: For one period: Find d n : time T0 T/2 *

Example: Shifted delta-train Leo Lam © A shifted “delta-train” Find d n : time T0 T/2 Complex coefficient!

Example: Shifted delta-train Leo Lam © A shifted “delta-train” Now as a series in exponentials: time T0 T/2 0 Same magnitude; add phase! Phase of Fourier spectra 

Example: Shifted delta-train Leo Lam © A shifted “delta-train” Now as a series in exponentials: 0 Phase 0 1/T Magnitude (same as non-shifted)

Example: Sped up delta-train Leo Lam © Sped-up by 2, what does it do? Fundamental frequency doubled d n remains the same (why?) For one period: time T/2 0 m=1 2 3 Great news: we can be lazy! The new T.

Lazy ways: re-using Fourier Series Leo Lam © Standard notation: “ ” means “a given periodic signal has Fourier series coefficients ” Given, find where is a new signal based on Addition, time-scaling, shift, reversal etc. Direct correlation: Look up table! Textbook Ch. 3.1 & everywhere online: ges/3/3d/Ece343_Fourier_series.pdf ges/3/3d/Ece343_Fourier_series.pdf

Graphical: Time scaling: Fourier Series Leo Lam © Example: Time scaling up (graphical) New signal based on f(t): Using the Synthesis equation: Fourier spectra 0 Twice as far apart as f(t)’s

Graphical: Time scaling: Fourier Series Leo Lam © Spectra change (time-scaling up): f(t) g(t)=f(2t) Does it make intuitive sense?

Graphical: Time scaling: Fourier Series Leo Lam © Spectra change (time scaling down): f(t) g(t)=f(t/2)

Fourier Series Table Leo Lam © Added constant only affects DC term Linear ops Time scale Same d k, scale  0 reverse Shift in time –t 0 Add linear phase term –jk   t 0 Fourier Series Properties:

Fourier Series: Fun examples Leo Lam © Rectified sinusoids Find its exponential Fourier Series: t 0 f(t) =|sin(t)| Expand as exp., combine, integrate

Fourier Series: Circuit Application Leo Lam © Rectified sinusoids Now we know: Circuit is an LTI system: Find y(t) Remember: +-+- sin(t) full wave rectifier y(t) f(t) Where did this come from? S Find H(s)!

Fourier Series: Circuit Application Leo Lam © Finding H(s) for the LTI system: e st is an eigenfunction, so Therefore: So: Shows how much an exponential gets amplified at different frequency s

Fourier Series: Circuit Application Leo Lam © Rectified sinusoids Now we know: LTI system: Transfer function: To frequency: +-+- sin(t) full wave rectifier y(t) f(t)

Fourier Series: Circuit Application Leo Lam © Rectified sinusoids Now we know: LTI system: Transfer function: System response: +-+- sin(t) full wave rectifier y(t) f(t)

Leo Lam © Summary Fourier Series circuit example