Signals and Systems EE235 Leo Lam © 2010-2012

Motivation Leo Lam © 2010-2012

Fourier Series: Dirichlet Conditon Condition for periodic signal f(t) to exist has exponential series: Weak Dirichlet: Strong Dirichlet (converging series): f(t) must have finite maxima, minima, and discontinuities in a period All physical periodic signals converge Weak Dirichlet: Otherwise you can’t solve for the coefficients! 3 Leo Lam © 2010-2012

End of Fourier Series We have accomplished: Next: Fourier Transform 4 Introduced signal orthogonality Fourier Series derivation Approx. periodic signals: Fourier Series Properties Next: Fourier Transform 4 Leo Lam © 2010-2012

Fourier Transform: Introduction Fourier Series: Periodic Signal Fourier Transform: extends to all signals Recall time-scaling: 5 Leo Lam © 2010-2012

Fourier Transform: Recall time-scaling: 6 Fourier Spectra for T, Fourier Spectra for T, for 2T, 6 Leo Lam © 2010-2012

Fourier Transform: Non-periodic signal: infinite period T 7 Fourier Spectra for T, for 2T, 7 Leo Lam © 2010-2012

Fourier Transform: Fourier Formulas: For any arbitrary practical signal And its “coefficients” (Fourier Transform): F(w) is complex Rigorous math derivation in Ch. 4 (not required reading, but recommended) Time domain to Frequency domain Weak Dirichlet: Otherwise you can’t solve for the coefficients! 8 Leo Lam © 2010-2012

Fourier Transform: Fourier Formulas compared: 9 Fourier transform Fourier transform coefficients: Fourier transform (arbitrary signals) Fourier series (Periodic signals): Fourier series coefficients: and 9 Leo Lam © 2010-2012

Fourier Transform (example): Find the Fourier Transform of What does it look like? If a <0, blows up phase varies with  magnitude varies with  10 Leo Lam © 2010-2012

Fourier Transform (example): Fourier Transform of Real-time signals magnitude: even phase: odd magnitude phase 11 Leo Lam © 2010-2012

Fourier Transform (Symmetry): Real-time signals magnitude: even – why? magnitude Even magnitude Odd phase Useful for checking answers 12 Leo Lam © 2010-2012

Fourier Transform/Series (Symmetry): Works for Fourier Series, too! Fourier series (periodic functions) Fourier transform (arbitrary practical signal) Fourier coefficients Fourier transform coefficients magnitude: even & phase: odd 13 Leo Lam © 2010-2012

Fourier Transform (example): Fourier Transform of F(w) is purely real F(w) for a=1 14 Leo Lam © 2010-2012

Summary Fourier Transform intro Inverse etc. Leo Lam © 2010-2012

Fourier Transform (delta function): Fourier Transform of Standard Fourier Transform pair notation 16 Leo Lam © 2010-2012

Fourier Transform (rect function): Fourier Transform of Plot for T=1? t -T/2 0 T/2 1 Define 17 Leo Lam © 2010-2012

Fourier Transform (rect function): Fourier Transform of Observation: Wider pulse (in t) <-> taller narrower spectrum Extreme case: <-> Peak=pulse width (example: width=1) Zero-Crossings: 18 Leo Lam © 2010-2012

Fourier Transform - Inverse relationship Inverse relationship between time/frequency 19 Leo Lam © 2010-2012

Fourier Transform - Inverse Inverse Fourier Transform (Synthesis) Example: 20 Leo Lam © 2010-2012

Fourier Transform - Inverse Inverse Fourier Transform (Synthesis) Example: Single frequency spike in w: exponential time signal with that frequency in t A single spike in frequency Complex exponential in time 21 Leo Lam © 2010-2012

Fourier Transform Properties A Fourier Transform “Pair”: f(t)  F() Re-usable! time domain Fourier transform Scaling Additivity Convolution Time shift 22 Leo Lam © 2010-2012

How to do Fourier Transform Three ways (or use a combination) to do it: Solve integral Use FT Properties (“Spiky signals”) Use Fourier Transform table (for known signals) 23 Leo Lam © 2010-2012

FT Properties Example: Find FT for: We know the pair: So: -8 0 8  G() 24 Leo Lam © 2010-2012

More Transform Pairs: More pairs: time domain Fourier transform 25 Leo Lam © 2010-2012

Periodic signals: Transform from Series Integral does not converge for periodic fns: We can get it from Fourier Series: How? Find x(t) if Using Inverse Fourier: So 26 Leo Lam © 2010-2012

Periodic signals: Transform from Series We see this pair: More generally, if X(w) has equally spaced impulses: Then: Fourier Series!!! 27 Leo Lam © 2010-2012

Periodic signals: Transform from Series If we know Series, we know Transform Then: Example: We know: We can write: 28 Leo Lam © 2010-2012

Summary Fourier Transform Pairs FT Properties Leo Lam © 2010-2012