Signals and Systems EE235 Leo Lam © 2010-2011

Climbing every mountain What do you call a teapot of boiling water on top of K2? A high-pot-in-use Leo Lam © 2010-2011

Extra Fourier Transform Returning ¾-term Examples Leo Lam © 2010-2011

Fourier Transform: Fourier Transform Inverse Fourier Transform: 4 Weak Dirichlet: Otherwise you can’t solve for the coefficients! 4 Leo Lam © 2010-2011

Low Pass Filter What is h(t)? (Impulse response) 5 Consider an ideal low-pass filter with frequency response H(w) w Looks like an octopus centered around time t = 0  Not causal…can’t build a circuit. 5 Leo Lam © 2010-2011

Low Pass Filter What is y(t) if input is: Ideal filter, so everything above is gone: y(t) Consider an ideal low-pass filter with frequency response H(w) w 6 Leo Lam © 2010-2011

Output determination Example Solve for y(t) Convert input and impulse function to Fourier domain: Invert Fourier using known transform: 7 Leo Lam © 2010-2011

Output determination Example Solve for y(t) Recall that: Partial fraction: Invert: 8 Leo Lam © 2010-2011

Describing Signals (just a summary) Ck and X(w) tell us the CE’s (or cosines) that are needed to build a time signal x(t) CE with frequency w (or kw0) has magnitude |Ck| or |X(w)| and phase shift <Ck and <X(w) FS and FT difference is in whether an uncountably infinite number of CEs are needed to build the signal. -B B w t x(t) X(w) 9 Leo Lam © 2010-2011

Describing Signals (just a summary) H(w) = frequency response Magnitude |H(w)| tells us how to scale cos amplitude Phase <H(w) tells us the phase shift H(w) cos(20t) Acos(20t+f) p/2 magnitude phase A -p/2 f 20 20 Leo Lam © 2010-2011

Convolution/Multiplication Summary The Pair: 11 Leo Lam © 2010-2011