Signals and Systems EE235 Lecture 26 Leo Lam © 2010-2012

Today’s menu Fourier Transform Leo Lam © 2010-2012

Convolution/Multiplication Example Given f(t)=cos(t)e–tu(t) what is F() Not a good Duality Example, but a great proof for the Multiplication properties! 3 Leo Lam © 2010-2012

More Fourier Transform Properties time domain Fourier transform Duality Time-scaling Multiplication Differentiation Integration Conjugation Dual of convolution 4 4 Leo Lam © 2010-2012

Fourier Transform Pairs (Recap) Review: 1 5 5 Leo Lam © 2010-2012

Fourier Transform and LTI System Back to the Convolution Duality: And remember: And in frequency domain Convolution in time h(t) x(t)*h(t) x(t) Time domain Multiplication in frequency H(w) X(w)H(w) X(w) Frequency domain input signal’s Fourier transform output signal’s Fourier transform 6 Leo Lam © 2010-2012

Fourier Transform and LTI (Example) Delay: h(t) LTI Time domain: Frequency domain (FT): Shift in time  Add linear phase in frequency 7 Leo Lam © 2010-2012

Fourier Transform and LTI (Example) Delay: Exponential response h(t) LTI Delay 3 Using Convolution Properties Using FT Duality 8 Leo Lam © 2010-2012

Fourier Transform and LTI (Example) Delay: Exponential response Responding to Fourier Series h(t) LTI Delay 3 Delay 3 9 Leo Lam © 2010-2012

Another LTI (Example) Given Exponential response What does this system do? What is h(t)? And y(t) if Echo with amplification LTI 10 Leo Lam © 2010-2012

Another angle of LTI (Example) Given graphical H(w), find h(t) What does this system do? What is h(t)? Linear phase  constant delay magnitude w phase 1 Slope=-5 11 Leo Lam © 2010-2012

Another angle of LTI (Example) Given graphical H(w), find h(t) What does this system do (qualitatively Low-pass filter. No delay. magnitude w phase 1 12 Leo Lam © 2010-2012

Another angle of LTI (Example) Given graphical H(w), find h(t) What does this system do qualitatively? Bandpass filter. Slight delay. magnitude w phase 1 13 Leo Lam © 2010-2012

Another angle of LTI (Example) Given graphical H(w), find h(t) What does this system do qualitatively? Bandpass filter. Slight delay. magnitude w phase 1 14 Leo Lam © 2010-2012

Summary Fourier Transforms and examples Leo Lam © 2010-2012

Low Pass Filter (extra examples) What is h(t)? (Impulse response) 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. 16 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 17 Leo Lam © 2010-2011

Output determination Example Solve for y(t) Convert input and impulse function to Fourier domain: Invert Fourier using known transform: Use the e-(jw0t)  2*pi*delta(w-w0) pair 18 Leo Lam © 2010-2011

Output determination Example Solve for y(t) Recall that: Partial fraction: Invert: 19 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) 20 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

Example (Fourier Transform problem) Solve for y(t) But does it make sense if it was done with convolution? 5 -5 w F(w) transfer function H(w) 1 -1 w 5 -5 w Z(w) = F(w) H(w) 5 -5 w = Z(w) =0 everywhere 22 Leo Lam © 2010-2011

Example (Circuit design with FT!) Goal: Build a circuit to give v(t) with an input current i(t) Find H(w) Convert to differential equation (Caveat: only causal systems can be physically built) ??? 23 Leo Lam © 2010-2011

Example (Circuit design with FT!) Goal: Build a circuit to give v(t) with an input current i(t) Transfer function: ??? Inverse transform! 24 Leo Lam © 2010-2011

Example (Circuit design with FT!) Goal: Build a circuit to give v(t) with an input current i(t) From: The system: Inverse transform: KCL: What does it look like? ??? Capacitor Resistor 25 Leo Lam © 2010-2011

Fourier Transform: Big picture With Fourier Series and Transform: Intuitive way to describe signals & systems Provides a way to build signals Generate sinusoids, do weighted combination Easy ways to modify signals LTI systems: x(t)*h(t)  X(w)H(w) Multiplication: x(t)m(t)  X(w)*H(w)/2p 26 Leo Lam © 2010-2011

Fourier Transform: Wrap-up! We have done: Solving the Fourier Integral and Inverse Fourier Transform Properties Built-up Time-Frequency pairs Using all of the above 27 Leo Lam © 2010-2011

Bridge to the next class Next class: EE341: Discrete Time Linear Sys Analog to Digital Sampling t continuous in time continuous in amplitude n discrete in time SAMPLING discrete in amplitude QUANTIZATION 28 Leo Lam © 2010-2011

Summary Fourier Transforms and examples Next, and last: Sampling! Leo Lam © 2010-2011