Signals and Systems EE235 Leo Lam © 2010-2011

Today’s menu Fourier Transform Leo Lam © 2010-2011

 x squared equals 9 x squared plus 1 equals y Find value of y 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

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 5 Leo Lam © 2010-2011

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 6 Leo Lam © 2010-2011

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 7 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 8 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) ??? 9 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! 10 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 11 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 12 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 13 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 14 Leo Lam © 2010-2011

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