Leo Lam © Signals and Systems EE235

Transformers Leo Lam ©

Today’s menu Fourier Transform Loads of examples

Fourier Transform: Leo Lam © Fourier Transform Inverse Fourier Transform:

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

Another angle of LTI (Example) Leo Lam © Given graphical H(), find h(t) What does this system do (qualitatively Low-pass filter. No delay. 6 magnitude   phase 0 0 1

Another angle of LTI (Example) Leo Lam © Given graphical H(), find h(t) What does this system do qualitatively? Bandpass filter. Slight delay. 7 magnitude   phase 0 1

Leo Lam © Summary Fourier Transforms and examples

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

Low Pass Filter Leo Lam © Consider an ideal low-pass filter with frequency response w 0 H() What is y(t) if input is: Ideal filter, so everything above is gone: y(t)

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

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

Describing Signals (just a summary) Leo Lam © C k and X() tell us the CE’s (or cosines) that are needed to build a time signal x(t) –CE with frequency  (or k 0 ) has magnitude |C k | or |X()| and phase shift <C k and <X() –FS and FT difference is in whether an uncountably infinite number of CEs are needed to build the signal. -B-BB  t x(t) X()

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

Example (Fourier Transform problem) Leo Lam © Solve for y(t) But does it make sense if it was done with convolution?  F() transfer function H() 01   = Z() =0 everywhere w Z() = F() H()

Example (Circuit design with FT!) Leo Lam © Goal: Build a circuit to give v(t) with an input current i(t) Find H() Convert to differential equation (Caveat: only causal systems can be physically built) 16 ???

Example (Circuit design with FT!) Leo Lam © Goal: Build a circuit to give v(t) with an input current i(t) Transfer function: 17 ??? Inverse transform!

Example (Circuit design with FT!) Leo Lam © 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? 18 ??? Capacitor Resistor

Fourier Transform: Big picture Leo Lam © 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()H() –Multiplication: x(t)m(t)  X()*H()/2 19

Fourier Transform: Wrap-up! Leo Lam © We have done: –Solving the Fourier Integral and Inverse –Fourier Transform Properties –Built-up Time-Frequency pairs –Using all of the above 20

Bridge to the next class Leo Lam © Next class: EE341: Discrete Time Linear Sys Analog to Digital Sampling 21 t continuous in time continuous in amplitude n discrete in time SAMPLING discrete in amplitude QUANTIZATION

Leo Lam © Summary Fourier Transforms and examples Next: Sampling and Laplace Transform