Leo Lam © Signals and Systems EE235 Lecture 26

Leo Lam © Today’s menu Fourier Transform

Convolution/Multiplication Example Leo Lam © Given f(t)=cos(t)e –t u(t) what is F()

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

Fourier Transform Pairs (Recap) Leo Lam © Review:

Fourier Transform and LTI System Leo Lam © 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() X()H() X() Frequency domain input signal’s Fourier transform output signal’s Fourier transform

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

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

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

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

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 11 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. 12 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. 13 magnitude   phase 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. 14 magnitude   phase 0 1

Leo Lam © Summary Fourier Transforms and examples

Low Pass Filter (extra examples) 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) 23 ???

Example (Circuit design with FT!) Leo Lam © Goal: Build a circuit to give v(t) with an input current i(t) Transfer function: 24 ??? 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? 25 ??? 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 26

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 27

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

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