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Leo Lam © 2010-2011 Signals and Systems EE235

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Leo Lam © 2010-2011 Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to the Fourier transform of the sinc function? A: "You're such a square!"

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Leo Lam © 2010-2011 Extra Fourier Transform Fourier Transform Examples

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Fourier Transform: Leo Lam © 2010-2011 4 Fourier Transform Inverse Fourier Transform:

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Low Pass Filter Leo Lam © 2010-2011 5 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.

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Low Pass Filter Leo Lam © 2010-2011 6 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)

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

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

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Describing Signals (just a summary) Leo Lam © 2010-2011 9 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()

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Describing Signals (just a summary) Leo Lam © 2010-2011 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

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Example (Fourier Transform problem) Leo Lam © 2010-2011 Solve for y(t) But does it make sense if it was done with convolution? 11 05 -5 F() transfer function H() 01 05 -5 = Z() =0 everywhere 05 -5 w Z() = F() H()

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Example (Circuit design with FT!) Leo Lam © 2010-2011 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) 12 ???

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Example (Circuit design with FT!) Leo Lam © 2010-2011 Goal: Build a circuit to give v(t) with an input current i(t) Transfer function: 13 ??? Inverse transform!

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Example (Circuit design with FT!) Leo Lam © 2010-2011 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? 14 ??? Capacitor Resistor

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Fourier Transform: Big picture Leo Lam © 2010-2011 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 15

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Fourier Transform: Wrap-up! Leo Lam © 2010-2011 We have done: –Solving the Fourier Integral and Inverse –Fourier Transform Properties –Built-up Time-Frequency pairs –Using all of the above 16

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Bridge to the next class Leo Lam © 2010-2011 Next class: EE341: Discrete Time Linear Sys Analog to Digital Sampling 17 t continuous in time continuous in amplitude n discrete in time SAMPLING discrete in amplitude QUANTIZATION

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

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