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Leo Lam © 2010-2012 Signals and Systems EE235
Transformers Leo Lam © 2010-2012 2
Example (Circuit design with FT!) Leo Lam © 2010-2012 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) 3 ???
Example (Circuit design with FT!) Leo Lam © 2010-2012 Goal: Build a circuit to give v(t) with an input current i(t) Transfer function: 4 ??? Inverse transform!
Example (Circuit design with FT!) Leo Lam © 2010-2012 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? 5 ??? Capacitor Resistor
Fourier Transform: Big picture Leo Lam © 2010-2012 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 6
Fourier Transform: Wrap-up! Leo Lam © 2010-2012 We have done: –Solving the Fourier Integral and Inverse –Fourier Transform Properties –Built-up Time-Frequency pairs –Using all of the above 7
Bridge to the next class Leo Lam © 2010-2012 Next class: EE341: Discrete Time Linear Sys Analog to Digital Sampling 8 t continuous in time continuous in amplitude n discrete in time SAMPLING discrete in amplitude QUANTIZATION
Leo Lam © 2010-2012 Summary Fourier Transforms and examples Next: Sampling and Laplace Transform
Sampling Leo Lam © 2010-2012 Convert a continuous time signal into a series of regularly spaced samples, a discrete-time signal. Sampling is multiplying with an impulse train 10 t t t multiply = 0 TSTS
Sampling Leo Lam © 2010-2012 Sampling signal with sampling period T s is: Note that Sampling is NOT LTI 11 sampler
Sampling Leo Lam © 2010-2012 Sampling effect in frequency domain: Need to find: X s () First recall: 12 timeT Fourier spectra 0 1/T
Sampling Leo Lam © 2010-2012 Sampling effect in frequency domain: In Fourier domain: 13 distributive property Impulse train in time impulse train in frequency, dk=1/Ts What does this mean?
Sampling Leo Lam © 2010-2012 Graphically: In Fourier domain: No info loss if no overlap (fully reconstructible) Reconstruction = Ideal low pass filter 0 X() bandwidth
Sampling Leo Lam © 2010-2012 Graphically: In Fourier domain: Overlap = Aliasing if To avoid Alisasing: Equivalently: 0 Shannon’s Sampling Theorem Nyquist Frequency (min. lossless)
Sampling (in time) Leo Lam © 2010-2012 Time domain representation cos(2 100t) 100 Hz Fs=1000 Fs=500 Fs=250 Fs=125 < 2*100 cos(2 25t) Aliasing Frequency wraparound, sounds like Fs=25 (Works in spatial frequency, too!)
Summary: Sampling Leo Lam © 2010-2012 Review: –Sampling in time = replication in frequency domain –Safe sampling rate (Nyquist Rate), Shannon theorem –Aliasing –Reconstruction (via low-pass filter) More topics: –Practical issues: –Reconstruction with non-ideal filters –sampling signals that are not band-limited (infinite bandwidth) Reconstruction viewed in time domain: interpolate with sinc function
Would these alias? Leo Lam © 2010-2012 Remember, no aliasing if How about: 0 1 013-3 NO ALIASING!
Would these alias? Leo Lam © 2010-2012 Remember, no aliasing if How about: (hint: what’s the bandwidth?) Definitely ALIASING! Y has infinite bandwidth!
Would these alias? Leo Lam © 2010-2012 Remember, no aliasing if How about: (hint: what’s the bandwidth?) -.5 0.5 Copies every.7 -1.5 -.5.5 1.5 ALIASED!
Leo Lam © 2010-2012 Summary Sampling and the frequency domain representations Sampling frequency conditions
Signals and Systems Fall 2003 Lecture #13 21 October The Concept and Representation of Periodic Sampling of a CT Signal 2. Analysis of Sampling.
Signal Processing in the Discrete Time Domain Microprocessor Applications (MEE4033) Sogang University Department of Mechanical Engineering.
ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete Objectives: Response to a Sinusoidal Input Frequency Analysis of an RC Circuit.
Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin EE445S Real-Time Digital Signal Processing Lab Spring.
Leo Lam © Signals and Systems EE235. Transformers Leo Lam ©
Advanced Computer Graphics (Spring 2006) COMS 4162, Lecture 3: Sampling and Reconstruction Ravi Ramamoorthi
Lecture 9: Fourier Transform Properties and Examples
Signal and System I Continuous-time filters described by differential equations + Recall in Ch. 2 Continuous time Fourier transform. LTI system.
Analogue and digital techniques in closed loop regulation applications Digital systems Sampling of analogue signals Sample-and-hold Parseval’s theorem.
Leo Lam © Signals and Systems EE235. Leo Lam © Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to.
Chapter 7 CT Signal Analysis : Fourier Transform Basil Hamed
Leo Lam © Signals and Systems EE235 Lecture 29.
Leo Lam © Signals and Systems EE235 Lecture 27.
First semester King Saud University College of Applied studies and Community Service 1301CT.
The Nyquist–Shannon Sampling Theorem. Impulse Train Impulse Train (also known as "Dirac comb") is an infinite series of delta functions with a period.
Sampling of Continuous Time Signal Section
Leo Lam © Signals and Systems EE235. Leo Lam © Futile Q: What did the monsterous voltage source say to the chunk of wire? A: "YOUR.
Sampling and Antialiasing CMSC 491/635. Abstract Vector Spaces Addition –C = A + B = B + A –(A + B) + C = A + (B + C) –given A, B, A + X = B for only.
Leo Lam © Signals and Systems EE235. Leo Lam © x squared equals 9 x squared plus 1 equals y Find value of y.
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