Leo Lam © Signals and Systems EE235

Transformers Leo Lam ©

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) 3 ???

Example (Circuit design with FT!) Leo Lam © 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 © 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 © 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 © 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 © 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 © Summary Fourier Transforms and examples Next: Sampling and Laplace Transform

Sampling Leo Lam © 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 © Sampling signal with sampling period T s is: Note that Sampling is NOT LTI 11 sampler

Sampling Leo Lam © Sampling effect in frequency domain: Need to find: X s () First recall: 12 timeT Fourier spectra 0 1/T

Sampling Leo Lam © 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 © Graphically: In Fourier domain: No info loss if no overlap (fully reconstructible) Reconstruction = Ideal low pass filter 0 X() bandwidth

Sampling Leo Lam © 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 © 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 © 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 © Remember, no aliasing if How about: NO ALIASING!

Would these alias? Leo Lam © Remember, no aliasing if How about: (hint: what’s the bandwidth?) Definitely ALIASING! Y has infinite bandwidth!

Would these alias? Leo Lam © Remember, no aliasing if How about: (hint: what’s the bandwidth?) Copies every ALIASED!

Leo Lam © Summary Sampling and the frequency domain representations Sampling frequency conditions