Leo Lam © 2010-2012

Signals and Systems

EE235

Leo Lam © 2010-2012

Transformers

2

Leo Lam © 2010-2012

Example (Circuit design with FT!)

• Goal: Build a circuit to give v(t) with an input current i(t)

• Find H(w)• Convert to differential equation• (Caveat: only causal systems can be physically

built)

3

???

Leo Lam © 2010-2012

Example (Circuit design with FT!)

• Goal: Build a circuit to give v(t) with an input current i(t)

• Transfer function:

4

???

)(

)()(

I

VH

Inverse transform!

Leo Lam © 2010-2012

Example (Circuit design with FT!)

• 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

???

)(

)()(

I

VH

Capacitor

Resistor

Leo Lam © 2010-2012

Fourier Transform: Big picture

• 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(w)H(w)– Multiplication: x(t)m(t) X(w)*H(w)/2p

6

Leo Lam © 2010-2012

Fourier Transform: Wrap-up!

• We have done:– Solving the Fourier Integral and Inverse– Fourier Transform Properties– Built-up Time-Frequency pairs– Using all of the above

7

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Bridge to the next class

• Next class: EE341: Discrete Time Linear Sys• Analog to Digital• Sampling

8

t

continuous in time

continuous in amplitude

n

discrete in timeSAMPLING

discrete in amplitudeQUANTIZATION

Leo Lam © 2010-2012

Summary

• Fourier Transforms and examples• Next: Sampling and Laplace Transform

Leo Lam © 2010-2012

Sampling

• 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 TS

Leo Lam © 2010-2012

Sampling

• Sampling signal with sampling period Ts is:

• Note that Sampling is NOT LTI

11

)()()(

n

nsss nTtnTxtx

sampler

Leo Lam © 2010-2012

Sampling

• Sampling effect in frequency domain:

• Need to find: Xs(w)• First recall:

12

timeT

Fourier spectra0

1/T

0 02 03002

Leo Lam © 2010-2012

Sampling

• 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?

Leo Lam © 2010-2012

Sampling

• Graphically:

• In Fourier domain:

• No info loss if no overlap (fully reconstructible)• Reconstruction = Ideal low pass filter

n sss T

nXT

X 21

)(

0

1( )

s

XT

X(w) bandwidth

Leo Lam © 2010-2012

Sampling

• Graphically:

• In Fourier domain:

• Overlap = Aliasing if • To avoid Alisasing:

• Equivalently:

n sss T

nXT

X 21

)(

0

Shannon’s Sampling TheoremNyquist Frequency (min. lossless)

Leo Lam © 2010-2012

Sampling (in time)

• Time domain representation

cos(2100t)100 Hz

Fs=1000

Fs=500

Fs=250

Fs=125 < 2*100

cos(225t)

Aliasing

Frequency wraparound, sounds like Fs=25

(Works in spatial frequency, too!)

Leo Lam © 2010-2012

Summary: Sampling

• 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

Leo Lam © 2010-2012

Would these alias?

• Remember, no aliasing if• How about:

0 1

0 1 3-3

NO ALIASING!

Leo Lam © 2010-2012

Would these alias?

• Remember, no aliasing if• How about: (hint: what’s the bandwidth?)

Definitely ALIASING!

Y has infinite bandwidth!

Leo Lam © 2010-2012

Would these alias?

• Remember, no aliasing if• How about: (hint: what’s the bandwidth?)

.7s 2 1.0B

-.5 0 .5

0.5B

-.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