M. Lustig, EECS UC Berkeley

EE123Digital Signal Processing

Lecture 3

M. Lustig, EECS UC Berkeley

A couple of things

• Read Ch 2 2.0-2.9 (2nd edition is fine)• Class webcasted in bcourses.berkeley.edu• Prof. Lustig’s office hours: TBA• Frank Ong

– M 4p-5, 212 Cory (this week I will cover his OH)– Lab Bash – Cancelled this week

• My office hours– Th 5-6pm, 212 Cory

• HW1 due this Friday (1/30), 11:59 pm– Submit on bCourses

• Lab0 due next Friday (2/6)

M. Lustig, EECS UC Berkeley

Discrete Time Systems

• Causality• Memoryless• Linearity• Time Invariance• BIBO stability

M. Lustig, EECS UC Berkeley

Discrete-Time LTI Systems

• The impulse response h[n] completely characterizes an LTI system “DNA of LTI”

discrete convolution

Sum of weighted, delayed impulse responses!

M. Lustig, EECS UC Berkeley

BIBO Stability of LTI Systems

• An LTI system is BIBO stable iff h[n] is absolutely summable

Cool DSP: Steganography

• Hide signals in other signals

• Example: hiding an image in a song

“Secret Message”

0110011…

Encode

0.2267 0.2268 0.2269 0.227 0.2271 0.2272 0.22730

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (s)

Am

plitu

de

Secretmessage

?

AudioSignal

Encode

CoverSignal

256x320

Let’s compare the two signals

Overall Signalsx[n] and y[n]look identical

(play the 2 clips)

Restricted signalsx5[n] and y5[n]very different!

0.2267 0.2268 0.2269 0.227 0.2271 0.2272 0.2273-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time (s)

Am

plitu

de

Original signal (x[n])Modified Signal (y[n])

= Signal y[n] at 5th

decimal place

0.2267 0.2268 0.2269 0.227 0.2271 0.2272 0.22730

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time (s)

Am

plitu

de

Original signal (x[n])Modified Signal (y[n])

How should we decode the secret message?

Decode

Rearrange

256x320

Bits toBytes

0 1 2 3 4 5 6 7

x 10-3

0

10

20

30

40

50

60

70

80

90

Time (s)

Byt

e M

essa

ge

Linear? Time-invariant? BIBO?

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Discrete-Time Fourier Transform (DTFT)Discrete-Time Fourier Transform (DTFT)

Why one is sumand the other integral?

Why use one over the other?Alternative

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Example 1:Example 1:w[n] “window”

DTFT:

Recall:

N-N

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Example 1 cont.Example 1 cont.

DTFT:

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Example 1 cont.Example 1 cont.

DTFT:

j

periodic sinc

-

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Example 1 cont.Example 1 cont.

from l’Hôpitalalso, Σx[n]

=1, why?

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Properties of the DTFTProperties of the DTFT

Periodicity:

if x[n] is real

Conjugate Symmetry:

Big deal for: MRI, Communications, more....

M. Lustig, EE123 UCBM. Lustig, EE123 UCB

Half Fourier Imaging in MRHalf Fourier Imaging in MR

k-space (Raw Data) Image

Discrete Fourier transform

Complete based on conjugate symmetryHalf the Scan time!

Complete based on conjugate symmetryHalf the Scan time!

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SSB ModulationSSB ModulationReal Baseband signal has conjugate symmetric spectrum

AM modulation

SSB-SC reduced power, half bandwidth

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SSBSSB

http://www.youtube.com/watch?v=y0qi9Fr2j6Y&list=PLA5FE5E811C57CF77

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Properties of the DTFT cont.Properties of the DTFT cont.

Time-Reversal

If x[n] = x[-n] and x[n] is real, then:

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Q: Suppose:

A: Decompose x[n] to even and odd functions

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Oops!

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Time-Freq Shifting/modulation:

Properties of the DTFT cont.Properties of the DTFT cont.

Good for MRI! Why

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Example 2Example 2What is the DTFT of:

High Pass Filter

See 2.9 for more properties