EE123 Digital Signal Processingee123/sp15/Notes/Lecture03_DTFT.pdf · • Read Ch 2 2.0-2.9 (2nd...
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M. Lustig, EECS UC Berkeley EE123 Digital Signal Processing Lecture 3
Transcript of EE123 Digital Signal Processingee123/sp15/Notes/Lecture03_DTFT.pdf · • Read Ch 2 2.0-2.9 (2nd...
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....
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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
