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Lecture 7 (Chapter 3)
Signals & SystemsSignals & SystemsFourier Response and Filtering
Ad t d f L t t f MITAdapted from: Lecture notes from MITHamid R. Rabiee
Ali Soltani
Fall 2011
Lecture 7 (Chapter 3)
The Eigenfunction Property of Complex Exponentialsg p y p p
h(t)ste stesH )(CT:
∞
∞−
−= dtethsH st)()(CT"System Function"
h[n]nz nzzH )(DT:
∞
∞
∞−=
−=n
nznhzH ][)(DT"System Function"
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Lecture 7 (Chapter 3)
Fourier Series: Periodic Signals and LTI Systemsg y
h(t)∞
= tjkeatx 0)( ω ∞
= tjkeajkHty 0)()( ωωh(t)−∞=
=k
keatx )( −∞=
=k
keajkHty )()( 0ω
kk ajkHa 0 )( ω→ )(00
0|)(|)( ωωω jkHjejkHjkH ∠=Includes both amplitude & phasegain""
So |ak|→|H(jkω0)||ak| or powers of signals get modified through filter/system
Includes both amplitude & phase
h[n] 0 0[ ] ( )jk jk tk
k Ny n H e a eω ω
=< >
= >=<
=Nk
njkkeanx 0][ ω
k
gain
jkk aeHa
""
)( 0ω→ )( 000 |)(|)(
ωωω jkeHjjkjk eeHeH ∠=Includes both amplitude & phase
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Lecture 7 (Chapter 3)
The Frequency Response of an LTI Systemq y p y
H(jω)tje ω tjejH ωω)(H(jω)e j )(
)( ωjs =CT Frequency response: dtethjH tjωω −∞
∞−= )()(
H(ejω)nje ω njj eeH ωω )(H(ej )e )(
)( ωjez =∞
DT Frequency response:nj
n
j etheH ωω −∞
−∞== )()(
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Lecture 7 (Chapter 3)
Frequency Shaping and Filteringq y p g g
By choice of H(jω) (or H(ejω)) as a function of ω, we can shape the frequency composition of the output
• Preferential amplification
• Selective filtering of some frequencies• Selective filtering of some frequencies
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Lecture 7 (Chapter 3)
Example #1: Audio Systemp y
For audio signals, the amplitude is much more important than the phase.
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Lecture 7 (Chapter 3)
Example #2: Frequency Selective Filtersp q y
Filter attenuates signals outside of the frequency range of interest
Lowpass Filters: Only amplitude shown here.
Note for DT:
)()( )2( πωω += jj eHeH
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Lecture 7 (Chapter 3)
Lowpass Filteringp g
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Lecture 7 (Chapter 3)
Highpass Filtersg p
Remember:)1( =− πnjn e
DTin frequency highest =π high frequency
high frequency
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Lecture 7 (Chapter 3)
Highpass Filteringg p g
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Lecture 7 (Chapter 3)
Bandpass Filtersp
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Lecture 7 (Chapter 3)
Bandpass Filteringp g
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Lecture 7 (Chapter 3)
Idealized Filters
CT
ωcCutoff frequency
Stopband Passband Stopband
DT
Note: |H| = 1 and ∠H = 0 for the ideal filters in the passbands, no need for the phase plot.
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no need for the phase plot.
Lecture 7 (Chapter 3)
Idealized Highpassg p
CT
DT
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Lecture 7 (Chapter 3)
Idealized Bandpassp
CT
lower cut-off upper cut-offDT
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Lecture 7 (Chapter 3)
Example #3: DT Averager/Smootherp g
]}1[][]1[{31][ +++−= nxnxnxny
FIR (Finite ImpulseResponse) filters
FIR (Finite Impulse Response) filters
]}1[][]1[{31][ +++−= nnnnh δδδ
p )p )
ωωωωω cos32
31]1[
31][)( +=++== −
∞
−∞=
− jj
n
njj eeenheH
LPF
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Lecture 7 (Chapter 3)
Example #4: Nonrecursive DT (FIR) filtersp ( )
][1
1][][1
1][ δ −++
=→−++
=−=−= kn
MNnhknx
MNny
M
Nk
M
Nk
)2/sin(]2/)1(sin[
11
11)( ]2/)[(
ωωωωω ++
++=
++= −
−=
−
−=−=
NMeMN
eMN
eH MNjM
Nk
jkj
NkNk
Rolls off at lower ω as M+N+1
increases
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Lecture 7 (Chapter 3)
Example #5: Simple DT “Edge” Detectorp p g
]}1[][{1][ −−= nxnxny
DT 2-point “differentiator”
]}1[][{21][
]}1[][{2
][
−−=
−−=
nnnh
nxnxny
δδ
)2/sin()]([2
2/2/2/2/ ωωωωω jjjj jeeeejj −−− =−
/ 21( ) (1 ) sin( / 2)2
j j jH e e jeω ω ω ω− −= − =
Passes high-frequency components|)2/sin(||)(| ωω =jeH
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Lecture 7 (Chapter 3)
Example #6: Edge enhancement using DT diffrentiatorp g g
Courtesy of Jason Oppenheim. Used with permission.
Courtesy of Jason Oppenheim UsedOppenheim. Used with permission.
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Lecture 7 (Chapter 3)
Example #7: A Filter Bankp
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