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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