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Prof. Nizamettin AYDIN

naydin@yildiz.edu.tr

http://www.yildiz.edu.tr/~naydin

Digital Signal Processing

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Lecture 13

Digital FilteringDigital Filtering

of Analog Signalsof Analog Signals

Digital Signal Processing

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READING ASSIGNMENTS

• This Lecture:– Chapter 6, Sections 6-6, 6-7 & 6-8

• Other Reading:– Recitation: Chapter 6

• FREQUENCY RESPONSE EXAMPLES

– Next Lecture: Chapter 7

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LECTURE OBJECTIVES

• Two Domains: Time & Frequency

• Track the spectrum of x[n] thru an FIR Filter: Sinusoid-IN gives Sinusoid-OUT

• UNIFICATION: How does Frequency Response affect x(t) to produce y(t) ?

D-to-AA-to-Dx(t) y(t)y[n]x[n] )( jeH

FIR

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TIME & FREQUENCY

FIR DIFFERENCE EQUATION is the TIME-DOMAIN

M

k

M

kk knxkhknxbny

00

][][][][

M

k

kjj ekheH0

ˆˆ ][)(

ˆ3ˆ2ˆˆ ]3[]2[]1[]0[)( jjjj ehehehheH

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Ex: DELAY by 2 SYSTEM

y[n]x[n] )( jeH

y[n]x[n] ][nh

]2[][for )( and ][ Find ˆ nxnyeHnh j

]2[][ nnh

}1,0,0{kb

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DELAY by 2 SYSTEM

k = 2 ONLY

y[n]x[n] ]2[ n

]2[][for )( and ][ Find ˆ nxnyeHnh j

M

k

kjj ekeH0

ˆˆ ]2[)(

)( jeHy[n]x[n] 2je

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GENERAL DELAY PROPERTY

ONLY ONE non-ZERO TERM for k at k = nd

dnjM

k

kjd

j eenkeH ˆ

0

ˆˆ ][)(

][][for )( and ][ Find ˆd

j nnxnyeHnh

][][ dnnnh

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FREQ DOMAIN --> TIME ??

• START with

kj bnheH or find and ][)(

)ˆcos(7)( ˆ2ˆ jj eeH

y[n]x[n] ][nh ?][ nh

y[n]x[n])( jeH

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FREQ DOMAIN --> TIME

EULER’s Formula)ˆcos(7)( ˆ2ˆ jj eeH )5.05.0(7 ˆˆˆ2 jjj eee

)5.35.3( ˆ3ˆ jj ee

]3[5.3]1[5.3][ nnnh

}5.3,0,5.3,0{kb

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PREVIOUS LECTURE REVIEW• SINUSOIDAL INPUT SIGNAL

– OUTPUT has SAME FREQUENCY– DIFFERENT Amplitude and Phase

• FREQUENCY RESPONSE of FIR– MAGNITUDE vs. Frequency– PHASE vs. Freq– PLOTTING

)(ˆˆ ˆ

)()( jeHjjj eeHeH

MAG

PHASE

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FREQ. RESPONSE PLOTS

• DENSE GRID (ww) from - to +–ww = -pi:(pi/100):pi;

•HH = freqz(bb,1,ww)– VECTOR bb contains Filter Coefficients– DSP-First: HH = freekz(bb,1,ww)

M

k

kjk

j ebeH0

ˆˆ )(

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PLOT of FREQ RESPONSE

ˆˆ )ˆcos22()( jj eeH RESPONSE at /3

}1,2,1{}{ kb

(radians)

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EXAMPLE 6.2

y[n]x[n])( jeH

ˆˆ )ˆcos22()( jj eeH

njj

j

eenx

eHny)3/(4/

ˆ

2][ and

known is )( when][ Find

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njj eenxny )3/(4/2][ when][ Find

EXAMPLE 6.2 (answer)

3/ˆat )( evaluate -Step One ˆ jeH

ˆˆ )ˆcos22()( jj eeH

3/ˆ@3)( 3/ˆ jj eeH

njjj eeeny )3/(4/3/ 23][ njj ee )3/(12/6

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EXAMPLE: COSINE INPUT

)cos(2][ and

known is )( when][ Find

43

ˆ

nnx

eHny j

ˆˆ )ˆcos22()( jj eeH

y[n]x[n])( jeH

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EX: COSINE INPUT (ans-1)

)cos(2][ when][ Find 43 nnxny

][][][

)cos(2

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)4/3/()4/3/(43

nxnxnx

een njnj

][][][)(][

)(][

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)4/3/(3/2

)4/3/(3/1

nynynyeeHny

eeHnynjj

njj

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EX: COSINE INPUT (ans-2)

)cos(2][ when][ Find 43 nnxny

)4/3/()3/()4/3/(3/2

)4/3/()3/()4/3/(3/1

3)(][

3)(][

njjnjj

njjnjj

eeeeHny

eeeeHny

)cos(6][

33][

123

)12/3/()12/3/(

nny

eeny njnj

ˆˆ )ˆcos22()( jj eeH

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SINUSOID thru FIR

• IF

• Multiply the Magnitudes

• Add the Phases

))(ˆcos()(][

)ˆcos(][

11 ˆ1

ˆ

1

jj eHneHAny

nAnx

)()( ˆˆ* jj eHeH

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LTI Demo with Sinusoids

FILTER

x[n]y[n]

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DIGITAL “FILTERING”

– SPECTRUM of x(t) (SUM of SINUSOIDS)– SPECTRUM of x[n]

• Is ALIASING a PROBLEM ?

– SPECTRUM y[n] (FIR Gain or Nulls)– Then, OUTPUT y(t) = SUM of SINUSOIDS

D-to-AA-to-Dx(t) y(t)y[n]x[n] )( jeH

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FREQUENCY SCALING

• TIME SAMPLING:– IF NO ALIASING:– FREQUENCY SCALING

D-to-AA-to-Dx(t) y(t)y[n]x[n] )( jeH

ˆ ˆ

sfsT ˆ

snTt

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11-pt AVERAGER Example

D-to-AA-to-Dx(t) y(t)y[n]x[n] )( jeH

ˆ ˆ

250 Hz

25 Hz ?))250(2cos())25(2cos()( 2

1 tttx

ˆ5

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211

ˆ

)ˆsin(11

)ˆsin()( jj eeH

10

0111 ][][

k

knxny

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D-A FREQUENCY SCALING

• RECONSTRUCT up to 0.5fs

– FREQUENCY SCALING

D-to-AA-to-Dx(t) y(t)y[n]x[n] )( jeH

ˆ ˆ

• TIME SAMPLING:

sf ˆ

ss ftnnTt

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TRACK the FREQUENCIES

D-to-AA-to-Dx(t) y(t)y[n]x[n] )( jeH

ˆ ˆ

Fs = 1000 Hz

• 0.5

• .05

• 0.5

• .05

• 250 Hz

• 25 Hz

NO new freqs

• 250 Hz

• 25 Hz )(

)(

05.0

5.0

j

j

eH

eH

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11-pt AVERAGER

NULLS or ZEROS

5.0ˆ 05.0ˆ

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EVALUATE Freq. Response

ˆ5

21

211

ˆ

)ˆsin(11

)ˆsin()( jj eeH

)5.0(5

21

211

ˆ

))5.0(sin(11

))5.0(sin()(

jj eeH

5.0ˆAt

5.2

)25.0sin(11

)75.2sin( je

5.00909.0 je

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EVALUATE Freq. Response

fs = 1000

MAG SCALE

PHASE CHANGE

)( 1000/)25(2jeH

)( 1000/)250(2jeH

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EFFECTIVE RESPONSE

DIGITAL FILTER

LOW-PASS FILTER

)( jeH

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FILTER TYPES

• LOW-PASS FILTER (LPF)– BLURRING

– ATTENUATES HIGH FREQUENCIES

• HIGH-PASS FILTER (HPF)– SHARPENING for IMAGES

– BOOSTS THE HIGHS

– REMOVES DC

• BAND-PASS FILTER

(BPF)

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B & W IMAGE

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B&W IMAGE with COSINE

FILTERED: 11-pt AVG

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FILTERED B&W IMAGE

LPF:BLUR

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ROW of B&W IMAGE

BLACK = 255

WHITE = 0

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FILTERED ROW of IMAGE

ADJUSTED DELAY by 5 samples