sp first l13
TRANSCRIPT
04/15/23 © 2003, JH McClellan & RW Schafer 1
Signal Processing First
Lecture 13Digital Filtering of Analog Signals
04/15/23 © 2003, JH McClellan & RW Schafer 3
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
21
)4/3/()4/3/(43
nxnxnx
een njnj
][][][)(][
)(][
21
)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
21
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