6.0 time/frequency characterization of signals/systems 6.1 magnitude and phase for signals
DESCRIPTION
6.0 Time/Frequency Characterization of Signals/Systems 6.1 Magnitude and Phase for Signals and Systems. Signals. : magnitude of each frequency component. : phase of each frequency component. Sinusoidals ( p.68 of 4.0). Signals. Example. See Fig. 3.4, p.188 and Fig. 6.1, p.425 of text. - PowerPoint PPT PresentationTRANSCRIPT
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6.0 Time/Frequency Characterization of Signals/Systems
6.1 Magnitude and Phase for Signals and Systems
jjj eYnyeHnheXnx
jYtyjHthjXtx
, , ,
, , ,
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Signals
jXjejXjX
jX : magnitude of each frequency component
jX : phase of each frequency component
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𝜔
−𝜔0
𝜔0
𝐼𝑚
𝑅𝑒
cos𝜔0(𝑡−𝑡 0) sin 𝜔0 𝑡
cos𝜔0𝑡
𝑥 (𝑡− 𝑡0 )↔𝑒− 𝑗 𝜔0𝑡 ⋅ 𝑋 ( 𝑗 𝜔)
Sinusoidals (p.68 of 4.0)
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SignalsExample
321 6cos3
24cos2cos
2
11 ttttX
See Fig. 3.4, p.188 and Fig. 6.1, p.425 of text
– change in phase leads to change in time-domain characteristics
– human auditory system is relatively insensitive to phase of sound signals
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SignalsExample : pictures as 2-dim signals signals dim-2 : , 21 ttx
1
2121
212121
2
2211
2211
,
,,
jX
dtedtettx
dtdtettxjjX
t
tjtj
ttj
– most important visual information in edges and regions of high contrast
– regions of max/min intensity are where different frequency components are in phase See Fig. 6.2, p.426~427 of text
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Two-dim Fourier Transform
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Two-dim Fourier Transform
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Systems
jHjXjY
jHjXjY
jHjXjY
log log log
jH : scaling of different frequency components
jH : phase shift of different frequency components
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SystemsLinear Phase (phase shift linear in frequency) means
constant delay for all frequency components
00
00
,
,0
0
njHeeHnnxny
tjHejHttxtynjj
tj
– slope of the phase function in frequency is the delay– a nonlinear phase may be decomposed into a linear
part plus a nonlinear part
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Time Shift (P.19 of 4.0)
Time Shift jXettx tjF 0
0
linear phase shift (linear in frequency) with amplitude unchanged
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Linear Phase for Time Delay
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Time Shift
𝜔
𝐼𝑚𝐼𝑚
𝐼𝑚𝑅𝑒
𝑅𝑒
𝑅𝑒
(P.20 of 4.0)
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SystemsGroup Delay at frequency ω
jHdd
common delay for frequencies near ω
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SystemsExamples– See Fig. 6.5, p.434 of text
Dispersion : different frequency components delayed by different intervals
later parts of the impulse response oscillates at near 50Hz
– Group delay/magnitude function for Bell System telephone lines
See Fig. 6.6, p.435 of text
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00 , nnkxnyttkxty
SystemsBode Plots
log vs. , log jHjH
ω not in log scale, finite within [ -, ]
Discrete-time Case
Condition for Distortionless
– Constant magnitude plus linear phase in signal band
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Distortionless
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6.2 Filtering
Ideal FiltersLow pass Filters as an example
– Continuous-time or Discrete-timeSee Figs. 6.10, 6.12, pp.440,442 of text
mainlobe, sidelobe, mainlobe width c
1
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1011
0
0𝜔𝜔
𝐹𝑡
𝑝 (𝑡)
𝐻 ( 𝑗 𝜔)
𝑃 ( 𝑗 𝜔)
𝐻 ( 𝑗 𝜔)
Filtering of Signals (p.55 of 4.0)
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Ideal FiltersLow pass Filters as an example– with a linear phase or constant delay
See Figs. 6.11, 6.13, pp.441,442 of text
– causality issue– implementation issues
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Realizable Lowpass Filter
00
0
𝜔
𝐹
𝑡
h1(𝑡) |𝐻1( 𝑗 𝜔)|
𝑇−𝑇
𝑇 2𝑇
h2(𝑡) 𝐹 {|𝐻 2( 𝑗 𝜔)|∠𝐻2( 𝑗 𝜔)
(p.58 of 4.0)
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Ideal FiltersLow pass Filters as an example– step response
n
m
tmhnsdhts ,
See Fig. 6.14, p.443 of text
overshoot, oscillatory behavior, rise time c
1
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Nonideal FiltersFrequency Domain Specification
(Lowpass as an example)
δ1(passband ripple), δ2(stopband ripple)
ωp(passband edge), ωs(stopband edge)
ωs - ωp(transition band)
phase characteristics
magnitudecharacteristics
See Fig. 6.16, p.445 of text
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δ1(passband ripple), δ2(stopband ripple)
ωp(passband edge), ωs(stopband edge)
ωs - ωp(transition band)
magnitudecharacteristics
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Nonideal FiltersTime Domain Behavior : step response
tr(rise time), δ(ripple), ∆(overshoot)
ωr(ringing frequency), ts(settling time)See Fig. 6.17, p.446 of text
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tr(rise time), δ(ripple), ∆(overshoot)
ωr(ringing frequency), ts(settling time)
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Nonideal FiltersExample: tradeoff between transition band width
(frequency domain) and settling time (time domain)
See Fig. 6.18, p.447 of text
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Nonideal FiltersExamples
2/sin
2/1sin1
1
11
2/
NMeMN
eH
knxMN
ny
MNjj
M
Nk
linear phaseSee Figs. 6.35, 6.36, pp.478,479 of text
A narrower passband requires longer impulse response
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Nonideal FiltersExamples– A more general form
32 , 0
32 ,33/2sin
k
kk
kbkh
knxbny
k
M
Nkk
See Figs. 6.37, 6.38, p.480,481 of textA truncated impulse response for ideal lowpass filter gives much sharper transitionSee Figs. 6.39, p.482 of textvery sharp transition is possible
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Nonideal FiltersTo make a FIR filter causal
Njjj eeHeH
Nnhnh
Nnnh
,0
1
1
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6.3 First/Second-Order Systems Described by Differential/Difference Equations
Continuous-time Systems by Differential Equations
Higher-order systems can usually be represented as combinations of 1st/2nd-order systems
– first-order
– second-order
txtydt
tdy
dt
tdybtkytx
dt
tydm
2
2
See Fig. 6.21, p.451 of text
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Discrete-time Systems by Differential Equations– first-order
– second-order
nxnayny 1
nxnyrnyrny 21cos2 2
See Fig. 6.26, 27, 28, p.462, 463, 464, 465 of text
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First-order Discrete-time system (P.461 of text)
)cos1
sin(tan)(
)cos21(1)(
][)1
1(][][
][
11)(
1 1
1
21
1
2
ωaaeH
ωaaeH
nua
anunhns
nuanh
aeeH
a, nxnayny
j
j
n
n
j
j
x[n]y[n]
Da
y[n-1]
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nanhanh )1)(0()0(
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…
Examples
(p.114 of 2.0)
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Second-order Discrete-time system (P.465 of text)
nurnnh
reeH
nuθnrnh
ereere
erθereH
nxnyrnθyrny
n
j
j
n
jjjj
jj
j
)1(
)1(1)( ,0
sin)1(sin
)(1 )(11
cos211)(
21cos2
2
2
2
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Problem 5.46, p.415 of text
truealso is 1 show
trueis when
2 ,1for true
)1(1]u[
)!1(!)!1(
textof P.385 5.13, example ,)(][
)1(1]u[)1(
11]u[
Fn
F
2Fn
Fn
kr
kr
r
en
rnrn
dedXjnnx
enn
en
rj
j
j
j
F
F
F
F
(p.84 of 5.0)
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Problem 6.59, p.508 of text
else ,0
10,1,...n ],[ ][when min
][][][-][
][)(21 (b)
][
][-][
)()( )( (a)
min)()(21
Nduration with FIR causal practical a :)(][
system ideal desired a :)(][
d2
2
Nd
20
d
1
0
2d
222
d
22
F
Fd
N-nhnh
nhnhnhnh
nedeE
ene
enhnh
eHeHeE
deHeH
eHnh
eHnh
nn
N
n
n
j
nj
n
nj
n
jjd
j
jjd
j
jd
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Problem 6.64, p.511 of text
(a)
(b)
(c)
(4.3.3 of Table 4.1)
2Mn 0,][
0n 0,][ causal, is ][
Mnabout symmetric:][][
0nabout symmetric:][
even and real ][even and real )(
)(][ ),(][
integer positive:M even, and real:)(
,)()(
FF
nh
nhnh
Mnhnh
nh
nheH
eHnheHnh
eH
eeHeH
r
r
rj
r
jrr
j
jr
jMjr
j
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