6.0 time/frequency characterization of signals/systems 6.1 magnitude and phase for signals
Post on 24-Feb-2016
31 Views
Preview:
DESCRIPTION
TRANSCRIPT
6.0 Time/Frequency Characterization of Signals/Systems
6.1 Magnitude and Phase for Signals and Systems
jjj eYnyeHnheXnx
jYtyjHthjXtx
, , ,
, , ,
Signals
jXjejXjX
jX : magnitude of each frequency component
jX : phase of each frequency component
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)
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
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
Two-dim Fourier Transform
Two-dim Fourier Transform
Systems
jHjXjY
jHjXjY
jHjXjY
log log log
jH : scaling of different frequency components
jH : phase shift of different frequency components
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
Time Shift (P.19 of 4.0)
Time Shift jXettx tjF 0
0
linear phase shift (linear in frequency) with amplitude unchanged
Linear Phase for Time Delay
Time Shift
𝜔
𝐼𝑚𝐼𝑚
𝐼𝑚𝑅𝑒
𝑅𝑒
𝑅𝑒
(P.20 of 4.0)
SystemsGroup Delay at frequency ω
jHdd
common delay for frequencies near ω
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
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
Distortionless
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
1011
0
0𝜔𝜔
𝐹𝑡
𝑝 (𝑡)
𝐻 ( 𝑗 𝜔)
𝑃 ( 𝑗 𝜔)
𝐻 ( 𝑗 𝜔)
Filtering of Signals (p.55 of 4.0)
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
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)
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
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
δ1(passband ripple), δ2(stopband ripple)
ωp(passband edge), ωs(stopband edge)
ωs - ωp(transition band)
magnitudecharacteristics
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
tr(rise time), δ(ripple), ∆(overshoot)
ωr(ringing frequency), ts(settling time)
Nonideal FiltersExample: tradeoff between transition band width
(frequency domain) and settling time (time domain)
See Fig. 6.18, p.447 of text
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
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
Nonideal FiltersTo make a FIR filter causal
Njjj eeHeH
Nnhnh
Nnnh
,0
1
1
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
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
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]
nanhanh )1)(0()0(
…
Examples
(p.114 of 2.0)
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
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)
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
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
top related