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