hengyong yu, phd associate professor department of...

EECE.3620 Signal and System I

Hengyong Yu, PhDAssociate Professor

Department of Electrical and Computer Engineering University of Massachusetts Lowell


)(tx

][nx

Ch.6. Time and Frequency Characterization of Signals and SystemsCh.6.1. Magnitude-phase representation of Fourier Transform

The magnitude 𝑋 𝑗𝜔 or 𝑋 𝑒𝑗𝜔 provides us

with the information about the relative magnitudes of the complex exponentials that make up the signal.

The phase angle ∢𝑋 𝑗𝜔 or∢𝑋 𝑒𝑗𝜔 provides

us with information concerning the relative phases of those complex exponentials.

In general, changes in the magnitude or phase functions can lead to changes in the time-domain characteristics of the signal.

𝑋 𝑗𝜔 = 𝑋 𝑗𝜔 𝑒𝑗∢𝑋 𝑗𝜔

𝑋 𝑒𝑗𝜔 = 𝑋 𝑒𝑗𝜔 𝑒𝑗∢𝑋 𝑒𝑗𝜔


)(ty)(tx )(th

Ch.6.2. Magnitude-phase representation of frequency response of LTI systems

[ ]y n[ ]x n [ ]h n

( ) ( ) ( )Y j H j X j

( ) ( ) ( )j j jY e H e X e

The magnitude 𝐻 𝑗𝜔 or 𝐻 𝑒𝑗𝜔 is commonly referred

as gain of the system.

The phase angle ∢𝐻 𝑗𝜔 or∢𝐻 𝑒𝑗𝜔 is usually referred to

as phase shift of the system.

∢𝑌 𝑗𝜔 = ∢𝐻 𝑗𝜔 + ∢𝑋 𝑗𝜔

∢𝑌 𝑒𝑗𝜔 = ∢𝐻 𝑒𝑗𝜔 + ∢𝑋 𝑒𝑗𝜔

EECE.3620 Signal and System I Ch.6.2.1. Linear and Nonlinear phase

When the phase shift at the frequency 𝜔 is a linear function of 𝜔, the system produces an output that is simply a time shift of the input.

Recall: 0( )j t

H j e

( ) 1 H j

0( ) ( )y t x t t

Continuous-time signal as an input

Response for a system with linear phase

∢𝐻 𝑗𝜔 = −𝜔𝑡0

EECE.3620 Signal and System I Ch.6.2.1. Linear and Nonlinear phase

When the phase shift is a nonlinear function, it will result in a change in the relative phases of the complex exponential functions. The output signal may look considerably different from the input signal.

Continuous-time signal as an input

Response for a system with nonlinear phase If a system have unity gain, the magnitude of

Fourier transform of the input is unchanged to such a system. Such a system is commonly called an all-pass system. The characteristic of an all-pass system is completely determined by its phase-shift characteristic.

EECE.3620 Signal and System I Ch.6.2.2. Group Delay

Recall: 0( )j t

H j e

0( ) ( )y t x t t

Group delay at a frequency 𝜔 is defined as

∢𝐻 𝑗𝜔 = −𝜔𝑡0

𝑡0 = −𝑑

𝑑𝜔∢𝐻 𝑗𝜔 = −

𝑑

𝑑𝜔−𝜔𝑡0 =

𝑑

𝑑𝜔𝜔𝑡0

𝜏(𝜔) = −𝑑

𝑑𝜔∢𝐻 𝑗𝜔

EECE.3620 Signal and System I Ch.6.2.2. Group Delay

Example 6.1 (Page 432) All-pass system

3

1

)()(i

i jHjH

2

2

1 2( )

1 2

i i i

i

i i i

j jH j

j j

f1 = 50, f2 =150, f3 =300Hz 2i if

1)( jHi

1

2

2 ( / )2 tan

1 ( / )

i i

i

)(1 jH )(2 jH )(3 jH

𝜔 is measured in radians per second𝑓 is measured in cycles per second (Hertz, Hz)

1 2 30.066, 0.033, 0.058

∢𝐻𝑖 𝑗𝜔 =


0 50 100 150 200 250 300 350 4000

0.02

0.04

0.06

0.08

0.1

0 50 100 150 200 250 300 350 400-20

-15

-10

-5

0

0 50 100 150 200 250 300 350 4001

1

1

1

1

Example 6.1 (Page 432) All-pass system

Ch.6.2.2. Group Delay

1)()(3

1

i

i jHjH

0 0.05 0.1 0.15 0.2-200

-100

0

100

200

( )h t

𝐻 𝑗𝜔 =

𝑖=1

3

∢𝐻𝑖 𝑗𝜔

𝜏(𝜔) = −𝑑

𝑑𝜔∢𝐻 𝑗𝜔

EECE.3620 Signal and System I clear all;close all;f1 = 50;w1 = f1*2*pi;f2 = 150; w2 = f2*2*pi;f3 = 300;w3 = f3*2*pi;ks1 = 0.066;ks2 = 0.033;ks3 = 0.058;od = 0.005*2*pi;omega = 0:od:400*2*pi;HW1 =(1+(j*omega./w1).^2-2*j*ks1*(omega./w1));HW1 = HW1./(1+(j*omega./w1).^2+2*j*ks1*(omega./w1));HW2 =(1+(j*omega./w2).^2-2*j*ks2*(omega./w2));HW2 = HW2./(1+(j*omega./w2).^2+2*j*ks2*(omega./w2));HW3 =(1+(j*omega./w3).^2-2*j*ks3*(omega./w3));HW3 = HW3./( 1+(j*omega./w3).^2+2*j*ks3*(omega./w3));HW = HW1.*HW2.*HW3;plot(omega/(2*pi),abs(HW));zoom on;figure(2)AH = unwrap(angle(HW));plot(omega/(2*pi),AH);zoom on;DH = -diff(AH);DH = DH./od;figure(3)plot(omega(2:length(omega))/(2*pi), DH);zoom on;

% inverse Fourier transformt = 0:0.00002:0.2;for n=1:length(t)

h(n) = 0;for m=1:length(omega)

h(n) = h(n)+HW(m)*exp(j*omega(m)*t(n))*od;endh(n) = h(n)./(2*pi);

endfigure(4)plot(t,real(h));zoom on;

Matlab codes for the plots of Example 6.1


1020log ( )H jDecibels (dB) scale:

Ch.6.2.3. Log-magnitude and phase plot

log ( ) log ( ) log ( )Y j H j X j

Bode Plots: Plot of 20𝑙𝑜𝑔10 𝐻(𝑗𝜔) and ∢𝐻(𝑗𝜔)vs. 𝑙𝑜𝑔10(𝜔) for continuous-time systems

Typical Bode Plot for the magnitude of a continuous-time system

Typical Bode Plot for the phase of a continuous-time system

If ℎ(𝑡) is real, then 𝐻(𝑗𝜔)is an even function of 𝜔, and ∢𝐻(𝑗𝜔) is an odd function of 𝜔. It is sufficient to only keep the part of 𝜔 > 0 in the plots.

20𝑙𝑜𝑔10 𝐻(𝑗𝜔) vs. 𝑙𝑜𝑔10(𝜔)

∢𝐻(𝑗𝜔) vs. 𝑙𝑜𝑔10(𝜔)

EECE.3620 Signal and System I Ch.6.2.3. Log-magnitude and phase plot

For discrete-time system, the magnitudes of Fourier transform and frequency response are often displayed in dB.However, a logarithmic frequency scale (Bode plot) is not typically used.

Typical representation of the magnitude of a discrete-time frequency response

Typical representation of the phase of a discrete-time frequency response

20𝑙𝑜𝑔10 𝐻(𝑒𝑗𝜔) vs. 𝜔

∢𝐻(𝑒𝑗𝜔) vs. 𝜔


c

1

c

)( jH

)(th

Ch.6.3. Time-domain Properties of Ideal Frequency-selective Filters

Continuous-time low-pass filter:

o Ideal lowpass filters have perfect frequency selectivity, and they introduce no phase distortion.

o As the bandwidth of the filter increases, the impulse response becomes narrower, and vice versa.

o The step responses overshoot their long-term final values and exhibit oscillatory behavior, frequently referred to as ringing.

( ) ( )

t

s t h d


3

1)( jeH

][nh

Ch.6.3. Time-domain Properties of Ideal Frequency-selective Filters

Discrete-time low-pass filter:

o Ideal lowpass filters have perfect frequency selectivity, and they introduce no phase distortion.

o As the bandwidth of the filter increases, the impulse response becomes narrower, and vice versa.

o The step responses overshoot their long-term final values and exhibit oscillatory behavior, frequently referred to as ringing.

[ ] [ ]n

m

s n h m

EECE.3620 Signal and System I Ch.6.4. Time-domain and Frequency-domain Aspects of Nonideal Filters

o The characteristics of ideal filters are not always desirable in practice.

o Ideal lowpass filters are noncausal, and they may not be implementable.

o Nonideal filters are of considerable practical importance, and such filters are frequently specified or quantified in terms of several parameters.

The students are encouraged to study how to characterize the non-ideal filters for filter designs.

EECE.3620 Signal and System I Ch.6.5. First-order and Second-order Continuous-time Systems

)()()( txtytydt

d

1 1 1( )

1 1H j

j j

First-order Continuous-time System Second-order Continuous-time System

)()()(2)( 22

2

2

txtytydt

dty

dt

dnnn

2

2 2

2

( )2

1

2 1

n

n n

n n

H jj j

j j

o Bode plot can be used to characterize the first and second-order continuous-time systems.

o We can use the bode plots of the first and second-order systems to approximate any complex system with rational frequency response

o The same is for discrete-time systems.

hengyong yu, phd associate professor department of...

Documents