EE2010/IM2004 Signals and Systems

Academic Year: Semester 1

Part I: Weeks 1-5

Lecturer: Prof. Ma Kai-Kuang Office: S2-B2C-83

Tel: 6790-6366 Email: ekkma@ntu.edu.sg

• Notes and the attached tutorial set are from Assoc/Prof Teh Kah Chan. • Refer to NTUlearn regularly for important announcement, including rules

and assessment criteria.

Subject Outline

• Signals and Systems

– Classification of Signals

– Elementary and Singularity Signals

– Operations on Signals

– Properties of Systems

• Linear Time-Invariant (LTI) Systems

– Continuous-Time and Discrete-Time LTI Systems

– Convolution

– LTI System Properties

– Correlation Functions

2

Textbook1. M. J. Roberts, Fundamentals of Signals and Systems,McGraw-Hill, International Edition, 2008. (TK5102.9.R646F)

References1. M. J. Roberts, Signals and Systems, McGraw-Hill, InternationalEdition, 2003. (TK5102.9.R63)2. A. V. Oppenheim and A. S. Willsky, Signals and Systems,Prentice-Hall, 2nd Edition, 1997. (QA402.P62)3. S. Haykin and B. V. Veen, Signals and Systems, Wiley, 2ndEdition, 2003. (TK5102.5.H419)4. S. S. Soliman and M. D. Srinath, Continuous and Discrete Signalsand Systems, Prentice-Hall, 2nd Edition, 1998. (TK5102.9.S686)5. B. P. Lathi, Linear Systems and Signals, Oxford University Press,1st Edition, 2002. (TK5102.5.L352)

3

Overviews of Signals and Systems

TransducerInput Signal

Input

Transmitter

Transmitted Signal

Channel and NoiseDistortion

Received Signal

TransducerOutput

OutputSignal

Receiver

Input Message

MessageOutput

Figure �� A typical signal and system example

�

Classi�cation of Signals

� Continuous�Time vs Discrete�Time Signal

� Continuous�Value vs Discrete�Value Signal

� Deterministic vs Random Signal

� Even vs Odd Signal

� Periodic vs Aperiodic Signal

� Energy�Type vs Power�Type Signal

�

Continuous�Time vs Discrete�Time Signal

� Continuous�Time �CT� Signal� A signal x�t� that is speci�ed for

all value of time t

� Discrete�Time �DT� Signal� A signal y�n� that is speci�ed only

for integer value of n

0 -1 75

t

20 14 63

t n

( )x y [n]

Figure �� Continuous�time vs discrete�time signal

�

Example �� Sketch the waveforms of the CT signal x�t� � t and DT

signal x�n� � n� respectively

(t ) x[n]x

-3

.50

5

-5-5

-1-2-323

0 1 2 3t n-2-1

1

Figure �� Examples of CT and DT signals

�

Continuous�Value vs Discrete�Value Signal

� Continuous�Value Signal� A signal x�t� whose amplitude can take

on any value

� Discrete�Value Signal� A signal y�t� whose amplitude can take on

only a �nite number of values0 0 Tt t

x( ) y ( )

T2

tt

Figure �� Continuous�value vs discrete�value signal

�

Example �� Sketch the waveforms of the continuous�value signal

x�t� � A sin���f�t� and discrete�value signal y�n� � ����n�

respectively

12

3-1

t

-2-3

1

-1

00

A

t n

( ) y[n]

-A

x

Figure �� Examples of continuous�value vs discrete�value signals

Deterministic vs Random Signal

� Deterministic Signal� A signal x�t� that can be mathematically

modeled explicitly as a function of time� i e � x�t� � A sin���f�t�

� Random Signal� A signal y�t� that is known only in terms of

probabilistic description� i e � noise

0 0

A

t-A

t

x( t) y ( )t

Figure �� Deterministic vs random signal

��

Even vs Odd Signal

� Even Signal� A signal xe�t� that satis�es the condition

xe�t� � xe��t�

� Odd Signal� A signal xo�t� that satis�es the condition

xo�t� � �xo��t�

00 t t

ex ( )t ( )toxFigure �� Even vs odd signal

��

� Any deterministic signal x�t� can be decomposed into sum of an

even and an odd signalx�t� � xe�t� � xo�t�

where

xe�t� �

���x�t� � x��t��

and

xo�t� �

���x�t�� x��t��

��

� The product of two even signals is an even signal

� The product of two odd signals is an even signal

� The product of an even signal and an odd signal is an odd signal

� Note that

Z T�

�T�xe�t�dt � �

Z T�

�

xe�t�dt

and

Z T�

�T�xo�t�dt � �

��

Example �� Show that the signal x�t� � A sin���f�t� is an odd signal

Since

x��t� � A sin���f���t��

� �A sin���f�t�

� �x�t�

hence� x�t� is an odd signal

0 t

)x (A

-A

t

Figure �� An odd signal example

��

Example 4: Find the even and odd components of the signalx(t) = cos(t) + sin(t) cos(t).

The even component of x(t) is

xe(t) =12

[x(t) + x(−t)]

=12

[cos(t) + sin(t) cos(t) + cos(−t) + sin(−t) cos(−t)]

= cos(t)

The odd component of x(t) is

xo(t) =12

[x(t) − x(−t)]

=12

[cos(t) + sin(t) cos(t) − cos(−t) − sin(−t) cos(−t)]

= sin(t) cos(t)

15

Example 5: Evaluate∫ T0

−T0x(t)dt where x(t) = t3 cos3(10t).

Since

x(−t) = (−t)3 cos3[10(−t)]

= −t3 cos3(10t)

= −x(t)

hence, x(t) is an odd signal. Thus,∫ T0

−T0

x(t)dt = 0

16

Periodic vs Aperiodic Signal

� Periodic Signal� A signal x�t� with a constant period � � T� ��

that

x�t� � x�t� T��� �� � t ��

For a discrete�time signal� the constant period is an integer

� � K� �� thatx�n� � x�n�K��� �� � n ��

� Aperiodic Signal� A signal y�t� or y�n� that does not satisfy the

above equation

��

6... ...

......0

1 32 4 5

n

6 7 0 1 2

0 2

0

0

0 0

254 7

3

t t

nnK

T

T T

x( )

x[ ] y [ ]

y( )t t

n

=0

Figure � Periodic vs aperiodic signal

��

Energy�Type vs Power�Type Signal

� Energy�Type Signal

� A signal x�t� or x�n� that has �nite energy� i e � � � Ex ���

where

CT signal� Ex �

Z�

��

jx�t�j�dt

DT signal� Ex �

�Xn��

jx�n�j�

� Power�Type Signal

� A signal x�t� or x�n� that has �nite power� i e � � � Px ���

where

�

CT signal� Px � limT��

�T

Z T��

�T��jx�t�j�dt

DT signal� Px � limK��

�

�K � �

KXn�K

jx�n�j�

� Note that if x�t� or x�n� is a periodic signal with period T� or

K�� respectively� then

CT signal� Px �

�T�

Z t��T�

t�

jx�t�j� dt

DT signal� Px �

�K�

k�K���Xnk

jx�n�j�

with any real value of t� and any integer value of k

��

Example �� Determine the energy and power of the periodic signal

x�t� � A cos���f�t��

Ex �

Z�

��

jx�t�j�dt

�

Z�

��

jA cos���f�t�j�dt � �

Px � limT��

�T

Z T��

�T��jx�t�j�dt �

�T�

Z T���

�T���jx�t�j�dt

�

�T�

Z T���

�T���jA cos���f�t�j�dt �

A��

Hence� x�t� is a power�type signal In general� power�type signals are

periodic signals

��

Example �� Determine the energy and power of the signal

y�t� � exp��jtj��Ey �

Z�

��

jy�t�j�dt

�

Z�

��

jexp��jtj�j�dt

� ��Z�

�

exp���t�dt � �

Py � limT��

�T

Z T��

�T��jy�t�j�dt

� limT��

�T�Ey � �

Hence� y�t� is an energy�type signal In general� energy�type signals

are aperiodic signals

��

Example �� Determine the energy and power of the discrete�time

periodic signal x�n� � A sin���n����

Ex �

�Xn��

jx�n�j�

�

�Xn��

jA sin���n���j�

� �

Px �

�K�

k�K���Xnk

jx�n�j� �

��

�Xn�jA sin���n���j�

�

A����

�� � �� � �� � ������

�

A��

Hence� x�n� is a power�type signal ��

Example � A simpli�ed transmitter model of a digital

communication system is shown in Figure ��� determine the

classi�cations of each signal

cos (2π f0 t n] ]nn]

n]

)AQuantizationSampling

Ideal

Discrete Noise

x (t) = x[ x[[y

[w

Figure ��� Transmitter model of a digital communication system

��

The waveforms of various signals are shown in Figure ��

� x�t� is a continuous�time� continuous�value� deterministic� even�

periodic� and power�type signal

� x�n� � x�nTs� is a discrete�time� continuous�value� deterministic�

even� periodic� and power�type signal

� �x�n� is a discrete�time� discrete�value� deterministic� even�

periodic� and power�type signal

� w�n� is a discrete�time� continuous�value� random� and aperiodic

signal

� y�n� is a discrete�time� continuous�value� random� and aperiodic

signal

��

[ ]nxt

0 0.

0.

0

t

n

n

n

[ ]ny

x( )

x [ ]n

Figure ��� Waveforms of various signals for Example

��

Elementary and Singularity Signals

� Exponential signal

x�t� � A exp �at�

� x�t� is growing if a � �

� x�t� is decaying if a � �

00

1

1

tt

a 0

exp( exp(

> a<

at)at)

0

Figure ��� Exponential signal

��

� Sinusoidal signal

x�t� � A cos ���f�t� �� or A sin ���f�t� ��

where A is the amplitude� f� is the frequency in Hertz� and � is

the phase angle in radians

� A sinusoidal signal is periodic with period T� � ��f�

T0

... ...t0

cosA

A (2π )f0 t

-A

Figure ��� CT sinusoidal signal

��

� The discrete time version of the sinusoidal signal is

x�n� � A cos�

��nK�

� ��

or A sin�

��nK�

� ��

where A is the amplitude� K� is a positive integer de�ned as the

fundamental period� and � is the phase angle in radians

[n]=x 8(cos π2 )

0 8. . . ..... ...

A

n

nA

-A

Figure ��� DT sinusoidal signal

�

� Complex exponential signal

A exp�j��f�t� � A cos ���f�t� � jA sin ���f�t�

� The magnitude of complex exponential signal is given by

jA exp�j��f�t�j � A

� The sinusoidal signal can be expressed as

A cos ���f�t� �� � �fA exp�j��f�t� exp�j��g

and

A sin ���f�t� �� � �fA exp�j��f�t� exp�j��g

��

Example ��� Sketch the function x�t� � � exp��at�� cos �����t�

for t � � Assume that a � �

5)

0 0.1 0.2 0.3 t

-5

(x tFigure ��� An exponentially damped sinusoidal signal

��

� The DT unit impulse �or Dirac Delta� function ��n� is de�ned as

��n� �

���

�� n � ��

�� n �� ��

δ n]

3 4 1 2-1. . . .

[

.

3

. .1

. . .421-1 0 0 3 nn

A

[ ]nδAFigure ��� DT impulse functions

��

� The CT unit impulse �or Dirac Delta� function ��t� is de�ned as

��t� �

���

�� t � ��

�� t �� ��

0 0

1

0T

A

) (t

t t

δ( )0TδAtFigure ��� CT impulse functions

��

� Properties of the CT impulse function

� Property �

Z�

��

��t�dt � �

� Property �

x�t�� � �t� T�� � x�T��� � �t� T��

� Property �

Z�

��

x�t�� � �t� T�� dt � x�T��

��

� The CT unit step function u�t� is de�ned as

u�t� �

���

�� t � ��

�� t � ��

0

1

t

t( )u

Figure ��� A CT unit step function

��

� The DT unit step function u�n� is de�ned as

u�n� �

���

�� n � ��

�� n � ��

n]u

.[

.n

. ......2 3

. .0

1

6541

Figure �� A DT unit step function

��

� The CT signum function sgn�t� is de�ned as

sgn�t� �

�������

�� t � ��

�� t � ��

��� t � ��

-10

1)

t

sgn(t

Figure ��� A CT signum function

��

� The DT signum function sgn�t� is de�ned as

sgn�n� �

�������

�� n � ��

�� n � ��

��� n � ��

......

-40

]1

-1-2-3-5-6

-1

.1 2 3 4 65 n

sgn[n

Figure ��� A DT signum function

��

� The CT unit rectangular function rect�t

T

is de�ned as

rect�

tT

��

���

�� jtj � T���

�� otherwise

( )rect T

0

1

22T T t

t

Figure ��� A CT unit rectangular function

�

� The DT unit rectangular function rect�n

K�

�assume that K is

even� is de�ned asrecth n

Ki

�

���

�� jnj � K���

�� otherwise

rect[ ]

..n

. . . .......0-1

1

12 2K K

nK

Figure ��� A DT unit rectangular function

��

� The sinc function sinc�t� is de�ned as

sinc�t� �

sin��t�

�t

-1-2

)

-3-4 0 1 2 3 4

1

sinc

t

t(

Figure ��� A sinc function

��

Example ��� The function x�t� � �� sinc�t� is sampled at every

Ts � ��� second interval to produce the sampled signal xs�t�� sketch

the waveforms for x�t� and xs�t�� respectively

xs�t� �

�Xn��

x�t�� ��t� nTs�

�

�Xn��

x�nTs�� ��t� nTs�

�

�Xn��

�� sinc�nTs�� ��t� nTs�

��

1 2 3 4

1 2 3 4

0

0

t

. . . . . . . .

5

5

-1-2-3-4

t

t

-4 -3 -2 -1

xs

) =t(x 5sinc(

)(

t)

Figure ��� Waveforms for x�t� and xs�t�

��

Operations on Signals

� Amplitude scaling� The operation Ax�t� �or Ax�n�� is to multiply

the amplitude of x�t� �or x�n�� by an amount A

3

(t) (t) (t)x

0 210 21 0 12

t t t

2

4

x x2

-2

-4-3

1.5Figure ��� Amplitude scaling of signals

��

� Time shifting� The operation x�t� T � �or x�n�K�� is to shift

x�t� �or x�n�� by an amount T �or K�

2

3 3 3

.10-1

0-1 10 0.5 2.5

0.5

0 1 2 30 1 2

0 2

2 2

t t t

n n n

t

x n[ ] x[n

t t+1)

x[n+1]

)(x (x ) (x

]1

Figure ��� Time shifting of signals

��

Example ��� Show that rect�t

T

� u�

t� T�

� u�

t� T�

=( )rect

0

0

0

1

1

1

22

2

T

2t

t

tT T

T

t

T

u( t + )2T

u( t

u( t + )2T u(t 2

T )

2T )

Figure ��� Example on time shifting operation

��

� CT time scaling� The operation x�t�a� is to scale x�t� by a

� It expands the function horizontally by a factor jaj

� If a � �� the function will be also time inverted

)2

0 2 0 4

t

-1

)

0

A A A

t t t

x (t /2)t (x(xFigure �� CT time scaling of signals

��

� DT time scaling� x�Kn� or x�n�K� where K is an integer

� x�Kn� � Time compression or decimation

� x�n�K� � Time expansion

[n]

0 0-1

3 4 5 6

24

.6

x

.

]

...2

65431 212

-2-2 4 6 83 421-1 0n n n

x 2n [n/2]x[Figure ��� DT time scaling of signals

��

Example ��� If x�t� � ���� rect�t

�

as shown in Figure ��� sketch

the waveform y�t� � ��x�t��

�

0-2 2

0.5

x

t

t)(Figure ��� Example of operations on signals

�

t( )x2

2t 2(x )2)=t(y

/2

)

-4 4

-2 6

0.5

0

-1

0-2 2

0

-1

t

t

t

x t(

Figure ��� Example of operations on signals

��

Continuous�Time and Discrete�Time Systems

� A system refers to any physical device �i e � communication

channels� �lters� that produces an output signal y�t� in response

to an input signal x�t�

H

H

x[

y

[y ]

x t)( t)(

n] n

Figure ��� Block diagram representation of a system

��

Properties of Systems

� Stability

� A system is said to be bounded�input bounded�output �BIBO�

stable if and only if every bounded input �i e � jx�t�j �� for

all t� or jx�n�j �� for all n� results in bounded output

� An example of a BIBO stable system

y�n� � rnx�n�u�n�� jrj � �

� An example of a BIBO unstable system

y�n� � rnx�n�u�n�� jrj � �

��

� Memory

� A system is said to possess memory if its output signal depends

on past or future values of the input signal

� An example of a system with memory

y�n� � x�n� � x�n� �� � x�n� ��

� A system is memoryless if its output signal depends only on the

present value of the input signal

� An example of a memoryless system

y�t� � x��t�

��

� Causality

� A system is causal if the present value of the output signal

depends only on the present or past values of the input signal

� An example of a causal system

y�n� �

���x�n� � x�n� �� � x�n� ���

� A system is noncausal if the present value of the output signal

depends on the future values of the input signal

� A noncausal system is not physically realizable in real time

� An example of a noncausal system

y�n� �

���x�n� �� � x�n� � x�n� ���

��

� Linearity

� A system is linear if the principle of superposition holds� i e � if

input signal is x��t� � a�x��t� � a�x��t�� then the output signal

is y��t� � a�y��t� � a�y��t� for any constants a� and a�

= =H1 2+a x a1( )t x2( )t 1a +y a( )t 2 ( )

)

ty2( )ty3x )t3( 1

11 H Hx y x y( ( ( (2 2t) t) t) tFigure ��� A linear system

��

Example ��� For the system as shown in Figure ��� determine

whether it is a linear system

H ( )y = x (t )2 t)(x t

Figure ��� A linear system example

��

In this case� the principle of superposition holds� hence it is a

linear system x )(t x ( )t2 H 2( )t2x)(ty =21 )(ty =

H )(ty =3 +a1x1( )t2 a2

1

2

y

x ( )t2= 1 +xa1 ( )t 2a x ( )t2

1x (2t)H

x )(t3

= 1 +a1 ( )t 2a ( )t2y

Figure ��� A linear system example

��

� Time Invariant

� A system is time invariant if for any delayed input x�t� T ��

the output is delayed by the same amount y�t� T �

( )x1 =

t)(y

t)(y1 = (y tt T

x

)H

H

x ( t T )

t( )Figure ��� A time invariant system

��

Example ��� For the system as shown in Figure �� with

y�t� � x�t� � c� where c is an arbitrary constant� determine

whether it is a time invariant system

H = +t t t) )(x )(y (x c

Figure ��� A time invariant system example

�

In this case� the system is time invariant

t)(1yx( t T )=t)(x1

t)(x

=

c

H +

H =t)(y )t(x +

( t T )y=x ( t T ) c

Figure �� A time invariant system example

��

� Linear Time Invariant �LTI�

� A system is linear time invariant if it satis�es both conditions

of linear and time invariance

� A LTI system can be analyzed in both time domain and

frequency domainH H1( )tx 1y ( )t 2( )ty

( )tx3 = 1 1 1) + 2a x T a( t 2

t

2)x T(t H ( )ty3 = 1 1 )1 + 2a y T a(t 2 2y T(t )

2( )xFigure ��� A LTI system

��

Example ��� Determine whether the system given by

y�t� � x��t� in Example �� is a LTI system

From Example ��� the system is linear However� the system is

not time invariant� hence it is not a LTI system

H( t T )x( t)x1 = (y T )t(2 t

y

T )x( t)1y =

( t)x H =t) x (2t)(

Figure ��� A non�LTI system example

��

Analysis of DT and CT LTI Systems

� Any LTI system can be uniquely de�ned by its impulse response

H

Hδ( h(

[]δ [ h

t) t)

n]n

Figure ��� Impulse response of a LTI system

��

� The output of any LTI system is the convolution of the input

signal and its impulse response

(t)

h[n]

x ( ) = *( )y x ( )h (

h

)ht t t t

x[n] = *[n]y x[n] [n]

Figure ��� System response of a LTI system

��

� The discrete time convolution �convolution sum� is de�ned as

y�n� � x�n� h�n� �

�Xm��

x�m�h�n�m�

� The continuous time convolution �convolution integral� is de�ned

as

y�t� � x�t� h�t� �

Z�

��

x���h�t� ��d�

��

Example ��� Sketch the waveform of y�n� � x�n� h�n� using the

graphical approach for convolution sum

0 0

1 1

]

2

1 1-1 22-1 n n

[x n] [h n

Figure ��� Example on convolution sum

��

y�n� � x�n� h�n� �

�Xm��

x�m�h�n�m�

m

m

m

n 1n 2

.

.1

-1

2

21

0 1

0

1

0

10-2 -1

2

1

-1

-1

-1

11

1

1

12

1

2

1

2

-2 0-1

-3 1

-3 -2 -1 0-1 21

0

0

m m

nmm

nmm

h[ h[n]

h[x [ ]m(i)n=-1,

(ii)x [ ]m h[ [y n]

[y n]

]

]

]n=0,

1 m

n2

��

[n]

y [n]

y [n]

x[m]

x[m]

x[m](iii)n=1,

(iv)n=2,

(v)n=3,

m

m

m

y

]

.

.

1 ..

1

1

1

0

-1

0

0 0-2 0

100

1

0 2

2

1

1

1

2

1

-1

-1

-1

-1 2 21

1

321

32

321

2

2

2

1

2 21 1

0 1 2 3 2

-1-2

-1

-1

-1 3

m m n

m m n

m m n

h[

h[

h[

1 ]

2 ]

3

Figure ��� Solution for example on convolution sum

��

Example ��� Show that ��n� �� x�n� � x�n� �� where x�n� is shown

in Figure ��

[n]x

n

21

1-1 2-2 0

Figure ��� Example on convolution with Delta function

�

Firstly� using the graphical approach Denote

y�n� � ��n� �� x�n� �

�Xm��

��m� ��x�n�m�

m

m

m

n+1n 1n 2

121

21

121

21

2 2

-1

x

0

-1-2

1

-1

-1 1 -1 1 2

.. .

...3 0

-2 -1

2

0

2

11

1 1

1 2

2

-20

0 -1 0 1 03 2-2

-1-2 032

m

nm

nmm

m

m

x [ [x

y [nx [δ [m(i)n=0,

(ii) =1,n δ [m y [n

] n ]m

2]

2] 1 ]

] ]

]

n

[��

2

2

m

m

m

2

2

22

21

2

2 2

. . .

..

. . .

1 2 3 2

0 2 3 4 -1 1

]

2 3

2 0 1 2 3 4 0-1 1

2

5 4

-1 1 -1 -2 1

11

-1 1 1

11

1

1

1

-1 1

1

-13 020 0

00 3

30 32

m n

m n

m nm

m

m

δ[m x [ y [n]

y [n]

= x[y [n] n

x [3

[4xδ[m(v)n=4,

δ[m(iv)n=3,

(iii) =2,n] 2 ]

]

]

]

]

Figure ��� Solution for example on convolution with Delta function

��

Alternatively� based on the de�nition of convolution sum� we have

y�n� � ��n� �� x�n� �

�Xm��

��m� ��x�n�m�

Since

��m� �� �

���

�� m � ��

�� m �� ��

Hence

y�n� �

�Xm��

��m� ��x�n�m�

� x�n� ����

Example �� Sketch the waveform of y�t� � x�t� x�t� using the

graphical approach for convolution integral

0-2 2

3)

t

x (t

Figure ��� Example on convolution integral

��

y�t� � x�t� x�t� �

Z�

��

x���x�t� ��d�

x( −τ)

-2 2 τ

τ

τ

τ

τ

3

3

3 3

3

0

0-2 2

0

2-2 t

-2

0

0

2 -4 0 t

y

x (−τ)

(i) t<-4,

+2-2

( )x

+2

x ( −τ)( )x

( )x

-2

τ

τ t

tt τ

)(

t t

t

��

τ

(

( −τ)

( −τ)

τ

τ

3

3

3

0

0-2

-2

2

36

36

36

-2 0-4

-4 02

2

<4,

-4 0

4

4

0 t

t

t

x

x

x

(ii) -4<

-2 +2

x( )

(iv)tx( )

-2 +2

y ( )t

y ( )t

y ( )tτt−τ)

x( )

+2-2

τ t

tτ>4,

(iii) 0<

t t

tt

tt

t<0,

t

Figure �� Solution for example on convolution integral

��

Example ��� Sketch the waveform of y�t� � x��t� x��t� using the

graphical approach for convolution integral

50

1

5 0

)A

ttA

( 1

1

x 2x( t) t

Figure ��� Example on convolution integral

��

y�t� � x��t� x��t� �

Z�

��

x����x��t� ��d�

-5

-5

-5

τ

τ

τ τ

τ

0

0

0 0

0

0

5

5

5 t

A

A

)A

A

x

x

x

A

A

t

t

(i) t

x ( −τ)

x ( −τ)

x

y

1

1

1

1

1

1

2

2

2

(τ)

(τ)

(−τ)

t

1

(τ)<0,

t

t

t

1

1 (t��

)ty

( )ty

( )ty

x 2(t τ)

x 2(t τ)A1

A1

x 2(t τ)

(

<10,

τ

τ

τ50

0

00

0

0

10

105

5

5

5

5

-5

-5

-5

1

1

1

2

2

2

1

1

5

5

5

t

t

t

A

A

A

t t

A

tt

t t

(ii) 0<

(iii) 5<

(iv) t >10,

1

1

1

x (τ)

x (τ)

x (τ)

A

A

1

1At <5,

t

Figure ��� Solution for example on convolution integral

��

Properties of Convolution

� Commutative

x��n� x��n� � x��n� x��n�

x��t� x��t� � x��t� x��t�

� Distributive

x��n� fx��n� � x��n�g � x��n� x��n� � x��n� x��n�

x��t� �x��t� � x��t�� � x��t� x��t� � x��t� x��t�

�

� Associativex��n� fx��n� x��n�g � fx��n� x��n�g x��n�

x��t� �x��t� x��t�� � �x��t� x��t�� x��t�

� Convolution with Delta function

x�n� ��n�K�� � x�n�K��

x�t� ��t� T�� � x�t� T��

��

Example ��� Show that x�t� ��t� T�� � x�t� T�� using the

de�nition of convolution integral

Based on the de�nition of convolution integral� we have

y�t� � x�t� ��t� T��

�

Z�

��

��� � T��x�t� ��d�

�

Z�

��

��� � T��x�t� T��d�

� x�t� T��Z�

��

��� � T��d�

� x�t� T����

Step Response of LTI Systems

� The step response is de�ned as the output of the system with the

unit step function as input signal

� Step response of a DT system

s�n� � u�n� h�n� �

�Xm��

h�m�u�n�m�

�

nXm��

h�m�

� Step response of a CT system

s�t� � u�t� h�t� �

Z�

��

h���u�t� ��d�

�

Z t��

h���d���

Example ��� Find the step response of the one�stage RC �lter as

shown in Figure ��� where the impulse response is given by

h�t� � �RC � exp�

� tRC

u�t� Ct

R( )u ( )ts

Figure ��� A simple one�stage RC �lter

��

In this case� the step response is given by

s�t� � u�t� h�t�

�

Z t��

h���d�

�

Z t��

�RC� exp

�

�RC

�u���d�

�

�RC

Z t�

exp

�

�RC

�d�

�

���

�� exp�

� tRC

� t � ��

�� t � ��

��

Properties of LTI Systems

� Memoryless LTI Systems

� A LTI system is memoryless if and only if its impulse

response is given by

DT system� h�n� � c��n�

CT system� h�t� � c��t�

where c is an arbitrary constant

� All memoryless LTI systems simply perform scalar

multiplication on the input��

� Causal LTI Systems

� A LTI system is causal if and only if its impulse response

satis�es the following condition

DT system� h�n� � �� for n � �

CT system� h�t� � �� for t � �

� A causal LTI system cannot generate an output before the

input is applied

��

� Stable LTI Systems

� A LTI system is BIBO stable if and only if its impulse response

satis�es the following condition

DT system�

�Xn��

jh�n�j � �

CT system�

Z�

��

jh�t�jdt � �

� An example of a stable LTI system

h�n� � nu�n�� jj � �

��

Example ��� Determine whether the system with impulse response

h�t� � exp��at�u�t� where a � � is �i� memoryless� �ii� causal� and

�iii� BIBO stable

�i� The system is not memoryless since h�t� �� c��t�

�ii� The system is causal since h�t� � � for t � �

�iii� The system is BIBO stable since

Z�

��

jh�t�jdt �

Z�

�

exp��at�dt

� ��a � �

��

System Interconnections

� Parallel Connection

+

+

h1

h2

+ h2h1

(

(x

(

(

(t) (t)

y(

t y

[n]y+

+h2[n]

h1[n]x[n]

x[n] +h1[n] h2[n] [n]y

t)

t)

t)

(t)

t)

)

x

Figure ��� Parallel connection of systems

�

� Cascade Connection

h1( h2( (y

y* h2(h1( )

(x

( t t) (

tt ) ))h2[n] [n]yh1[n]x [n]

x [n] *h1[n] h2[n] [n]y

t)

x t) t)

tFigure ��� Cascade connection of systems

�

Example ��� Determine the equivalent impulse response h�n� of the

overall system as shown in Figure ��� where h��n� � u�n��

h��n� � u�n� ��� u�n�� h��n� � ��n� ��� and h��n� � nu�n�

[ ]nx [ ]ny

[ ]nh1

h [ ]n2

[ ]n

+

h3

[ ]nh4

+

+

Figure ��� Example on interconnections of systems

�

The resultant overall system impulse response is

h�n� � fh��n� � h��n�g h��n�� h��n�

� fu�n� � u�n� ��� u�n�g ��n� ��� nu�n�

� u�n� �� ��n� ��� nu�n�

� u�n�� nu�n�

� f�� ngu�n��

Di�erential and Di�erence Equations

� Block diagram representation of di�erence equation

y�n� � x�n�� �y�n� �� � �y�n� ��

]

+

+

+3

2

D

D

x yn n[ ] [

Figure ��� Block diagram representation of di�erence equation

�

� Block diagram representation of di�erential equation

y�t� � x�t���

��

ddty�t���

��

d�dt�y�t�

)+

+

45

12

ddt

ddt

) y(x t t(

Figure ��� Block diagram representation of di�erential equation

�

Example ��� Find the block diagram representation of di�erential

equation for the simple one�stage RC low�pass �lter as shown in

Figure ��

R

)C( yt)x (t

Figure ��� A simple one�stage RC �lter

�

In this case� the input and output relation of the �lter is given by

x�t� � RC �

ddty�t� � y�t�

y�t� � x�t��RC �

ddty�t�

+ y

RC ddt

t( )x ( )t

Figure �� Block diagram representation of di�erential equation for

the RC �lter

�

Correlation Function

� The correlation function is a mathematical expression of how

correlated two signals are as a function of how much one of them

is shifted

� The correlation function between two functions is a function of

the amount of shift

� Two types of correlation functions

� Autocorrelation function

� Cross correlation function�

� Autocorrelation function

� The autocorrelation is the correlation of a function with itself

� For an energy�type signal x�n� or x�t�

DT signal� Rxx�m� �

�Xn��

x�n�x��n�m�

CT signal� Rxx��� �

Z�

��

x�t�x��t� ��dt

where x��t� denotes the complex conjugation of x�t�

� For a power�type signal x�n� or x�t�

DT signal� Rxx�m� � limK��

�

�K � �

KXn�K

x�n�x��n�m�

CT signal� Rxx��� � limT��

�T

Z T��

�T��x�t�x��t� ��dt

�

� For an energy�type signal x�n� or x�t�

DT signal� Ex � Rxx���

CT signal� Ex � Rxx���

� For a power�type signal x�n� or x�t�

DT signal� Px � Rxx���

CT signal� Px � Rxx���

� Properties of autocorrelation function

� The peak of autocorrelation function occurs at the zero shift

DT signal� Rxx��� � Rxx�m�

CT signal� Rxx��� � Rxx���

� Autocorrelation functions are even functions

DT signal� Rxx�m� � Rxx��m�

CT signal� Rxx��� � Rxx����

� A time shift in the signal does not change its autocorrelation

function� i e � the autocorrelation functions of x�t� and x�t� T �

are the same

���

Example ��� Find the autocorrelation function and power of the

sinusoidal signal x�t� � A sin���f�t�

Since x�t� is a power�type signal� the autocorrelation function is

given by

Rxx��� � limT��

�T

Z T��

�T��x�t�x��t� ��dt

� limT��

�T

Z T��

�T��A� � sin���f�t�� sin���f��t� ���dt

� limT��

�T

Z T��

�T��A�

��cos���f���� cos���f���t� ���� dt

�

A��� cos���f���

���

The power of signal x�t� is given by

Px � Rxx���

�

A��

Alternatively� based on the de�nition of power� we have

Px � limT��

�T

Z T��

�T��jx�t�j�dt

�

�T�

Z T���

�T���jx�t�j�dt

�

�T�

Z T���

�T���jA sin���f�t�j� dt

�

A��

���

Example ��� Find the autocorrelation function and power of the

sinusoidal signal y�t� � A sin ���f��t� T ��� where T is an

arbitrary constant delay

Denote � � ��f�T � we have

y�t� � A sin���f�t� ��f�T �

� A sin���f�t� ��

Since y�t� is a power�type signal� the autocorrelation function is

given by

Ryy��� � limT��

�T

Z T��

�T��y�t�y��t� ��dt

���

� limT��

�T

Z T��

�T��A� � sin���f�t� ��� sin���f��t� ��� ��dt

� limT��

�T

Z T��

�T��A�

��cos���f���� cos���f���t� ��� ���� dt

�

A��� cos���f���

Comparing with the results with Example ��� we conclude that

the autocorrelation functions of x�t� and x�t� T � are the same

���

Example ��� Find the autocorrelation function of the signal x�n�

as shown in Figure �� using the graphical approach

[n]x

n

1

2

0 1

Figure ��� Example on autocorrelation function of a DT signal

���

Since x�n� is an energy�type signal�

Rxx�m� �

�Xn��

x�n�x��n�m� �

�Xn��

x�n�x�n�m�

m 1 m+1m

.

.

1

0 1

0

-2

-1-2

0 2

0

21

-1

2

xx

2

21

12

21

21

21

642

642

-2

-2 -1 1

11 0

-1

-1

2

1 2

2-1 0

0 22 -11

1

n

n

n

n

n

n

m

m

(ii)

(i)m=2,

x[n]

x[n]

x[n] n+1

x n+2

x[

R

R=1,m

]n+m

0

[ ]x

[ ][m]

[m]

xx

���

]m

[ ]m

[ ]m

.

..

.

0

2

2

-2

-2

12

12

12

[

1

xx

6

2

2

6

64

1 2 32

0

-2

24

42

21

12

1-1

-1

-1

-1 1

1 12-1 -2 -1 21

-2 -1 0 1 2 -1 0 21

0 0 210-1

00 m

m

m

n

n

n

n

n

n

(iii)

(iv)

(v)

R

R

R

x [=-1,m

x [m=0,

x [=-2,m

x

x

x [n

n

n]

]

] n]

n

n

1][

[ ]2

xx

xx

Figure ��� Solution for example on autocorrelation function

���

� Cross correlation function

� The cross correlation is the correlation of two di�erent functions

� For energy�type signals x�n� and y�n� �or x�t� and y�t��

DT signal� Rxy�m� �

�Xn��

x�n�y��n�m�

CT signal� Rxy��� �

Z�

��

x�t�y��t� ��dt

� For power�type signals x�n� and y�n� �or x�t� and y�t��

DT signal� Rxy�m� � limK��

�

�K � �

KXn�K

x�n�y��n�m�

CT signal� Rxy��� � limT��

�T

Z T��

�T��x�t�y��t� ��dt

���

Example �� Find the cross correlation function between the two

signals x�t� � exp�j��f�t� and y�t� � exp�j���f�t�

Since x�t� and y�t� are power�type signals�

Rxy��� � limT��

�T

Z T��

�T��x�t�y��t� ��dt

� limT��

�T

Z T��

�T��exp�j��f�t�� exp��j���f��t� ���dt

� limT��

�T

Z T��

�T��exp��j��f�t�� exp��j��f���dt

� exp��j��f��� limT��

�T

Z T��

�T���cos���f�t�� j sin���f�t�� dt

� �

��

Example ��� Find the cross correlation function between the two

signals x�t� and y�t� as shown in Figure �� using the graphical

approach

2

2

20

2

-2

)

-2 -2 0t t

t ( )x ty(

Figure ��� Example on cross correlation function of CT signals

���

Since x�t� and y�t� are energy�type signals�

Rxy��� �

Z�

��

x�t�y��t� ��dt �

Z�

��

x�t�y�t� ��dt

2

-2 2

τ

τ

2

0

2

-20

2

-22

0 2-2-2

0

-8

0

(ii) 2<x

<4,

+2

2+2

τ

ττ

(τ)

(τ)

4-2

42

0

2τ τ

(i) >4,

2τ τ

τ

tt

t

t

Rxy

Rxy

x

+τ)

t+τ)

+2τ

+τ)(t) (y t

ty (

t(y

τ

τ

)t(x

)(���

)(x

t)(x

t)(x

t)(x

τ

τ

τ-2

τ-2

20-2

-2

t

0

τ

2

0-2 2

0

-8

-8

-8

-8

2

2

2

-2

-2

(iii)

(iv)

(v)

0< <2,

-2< <0,

-4< <-2,

(vi) <-4,

2

2+2

+2

τ+2

2

2τ

τ

τ

ττ

τ

τ

(τ)2

-2

0 2

0

2

2

2

4

4

4

42

0-2

-2-4 0

-2-4

(τ)

8(τ)

(τ)8

τ

t

t

t

t

Rxy

Rxy

Rxy

Rxy

( ty +τ)

( ty +τ)

( ty +τ)

( ty +τ)τ

τ

8

+2τ

τ

τ

τ

Figure ��� Solution for example on cross correlation function

���

1

NANYANG TECHNOLOGICAL UNIVERSITY SCHOOL OF ELECTRICAL & ELECTRONIC ENGINEERING

EE2010/IM2004 SIGNALS AND SYSTEMS

TUTORIAL 1

Q1.1: Determine the even and odd components of the DT function Sketch the waveforms of xe[n], xo[n], and x[n].

Q1.2: Determine whether the following signals are energy-type or power-type signals:

(a)

(b)

Q1.3: A sinusoidal signal is passed through a half-wave rectifier circuit to produce:

(a) Sketch the waveforms of x(t) and y(t), respectively. (b) Determine the energy and power levels of x(t) and y(t), respectively.

≤≤−

= otherwise. ,0

,22for ,)(

|| tetx

t

.)1(][ nny −=

.)1(][ nnnx −+=

)20sin(2)( ttx π=

>

= otherwise. ,0

,0)( if ),()(

txtxty

.1, ,2, (b) :3.1.1, (b) .0,1 (a) :2.1

.][,)1(][ :1.1:Answers

4

=∞==∞=

=∞==−=

=−=

yyxx

yyxx

on

e

PEPEQPEPeEQ

nnxnxQ

2

NANYANG TECHNOLOGICAL UNIVERSITY SCHOOL OF ELECTRICAL & ELECTRONIC ENGINEERING

EE2010/IM2004 SIGNALS AND SYSTEMS

TUTORIAL 2

Q2.1: Assume that sketch the following waveforms and evaluate

(a)

(b)

(c)

Q2.2: Sketch the waveforms of

and

Q2.3: Assuming that the signal v(t) is an energy-type signal and its energy is denoted as Ev, determine the energy levels of the following signals as a function of Ev.

(a)

(b)

)]()([)( where,2

4)( and )()( 0

1

10 TtututtvtxtynTtvtx

n−−=

+

−=−= ∑−=.20 =T

),2sin()( 0tftv π= ∫∞

∞−.)( dtty

./1 where,2

rect)()( 000

fTTttvtw =

×=

( ).)(sgn)( twtx =

∑∞

−∞=

−×=n

nTttxty ).4/()()( 0δ

).(3)( tvtx −=

).3()( −= tvty

. (b) .9 (a) :3.2

.0)( :1.2

:Answers

-

vyvx EEEEQ

dttyQ

==

=∫∞

∞

3

NANYANG TECHNOLOGICAL UNIVERSITY SCHOOL OF ELECTRICAL & ELECTRONIC ENGINEERING

EE2010/IM2004 SIGNALS AND SYSTEMS

TUTORIAL 3

Q3.1: Evaluate the convolution sum

Q3.2: For the system as shown in Figure Q3.2, evaluate the system output y(t) where

Figure Q3.2.

Q3.3: Determine the properties of the system shown in Figure Q3.3 in terms of linearity and time invariance.

Figure Q3.3.

−=

=∗= otherwise. ,0

.1,0,1for ,1][ where][][][

nnxnxnxny

.4

2rect )( and 2

3rect 2

1rect )( 21

−

=

−

−

−

=tAtxtAtAtx

invariant. not timebut linear is system The :3.3:Answers

Q

4

NANYANG TECHNOLOGICAL UNIVERSITY SCHOOL OF ELECTRICAL & ELECTRONIC ENGINEERING

EE2010/IM2004 SIGNALS AND SYSTEMS

TUTORIAL 4

Q4.1: Radio signals can travel through a wireless channel by more than one path, with different time delays and attenuations (known as channel fading). Consider a three-path case with a system impulse response given by

Assume that the input signal is given by

(a) Determine whether the system is memoryless, causal, and stable. (b) Determine the system output y(t) and sketch its waveform. (c) Express the energy of y(t) as a function of the energy of x(t).

Q4.2: Evaluate the step response of the system with impulse response given by

Q4.3: (a) Find the equivalent impulse response h[n] of the overall system as shown in Figure Q4.3, where

(b) Determine whether the overall system is memoryless, causal, and stable. (c) Determine the step response of the overall system.

Figure Q4.3.

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NANYANG TECHNOLOGICAL UNIVERSITY SCHOOL OF ELECTRICAL & ELECTRONIC ENGINEERING

EE2010/IM2004 SIGNALS AND SYSTEMS

TUTORIAL 5

Q5.1: Show that the cross-correlation function between any energy-type signal x[n] and the delta function is equal to x[-m].

Q5.2: Find the cross-correlation function between the two signals

Figure Q5.2.

Q5.3: Consider two complex-valued signals given by

(a) Sketch the amplitude plots of x(t) and y(t), respectively. (b) Determine the power levels of x(t) and y(t), respectively. (c) Find the cross-correlation function of x(t) and y(t). (d) Comment on the result obtained in part (c).

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