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Lecture 2:

Discrete-Time Signals & Systems

Reza Mohammadkhani, Digital Signal Processing, 2015University of Kurdistan eng.uok.ac.ir/mohammadkhani

Signal Definition and Examples 2

� Signal: any physical quantity that varies with time,

space, or any other independent variable/variables

Example 1: speech signals

� Intensity of brightness: � �, �Example 2: Black & White Picture

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� �, �

�

�

� Black and white video: � �, �, �

� � �, � � �� �, � , � �, � , �� �, � �Example 3: Color Picture

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� �, � � �, � �� �, �

… full color image5

Signal Types6

� Continuous-Time vs Discrete-Time

� Continuous-Valued vs Discrete-Valued

� Analog: continuous in both time and amplitude

� Digital: discrete in both time and amplitude

� Deterministic vs Random

Discrete-Time Signal7

� A sequence of numbers defined for every integer

value of variable n: � = � , −∞ < < ∞� Can be sampled from a CT signal:� = �� � , −∞ < < ∞

where � is sampling period.

Discrete-Time Signal Representations8

… DT sampled from a CT signal9

Some Elementary DT Signals10

� Unit sample (impulse) sequence

� ≜ �1 = 00 ≠ 0� Unit step sequence

� ≜ �1 ≥ 00 < 0� Exponential sequences� = ���� Sinusoidal� = � cos "# + %

Basic Operations11

� Time-shifting: � − #� Multiplication: �� � …

� Example

� More generally

Types of DT signals12

&' = ( � )'

�*+',' = lim0→'

123 + 1 ( � )

0

�*+0� Energy signal vs Power signal

� Energy signal: 0 < &' < ∞� Power signal: 0 < ,' < ∞

Types of DT signals (2)13

� Even and Odd signals

� Any arbitrary signal � � Even: ∀ ∈ ℤ: � − = � � Odd: ∀ ∈ ℤ: � − = −�

� Periodic vs Aperiodic signals

� Periodic: ∀ ∈ ℤ, ∃ 9 ∈ ℤ � + 9 = �

Discrete-Time Systems14

Discrete-Time System15

� � � �

Examples16

� Ideal Delay System

� = � − # −∞ < < ∞� Moving Average

� = 13: + 3) + 1 ( � − ;0<

=*+0>

� Memoryless Systems

System Properties17

� Linear Systems

� Additivity property:

� �: + �) = � �: } + �{�) = �: + �) � Homogeneity or scaling property� A� = A� � = A�

� Time-Invariant Systems

If � � = � then � � − # = � − #

� = � ) for all integer values of n

System Properties (2)18

� Causality

System output sequence value at every = # only depends on the input sequence values for ≤ #.� forward difference system� = � + 1 − � � backward difference system

� � = � − � − 1� Stability

If � ≤ CD < ∞ for all then � ≤ CE < ∞ for all Bounded-Input Bounded Output (BIBO)

Linear Time-Invariant Systems19

LTI Systems20

Linearity:

Time-invariant:

Example 21

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� Convolution is Commutative:

� Convolution is Distributive:

Properties of LTI systems23

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

h1[n]

x[n] y[n]

h2[n]

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

� Cascade connection of LTI Systems:

Properties of LTI systems (2)24

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

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

h1[n]∗h2[n]x[n] y[n]

Properties of LTI systems (3)25

� Causality of LTI systems:

ℎ ; = 0 for ; < 0

� Stability of LTI systems:

Impulse response is absolute summable:

( ℎ ;'

=*+'< ∞

� � = � ∗ ℎ = ∑ � ; ℎ − ;'=*+'� = ℎ ∗ � = ∑ ℎ ; � − ;'=*+'

Properties of LTI systems (4)26

� Memoryless LTI system

ℎ = I� � � = � ∗ ℎ = ∑ � ; ℎ − ;'=*+'� = ℎ ∗ � = ∑ ℎ ; � − ;'=*+'

� Finite Impulse Response (FIR) systems

� Infinite Impulse Response (IIR) systems

� Example: ℎ = 1/2 ��

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Linear Constant-Coefficient

Difference Equations28

Linear Constant-Coefficient Difference Equations

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� An important class of LTI systems

The output is not uniquely specified for a given input

� The initial conditions are required

� Linearity, time invariance, and causality depend on

the initial conditions

� If initial conditions are assumed to be zero, system is

linear, time invariant, and causal

Example30

� Difference Equation Representation

� ∑ A=� − ;:=*# = ∑ KL� − M#L*#

Solution method 131

Homogeneous equation:

Recursive Computation 32

Frequency-Domain Representations of

Discrete-Time Signals & Systems33

Discrete-Time Fourier Transform34

� Complex exponential eignfunction

� H(ejω) is a complex function of frequency

� Specifies amplitude and phase change of the input

[ ] njenx ω=[ ] [ ] [ ] [ ] )( knj

kk

ekhknxkhny −∞

−∞=

∞

−∞=∑∑ =−= ω

[ ] [ ] ( ) njjnjkjk

eeHeekhny ωωωω =

= −∞

−∞=∑

( ) [ ] kjk

j ekheH ωω −∞

−∞=∑=

Frequency Response35

� If input signals can be represented as a sum of

complex exponentials

� we can determine the output of the system

� Different from continuous-time frequency response

� Discrete-time frequency response is periodic with 2π

[ ] ∑ ωα=k

nj

kkenx

[ ] ( )∑ ωωα=k

njj

kkk eeHny

( )( ) [ ] ( ) [ ] [ ] kjk

kjrk2j

k

kr2j

k

r2j ekheekhekheH ω−∞

−∞=

ω−π−∞

−∞=

π+ω−∞

−∞=

π+ω ∑∑∑ ===

( )( ) ( )ωπ+ω = jr2j eHeH

Discrete-Time Fourier Transform36

� X(ejω) is the Fourier spectrum of the sequence x[n]

� It specifies the magnitude and phase of the sequence

� The phase wraps at 2π hence is not uniquely specified

� The frequency response of a LTI system is the DTFT of the impulse response

( ) [ ] transform) (forward enxeX njn

j ω−∞

−∞=

ω ∑=

[ ] ( ) transform) (inverse deeX2

1nx njj ω

π= ∫

π

π−

ωω

( ) [ ] [ ] ( ) ωπ

== ∫∑π

π−

ωωω−∞

−∞=

ω deeH2

1nh and ekheH njjkj

k

j

DTFT Pair37

( ) [ ] [ ] ( ) ωπ

== ∫∑π

π−

ωωω−∞

−∞=

ω deeX2

1nx and enxeX njjnj

n

j

Existence of DTFT38

� For a given sequence the DTFT exist if the infinite sum convergence

Or

� So the DTFT exists if a given sequence is absolute summable

� All stable systems are absolute summable and have DTFTs

( ) [ ] njn

j enxeX ω−∞

−∞=

ω ∑=

( ) ω∞

DTFT Properties39

Symmetric Sequence and Functions40

Conjugate-symmetricConjugate-antisymmetric

Sequence

Function

[ ] [ ]nxnx *ee −= [ ] [ ]nxnx *oo −−=

[ ] [ ] [ ]nxnxnx oe += [ ] [ ] [ ]( )nxnx21

nx *e −+= [ ] [ ] [ ]( )nxnx2

1nx *o −−=

( ) ( )ω−ω = j*eje eXeX ( ) ( )ω−ω −= j*ojo eXeX

( ) ( ) ( )ωωω += jejoj eXeXeX ( ) ( ) ( )[ ]ω−ωω += j*jje eXeX2

1eX ( ) ( ) ( )[ ]ω−ωω −= j*jjo eXeX

2

1eX

References41

� D. Manolakis and V. Ingle, Applied Digital Signal

Processing, Cambridge University Press, 2011.

� Miki Lustig, EE123 Digital Signal Processing, Lecture

notes, Electrical Engineering and Computer Science,

UC Berkeley, CA, 2012. Available at:http://inst.eecs.berkeley.edu/~ee123/fa12/

� Güner Arslan, EE351M Digital Signal Processing,

Lecture notes, Dept. of Electrical and Computer

Engineering, The University of Texas at Austin, 2007.

Available at:www.ece.utexas.edu/~arslan/351m.html

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