CHAPTER 2 Discrete-Time Signals and Systems in the Time-Domain YANG Jian [email protected] School of Information Science and Technology Yunnan University

CHAPTER 2 Discrete-Time Signals and Systems

in the Time-Domain

CHAPTER 2 Discrete-Time Signals and Systems

in the Time-Domain

YANG Jian

[email protected]

School of Information Science and Technology

Yunnan University

云南大学滇池学院课程：现代信号处理数字信号处理 Digital Signal Processing 2

OutlineOutline

• Discrete-Time Signals

• Typical Sequences and Sequence Representation

• The Sampling Process

• Discrete-Time Systems

• Time-Domain Characterization of LTI Discrete-Time Systems

• Finite-Dimensional LTI Discrete-Time Systems

• Correlation of Signals

• Summary


Discrete-Time SignalsDiscrete-Time Signals

• Basic signals

– Unit sample or unit impulse sequence

– Unit step sequence

– Exponential sequence

• Signal classification

– Continuous-time / discrete-time signals

– Deterministic / random signals

– Energy signals

• signals with finite energy

– Power signals

• signals with finite power

– Energy signals have zero power, and power signals have infinite energy



• Time-Domain Representation

– Sequence of numbers:

• — sequence

• — samples

• — sample value or nth samples, a real or complex value

– Figure of sequence:

• is defined only for integer value of

( )x n

n

( )x n

( ) ,0.3,0.76,0,1, 2,0.92,x n

( )x n n



• Operation on sequences

– Basic operation

• Adder / Subtraction:

• Scalar multiplication ( gain / attenuation ):

• Delay / Advance:

– Combination of Basic Operations

• Multiplier:

• Linear combination:

1 2( ) ( ) ( )x n x n y n

1 2( ) ( ) ( )x n x n y n

( ) ( )Ax n y n

0( ) ( )x n n y n

1 1 2 2( ) ( 3) ( )a x n a x n y n



• Operation on sequences

– Sampling Rate Alteration ( special operations of for discrete-time signals )

• Up-sampling:

• Down-sampling:

( / ), 0, , 2 , ,( )

0, ,

x n L n L Ly n

otherwise

( ) ( )y n x nM



• Classification of Sequences

– The number of sequences: finite / infinite

• Finite-length sequences:

– Symmetry

• conjugate-symmetric ( even ):

• conjugate-antisymmetric ( odd ):

– Periodity: periodic / aperiodic

• Periodic sequence:

1 2( ) 0, x n n N and n N

( ) ( ), , integer.x n x n kN for all n k is any

( ) ( )x n x n

( ) ( )x n x n




– Energy and Power Signals

2

2

: ( )

1: lim ( )

2 1

xn

K

kn K

energy x n

power P x nK

: < ,

: ,x

x

energy signals P

power signals P




– Other types of Classification

• Bounded:

• Absolutely summable:

• Square-summable:

( ) xx n B

( )n

x n

2( )

n

x n


Typical Sequences and Sequence Representation


• Some Basic Sequences

– Unite sample sequence:

• An arbitrary sequence can be represented by unite sample sequence in time-domain

– Unite step sequence:

1, 0( )

0, 0

nn

n

1, 0( )

0, 0

nn

n

( ) ( ), ( ) ( ) ( 1)n

k

n k n n n




• Sinusoidal and Exponential Sequences

– The real sinusoidal sequence:

– The exponential sequence:

• The sinusoidal sequence are periodic of period N as long as is an integer multiple of . The smallest possible N is the fundamental period of the sequence.

0( ) cos( ), x n A n n

0 0 0 0

0 0

( ) ( )

0 0

( )

cos( ) sin( )

j n n j nn

n n

x n A Ae A e e

A e n j A e n

0 N 2




• Some Typical Sequences

– Regular window sequence:

– Real exponential sequence:

• Representation of an Arbitrary Sequence

– An arbitrary sequence can be represented as a weight sum of basic sequence and its delayed version.

1, 0 1( )

0, R

n Nw n

otherwise

( ) ( )nx n a n

( ) ( ) ( )k

x n x k n k


The Sampling ProcessThe Sampling Process

• Uniform sampling:

– Often the discrete-time sequence is developed by uniformly sampling a continuous-time signal : ( )ax t

( ) ( )ax n x nT

• the sampling

frequency

• the sampling

angular frequency

1 ,TF T

2 ,T TF


The Sampling ProcessThe Sampling Process

• Aliasing:

– When , a continuous-time sinusoidal signal of higher frequency would acquire the identity of a sinusoidal sequence of lower frequency after sampling.

e.g.

2T MAX


Discrete-Time SystemDiscrete-Time System

• Discrete-time system

• Simple Discrete-Time Systems

– The accumulator

– The M-point moving-average filter

– The factor-of-L interpolator

( ) [ ( )] -y n H x n n

H [ ] Output y(n)Input x(n)



• Classification of Discrete-Time System

– Linear system:

– Shift-Invariant System:

– LTI System:

The linear time-invariable discrete-time system satisfies both the linear and the time-invariable properties.

1 1 2 2

1 2 1 2

( ) ( ), ( ) ( ),

( ) ( ) ( ) ( )

if x n y n x n y n

then x n x n y n y n

0 0 ( ) ( ), ( ) ( )if x n y n then x n n y n n




– Causal System:

In a causal discrete-time system, the th output sample

depends only on input samples for and

does notdepend on input samples for .

1 1 2 2

1 2

1 2

( ) ( ) ( ) ( )

{ ( ) ( ), }

{ ( ) ( ), }

if u n y n and u n y n

then u n u n for n N

implies also that y n y n for n N

0n

0( )y n ( )x n0n n

0n n




– Stable System:

Definition of bounded-input, bounded-output ( BIBO ) stable.

• Passive and Lossless Systems

– The passivity:

– The losslessness: the same energy

x( ) ,

( ) ,

x

y

if n B n

then y n B n

2 2( ) ( )

n n

y n x n



• Impulse and Step Responses

– Input sequence → output sequence

– Impulse response :

– Step response :

( )h n ( ) ( )n h n

( )s n ( ) ( )n s n


Time-Domain Characterization of LTI Discrete-Time Systems


• Input-Output Relationship

– The response y(n) of the LTI discrete-time system to x(n) will be given by the convolution sum:

– The operation

• Step 1, time-reverse:

• Step 2, shift n sampling period:

• Step 3, product:

• Step 4, summing all samples:

( ) ( ) ( ) ( ) ( ) ( ) ( )k k

y n x k h n k x n k h k x n h n

( ) ( )h k h k

( ) ( )h k h n k

( ) ( ) ( )x k h n k v k

( ) ( ) ( )k k

v k x k h n k




• Some useful properties of the convolution operation

– Commutative:

– Associative for stable and single-sided sequences:

– Distributive:

1 2 2 1( ) ( ) ( ) ( )x n x n x n x n

1 2 3 1 2 3( ) [ ( ) ( )] [ ( ) ( )] ( )]x n x n x n x n x n x n

1 2 3 1 2 1 3( ) [ ( ) ( )] ( ) ( ) ( ) ( )]x n x n x n x n x n x n x n




• Simple Interconnection Schemes

– Cascade Connection:

– Parallel Connection:

– Inverse System:

1 2( ) ( ) ( )h n h n h n

1 2( ) ( ) ( )h n h n h n

1 2( ) ( ) ( )h n h n n

1 2 2 1 1 2( ) ( ) ( ) ( ) ( ) ( )h n h n h n h n h n h n

1( )h n

2( )h n

1 2( ) ( )h n h n




• Stability Condition in Terms of the Impulse Response

– An LTI digital filter is BIBO stable if only if its impulse response sequence is absolutely summable, i.e.:

• Causality Condition in Terms of the Impulse Response

– An LTI discrete-time system is causal if and only if its impulse response is a causal sequence satisfying the condition:

( )h n

( )n

S h n

( ) 0, 0h k for k


Finite-Dimensional LTI Discrete-Time Systems


• The difference equation:

– An important subclass of LTI discrete-time systems is characterized by a linear constant coefficient difference equation of the form:

– The order of the system is given by max( N, M )

0 0

( ) ( )N M

k kk k

d y n k p x n k




• Total Solution Calculation

– The complementary solution

• The homogeneous difference equation:

• The characteristic equations:




• Total Solution Calculation

– The particular solution is of the same form as specified input .

– The total solution:

( )py n

( )x n

( ) ( ) ( )c py n y n y n




• Zero-Input Response and Zero-State Response

– zero-input response = complementary solution with initials;

– zero-state response = the convolution sum of x(n) and h(n).

( ) 0 ( ),

( ),

: ( ) ( )

zi

zs

zi zs

if x n the solution is y n

and if applying the specified input with

all initial conditions set to zero the solution is y n

then the total solution is y n y n




• Impulse Response Calculation




• Impulse Response Calculation

– The solutions




• Location of Roots of Characteristic Equation for BIBO Stability

– A casual LTI system characteristic of a linear constant coefficient difference equation is BIBO stable, if the magnitude of each of the roots its characteristic equation is less than 1.

– The necessary and sufficient condition:

1k




• Classification of LTI System

– Based on impulse response length

• Finite impulse response ( FIR ):

• Infinite impulse response ( IIR ):

1 2 1 2( ) 0,h n for n N and n N , with N N

2

1

( ) ( ) ( )N

k N

y n h k x n k

0

( ) ( ) ( )n

k

y n x k h n k




• Classification of LTI System

– Based on the output calculation process

• Non-recursive system:

If the output sample can be calculated sequentially, knowing only the present and pass input samples.

• Recursive system:

If the computation of the output involves past output samples.

– Remarks:

• FIR — Non-recursive

• IIR — Recursive


Correlation of SignalsCorrelation of Signals

• Definitions

– A measure of similarity between a pair of energy signals, x(n) and y(n), is given by the cross-correlation sequence defined by:

– The autocorrelation sequence of x(n) is given by:

( ) ( ) ( ), 0, 1, 2,xyn

r l x n y n l l

( ) ( ) ( ) ( ) ( ) ( )yx xyn m

r l y n x n l y m l x m r l

( ) ( ) ( )xxn

r l x n x n l

( ) ( ) [ ( )] ( ) ( )xyn

r l y n x l n y l x l



• Properties of Autocorrelation and Cross-correlation Sequences

– Set and as energies of the sequences x(n) and y(n) , then we can get

or equivalently

– If y(n) = x(n), then

• The sample value of the autocorrelation sequence has its max value at zero lag ( l = 0 ).

(0) 0xx xr (0) 0yy yr

2(0) (0) ( ) 0xx yy xyr r r l

( ) (0) (0)xy xx yy x yr l r r

( ) (0)xy xx xr l r



• Properties of Autocorrelation and Cross-correlation Sequences

– If , where N is integer and b>0 is an arbitrary number. In this case , so

( ) ( )y n bx n N

(0) ( ) (0)xx xy xxbr r l br

2y xb



• Normalized Forms of Correlation:

• Correlation Computation for Power and Periodic Signals

– Power signals:

– Periodic signals:

( )( )( ) , ( )

(0) (0) (0)xyxx

xx xyxx xx yy

r lr ll l

r r r

1 1( ) lim ( ) ( ), ( ) lim ( ) ( )

2 1 2 1

K K

xy xxK Kn K n K

r l x n y n l r l x n x n l K K

0 0

1 1( ) ( ) ( ), ( ) ( ) ( )

N N

xy xxn n

r l x n y n l r l x n x n lN N


SummarySummary

• The LTI system has numerous applications in practice.

• The LTI system can be described by an input-output relation composed of a linear constant coefficient difference equation.

• The LTI discrete-time system is usually classified in terms of the length of its impulse response.

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CHAPTER 2 Discrete-Time Signals and Systems in the Time-Domain YANG Jian [email protected] School of Information Science and Technology Yunnan University