time - domain analysis of discrete - time systems the first half of chapter 3

Time - Domain Analysis Of

Discrete - Time Systems

The first half of Chapter 3

3.1. Introduction

Discrete-TimeSignal

Discrete-TimeSignal

A Sequence of numbers inherently Discrete-Time sampling Continuous-Time signals, ex)

A Sequence of numbers inherently Discrete-Time sampling Continuous-Time signals, ex) ][],[ nynx

Discrete-TimeSystem

Discrete-TimeSystem

Inputs and outputs are Discrete-Time signals.A Discrete-Time(DT) signal is a sequence of numbersDT system processes a sequence of numbers to yield another sequence .

Inputs and outputs are Discrete-Time signals.A Discrete-Time(DT) signal is a sequence of numbersDT system processes a sequence of numbers to yield another sequence .

][nx][ny

2Signal and System II

3.1-0 Discrete signals and systems

3.1. Introduction

Signal and System II 3

Discrete-TimeSignal by sampling CT signal

Discrete-TimeSignal by sampling CT signal

T = 0.1 : Sampling interval n : Discrete variable taking on integer values. conversion to process CT signal by DT system.

T = 0.1 : Sampling interval n : Discrete variable taking on integer values. conversion to process CT signal by DT system.

][)(

)(1.0 nxeenTx

etxnnT

t

CD

DC ,

3.1. Introduction


3.1-1 Size of DT Signal3.1-1 Size of DT Signal

Energy Signal

(3.1)

Signal amp. 0 as

Power Signal

(3.2)

for periodic signals, one period time averaging.

• A DT signal can either be an energy signal or power signal• Some signals are neither energy nor power signal.

Energy Signal

(3.1)

Signal amp. 0 as

Power Signal

(3.2)

for periodic signals, one period time averaging.

• A DT signal can either be an energy signal or power signal• Some signals are neither energy nor power signal.

n

x nxE2][

n xE

N

NN

x nxN

P2][

12

1lim

3.1. Introduction


Example 3.1 (1)Example 3.1 (1)

3.1. Introduction


Example 3.1 (2)Example 3.1 (2)

3.2. Useful Signal Operations


Shifting

Shifting

]5[][ nxnxS

Left shift Advance

Right shift Delay

Mnn Mnn 0M

Figure 3.4 time shift



Time Reversal

,

anchor point : n = 0

cf)

Time Reversal

,

anchor point : n = 0

cf)

][][ nxnxr nn

][nx

Figure 3.4 time reversal



Example 3.2

1.

2.

Example 3.2

1.

2.

)]([][][ knxknxnkx

][][][)(

knxknxnxreverse

nn

kadvance

knn

)]([][][)(

knxnxnxkdelay

knn

reverse

nn



Sampling Rate Alteration : Decimation and Interpolation decimation ( down sampling )

, M must be integer values (3.3)

Sampling Rate Alteration : Decimation and Interpolation decimation ( down sampling )

, M must be integer values (3.3)][][ Mnxnxd



interpolation ( expanding )

L must be integer value more than 2. (3.4)

interpolation ( expanding )

L must be integer value more than 2. (3.4)

otherwise

LLnLnxnxe

,....,2,,0

0

]/[][

3.3. Some Useful DT Signal Models


3.3-1 DT Impulse Function3.3-1 DT Impulse Function

00

01][

n

nn (3.5)



3.3-2 DT Unit Stop Function3.3-2 DT Unit Stop Function

00

01][

n

nnu (3.6)



Example 3.3 (1)

for all

Example 3.3 (1)

for all

][][][][ 321 nxnxnxnx

]8[2])11[]5[(4])5[][( nnunununun

n



Example 3.3 (2)

Example 3.3 (2)



3.3-3 DT Exponential γn3.3-3 DT Exponential γn

CT Exponential

( or )

ex)

( or )

Nature of

CT Exponential

( or )

ex)

( or )

Nature of

te

tte er )ln( tte )7408.0(3.0 tt e 386.14

nne e )ln( n

aajba

nnjbanjban

eeee

eeee

1

)()(



plane plane

exp. dec. LHP Inside the unit circle

exp. inc. RHP Outside the unit circle

osc. Imaginary axis Unit circle

0a

0a

0a

n

n

1



3.3-4 DT Sinusoid cos(Ωn+Θ) 3.3-4 DT Sinusoid cos(Ωn+Θ)

DT Sinusoid : : Amplitude : phase in radians : angle in radians : radians per sample : DT freq. (radians/2π) per sample or cycles for sample

, : period (samples/cycle)

DT Sinusoid : : Amplitude : phase in radians : angle in radians : radians per sample : DT freq. (radians/2π) per sample or cycles for sample

, : period (samples/cycle)

)2cos()cos( Fncncc

nF

0

1

NF 0N



About Figure 3.10

(cycles/sample)

, same freq.

Sampled CT Sinusoid Yields a DT Sinusoid CT sinusoid

About Figure 3.10

(cycles/sample)

, same freq.

Sampled CT Sinusoid Yields a DT Sinusoid CT sinusoid

)412

cos(

n12

24

1

2

F

)cos()cos( nn

,coscos][cos nwnTnxwtsampling

SecondsT wT



3.3-5 DT Complex Exponential ejΩn3.3-5 DT Complex Exponential ejΩn

freq :

a point on a unit circle at an angle of

freq :

a point on a unit circle at an angle of

njne

njnenj

nj

sincos

sincos

nr

ree jnj

,1n

3.4. Examples of DT Systems


3.4-03.4-0

Example 3.4 (Savings Account) input : a deposit in a bank regularly at an interval T. output : balance interest : per dollar per period T. balance y[n] = previous balance y[n-1] + interest on y[n-1] + deposit x[n]

or

(3.9a) delay operator form (causal)

withdrawal : negative deposit, loan payment problem : or (3.9b) advance operator form (noncausal)

Example 3.4 (Savings Account) input : a deposit in a bank regularly at an interval T. output : balance interest : per dollar per period T. balance y[n] = previous balance y[n-1] + interest on y[n-1] + deposit x[n]

or

(3.9a) delay operator form (causal)

withdrawal : negative deposit, loan payment problem : or (3.9b) advance operator form (noncausal)

][]1[)1(][]1[]1[][ nxnyrnxnrynyny

ranxnayny 1],[]1[][

1:]1[][]1[ nnnxnayny

][nx][ny

r

My ]0[ Mx ]0[][nx



Block diagram

(a)addition, (b)scalar multiplication, (c)delay, (d)pickoff node

Block diagram

(a)addition, (b)scalar multiplication, (c)delay, (d)pickoff node



Example 3.5 (Scalar Estimate)

: th semester : students enrolled in a course requiring a certain textbook : new copies of the book sold in the th semester.

¼ of students resell the text at the end of the semester. book life is three semesters.

(3.10a)

(3.10b)

Block diagram

Example 3.5 (Scalar Estimate)

: th semester : students enrolled in a course requiring a certain textbook : new copies of the book sold in the th semester.

¼ of students resell the text at the end of the semester. book life is three semesters.

(3.10a)

(3.10b)

Block diagram

n n

n][nx][ny

][]2[16

1]1[

4

1][ nxnynyny

]2[][16

1]1[

4

1]2[ nxnynyny

][]2[16

1]1[

4

1][ nxnynyny

(3.10c)



Example 3.6 (1) (Digital Differentiator) Design a DT system to differentiate CT signals. Signal bandwidth is below 20kHz ( audio system )

Example 3.6 (1) (Digital Differentiator) Design a DT system to differentiate CT signals. Signal bandwidth is below 20kHz ( audio system )

)(][),(][ nTynynTxnx

))1(()(1

lim)(

,)(

0TnxnTx

Tdt

dxnTy

nTtdt

dxty

TnTt

: backward difference



Example 3.6 (2)

: Sufficiently small

Example 3.6 (2)

: Sufficiently small

]1[][1

]1[][1

lim][0

nxnxT

nxnxT

nyT

T

sfreqhighest

T 25000,40

1

.2

1



Example 3.6 (3)

< in case of >

Forward difference

Example 3.6 (3)

< in case of >

Forward difference

ttx )(

11)1(1

][

0)(][

nTnnTT

ny

tnTttxnxnTtnTt

][]1[1

][ nxnxT

ny



Example 3.7 ( Digital Integrator ) Design a digital integrator as in example 3.6

: accumulator

block diagram : similar to that of Figure 3.12 ( )

Example 3.7 ( Digital Integrator ) Design a digital integrator as in example 3.6

: accumulator

block diagram : similar to that of Figure 3.12 ( )

0],[lim][

)(lim)(

,)()(

0

0

TkxTny

TkTxnTy

nTtdxty

kT

kT

t

n

k

kxT ][

][]1[][ nTxnyny

1a



Recursive & Non-recursive Forms of Difference Equation

(3.14a), (3.14b)

Kinship of Difference Equations to Differential Equations Differential eq. can be approximated by a difference eq. of the same order. The approximation can be made as close to the exact answer as possible by choosing sufficiently small value for .

Order of a Difference Equation The highest – order difference of the output or input signal, whichever is higher.

Analog, Digital, CT & DT Systems DT, CT the nature of horizontal axis

Analog, Digital the nature of vertical axis.-DT System : Digital Filter-CT System : Analog Filter-C/D, D/C : A/D, D/A

Recursive & Non-recursive Forms of Difference Equation

(3.14a), (3.14b)

Kinship of Difference Equations to Differential Equations Differential eq. can be approximated by a difference eq. of the same order. The approximation can be made as close to the exact answer as possible by choosing sufficiently small value for .

Order of a Difference Equation The highest – order difference of the output or input signal, whichever is higher.

Analog, Digital, CT & DT Systems DT, CT the nature of horizontal axis

Analog, Digital the nature of vertical axis.-DT System : Digital Filter-CT System : Analog Filter-C/D, D/C : A/D, D/A

k

kxTny ][][ ][]1[][ nTxnyny

T



Advantage of DSP1. Less sensitive to change in the component parameter values.2. Do not require any factory adjustment , easily duplicated in volume,

single chip (VLSI)3. Flexible by changing the program4. A greater variety of filters5. Easy and inexperience storage without deterioration6. Extremely low error rates, high fidelity and privacy in coding7. Serve a number of inputs simultaneously by time-sharing,

easier and efficient to multiplex several D. signals on the samechannel

8. Reproduction with extreme reliability

Advantage of DSP1. Less sensitive to change in the component parameter values.2. Do not require any factory adjustment , easily duplicated in volume,

single chip (VLSI)3. Flexible by changing the program4. A greater variety of filters5. Easy and inexperience storage without deterioration6. Extremely low error rates, high fidelity and privacy in coding7. Serve a number of inputs simultaneously by time-sharing,

easier and efficient to multiplex several D. signals on the samechannel

8. Reproduction with extreme reliability

Disadvantage of DSP1. Increased system complexity (A/D, D/A interface)2. Limited range of freq. available in practice (about tens of MHz)3. Power consumption

Disadvantage of DSP1. Increased system complexity (A/D, D/A interface)2. Limited range of freq. available in practice (about tens of MHz)3. Power consumption



3.4-1 Classification of DT Systems3.4-1 Classification of DT Systems

Linearity & Time Invariance

Systems whose parameters do not change with time (n)If the input is delayed by K samples, the output is the same as before but delayed by K samples.

Causal & Noncausal Systems output at any instant depends only on the value of the input for

Invertible & Noninvertible Systems is invertible if an inverse system exists s.t. the cascade of and results in an identity system Ex) unit delay unit advance ( noncausal ) Cf)

Linearity & Time Invariance

Systems whose parameters do not change with time (n)If the input is delayed by K samples, the output is the same as before but delayed by K samples.

Causal & Noncausal Systems output at any instant depends only on the value of the input for

Invertible & Noninvertible Systems is invertible if an inverse system exists s.t. the cascade of and results in an identity system Ex) unit delay unit advance ( noncausal ) Cf)

22112211 ykykxkxk

][][ nxeny n ][1 nx ][][ 012 Nnxnx

kn ][nx kn

S iS SiS

Ex) for and

][][],[cos][],[][ nxnynxnyMnxny



Stable & Unstable Systems (1) The condition of BIBO ( Boundary Input Boundary Output ) and external stability

If is bounded, then , and

Clearly the output is bounded if the summation on the right-hand side is bounded; that is

Stable & Unstable Systems (1) The condition of BIBO ( Boundary Input Boundary Output ) and external stability

If is bounded, then , and

Clearly the output is bounded if the summation on the right-hand side is bounded; that is

mm

m

mnxmhmnxmhny

mnxmhnxnhny

][][][][][

][][][][][

][nx 1][ Kmnx

m

mhKny ][][ 1

2][ Knh

n



Stable & Unstable Systems (2) Internal ( Asymptotic ) Stability For LTID systems,

and

Since the magnitude of is 1, it is not necessary to be considered. Therefore in case of

if as ( stable )

if as ( unstable )

if for all ( unstable )

Stable & Unstable Systems (2) Internal ( Asymptotic ) Stability For LTID systems,

and

Since the magnitude of is 1, it is not necessary to be considered. Therefore in case of

if as ( stable )

if as ( unstable )

if for all ( unstable )

je njnn e

nje n

0,1 n

n ,1

1,1 n

n

n

n



Memoryless Systems & Systems with Memory Memoryless : the response at any instant depends at most on the input at the same instant .

Ex)

With memory : depends on the past, present and future values of the input.

Ex)

Memoryless Systems & Systems with Memory Memoryless : the response at any instant depends at most on the input at the same instant .

Ex)

With memory : depends on the past, present and future values of the input.

Ex)

nn

][sin][ nxny

][]1[][ nxnyny

3.5. DT System Equations


Difference Equations

(3.16)

order :

Causality Condition causality : if not, would depend on if ,

(3.17a) delay operator form

(3.17b)

Difference Equations

(3.16)

order :

Causality Condition causality : if not, would depend on if ,

(3.17a) delay operator form

(3.17b)

][]1[]1[

][][]1[]1[][

11

11

nxbnxbMnxb

MnxbnyanyaNnyaNny

NNMN

MNNN

),max( MN

NM ][ Nny ][ Mnx

NM

][]1[]1[

][][]1[]1[][

11

011

nxbnxbNnxb

NnxbnyanyaNnyaNny

NN

NN

][]1[]1[

][][]1[]1[][

11

011

NnxbNnxbnxb

nxbNnyaNnyanyany

NN

NN



3.5-1 Recursive (Iterative) Solution ofDifference Equations

3.5-1 Recursive (Iterative) Solution ofDifference Equations

: • initial conditions • input • for causality

Example 3.8 (1)

: • initial conditions • input • for causality

Example 3.8 (1)

][]1[]1[

][][]2[]1[][

11

021

NnxbNnxbnxb

nxbNnyanyanyany

NN

N

(3.17c)

]0[y N]0[x0][ nx



Example 3.8 (2) Example 3.8 (2)



Example 3.9

Closed – form solution

Example 3.9

Closed – form solution



Operation Notation Differential eq. Operator D for differentiation Difference eq. Operator E for advancing a sequence.

(3.24a) (3.24b)

(3.24c) (3.24d)

Operation Notation Differential eq. Operator D for differentiation Difference eq. Operator E for advancing a sequence.

(3.24a) (3.24b)

(3.24c) (3.24d)

][][][]1[][]1[

]2[][],1[][ 2

nExnaynEynxnayny

nxnxEnxnEx

][][)( nExnyaE

]2[][16

1]1[

4

1]2[ nxnynyny

][][)16

1

4

1( 22 nxEnyEE

][)(][)( 11

1011

1 nxbEbEbEbnyaEaEaE NNNN

NNNN

][][][][ nxEPnyEQ

NNNN aEaEaEEQ

1

11][

NNNN bEbEbEbEP

1

110][



Response of Linear DT Systems

(3.24a)

(3.24b)

LTID system General solution = ZIR (zero input response) + ZSR (zero state response)

Response of Linear DT Systems

(3.24a)

(3.24b)

LTID system General solution = ZIR (zero input response) + ZSR (zero state response)

][)(

][)(

11

10

11

1

nxbEbEbEb

nyaEaEaE

NNNN

NNNN

][][][][ nxEPnyEQ

time - domain analysis of discrete - time systems the first half of chapter 3

Documents

power signal

introduction signal

useful dt signal models

signal amp

discretetimedt signal

sampling ct signal t

discretetime signals

time shift slide