Discrete-time Systems in the Time Domain Domain Chapter 4 Chapter - - PowerPoint PPT Presentation

Discrete-time Systems in the Time Domain Domain

Chapter 4 Chapter 4 Sections 4.1 - 4.7

• Dr. Iyad Jafar

Outline Outline

Definition Examples Classification of Discrete-time Systems Impulse and Step Responses Linear Time-Invariant Systems (LTI)

F D l LTI D T

Finite-Dimensional LTI Discrete-Time

Systems y

2

Def Definition nition

A discrete-time system is a process that transforms a

given input sequence x[n] into another output sequence g p q p q y[n] with desirable properties.

y[n] T(x[n])

• Discrete-time systems may have more then one input

y y p and/or output

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Exam Examples ples

Accumulator

The output y[n] at any instant of time n is the sum of all p y[ ] y previous samples up to instant n. h h f h

• r, the output at instant n is the

sum of the current sample and the previous output y[n-1]

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Exam Examples ples

The M-point Moving Average (MA) Filter

The output y[n] at any instant n is the average of the p y[ ] y g current sample x[n] and the previous M-1 samples. I d f h d h It is used for smoothing rapid variations in the input sequence.

1.5 x[n] y[n] 0.5 1 [ n ] y[ ]

• 0.5

x[

M=3

5

5 10 15 20

• 1

n

Exam Examples ples

The M-point Moving Average (MA) Filter

Example 3.1: Show that y[n] can be written as p y[ ] y[n] = y[n-1] + 1/M (x[n] – x[n-M])

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Exam Examples ples

The Linear Interpolator

Used to estimate sample values between pairs of adjacent p p j sample values of a discrete-time sequence. Factor-of-2 Interpolator Factor of 2 Interpolator Factor-of-3 Interpolator

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Exam Examples ples

The Linear Interpolator

U li

3 4 ] 3 4 ]

Up-sampling

1 2 x[ n ] 1 2 y[ n ] 2 4 6 8 10 12 n 2 4 6 8 10 12 n

Interpolation

3 4 1 2 z[ n ]

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2 4 6 8 10 12 n

Classification of Discre Classification of Discrete-time Syst

• time Systems

ems

Linearity

In a linear system, if y1[n] is the output due to x1[n] and y , y1[ ] p

1[ ]

y2[n] is the output due to x2[n], then for an input the output is given by for all possible values of α, β and n Example 3.2: a) y[n] = n x[n] ) y[ ] [ ] b) [ ] = ( [ ])2 b) y[n] = (x[n])2

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Classification of Discre Classification of Discrete-time Syst

• time Systems

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Shift-invariant

For a shift-invariant system, if y1[n] is the output due to y , y1[ ] p the input x1[n], then the output to an input sequence is basically where no is an integer. If the independent variable is time, the system is called time-invariant system. y Generally, in time-invariant systems, the output of the system does not depend on the time at which the input is y p p applied.

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Classification of Discre Classification of Discrete-time Syst

• time Systems

ems

Shift-invariant

Example 3.3: Determine whether the following p g systems are time-invariant. a) y[n] = x[n] – x[n-3] a) y[n] x[n] x[n 3] b) y[n] = n x[n] c)

x[n / L] , n = 0, L, 2L, 3L ... y[n] 0 , otherwise

• ⎧

⎨ ⎩

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Classification of Discre Classification of Discrete-time Syst

• time Systems

ems

Causality

A system is said to be causal if the computation of the y p

• utput sample no depends on the input samples in x[n]

for n ≤ no.

• Example

3 4: Determine whether the following Example 3.4: Determine whether the following systems are causal. a) y[n] = x[n] – x[n-5] b) y[n] = x[n] - 4 x[n+5] c) y[n] = x[3 – n] ) y[ ] [ ]

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Classification of Discre Classification of Discrete-time Syst

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Stability

There are many definitions for stability. In this course, a y y , system is said to be bounded-input bounded-output (BIBO) if for every bounded-input sequence ( ) y p q the output is also bounded. E ample 3 5: Th l t i t BIBO t It Example 3.5: The accumulator is not a BIBO system. It is not stable when the input is x[n] = u[n].

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Classification of Discre Classification of Discrete-time Syst

• time Systems

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Passive and Lossless Systems

A discrete-time system is said to be passive if y p

2 2

y[n] x[n]

• ∑

∑

and it is said to be lossless if

n n

• 2

2 n n

y[n] x[n]

• ∑

∑

Example 3.6: Consider the system y[n] = a x[n – N], N > 0.

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Classification of Discre Classification of Discrete-time Syst

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Memory Systems

A system is a memory (dynamic) system if the y y ( y ) y computation of the output requires storing samples from the input. Otherwise, it is a static system. p , y y[n] = 2 x[n] y[n] = x[n] + x[n 2] y[n] = x[n] + x[n-2]

Recursive Systems

A system is said to be recursive when the computation of the output sample requires the knowledge of the previous output sample(s). y [n] = y[n-3] + 6 x[n] y y y[n] = x[n] - x[n-3]

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Im Impulse pulse and St and Step R ep Responses sponses

Systems are usually characterized by stimulating them

with typical signals. The response of the system reflects yp g p y the properties of the system for similar sequences.

The response of a discrete-time system to a unit sample δ[ ] i

ll d h i l sequence δ[n] is called the unit sample response or simply, the impulse response, and is denoted by h[n].

The response of a discrete-time system to a unit step

p y p sequence μ[n] is called the unit step response or simply, the step response, and is denoted by s[n]. p p , y [ ]

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Im Impulse pulse and St and Step R ep Responses sponses

Example 3.7: a) y[n] = α x[n] + α x[n+1] + α x[n-1] + α x[n-2] a) y[n]

α1 x[n] + α2 x[n+1] + α3 x[n-1] + α4 x[n-2] The impulse response of the system is obtained by setting x[n] = δ[n] x[n] = δ[n] h[n] = α δ[n] + α δ[n+1] + α δ[n 1] + α δ[n 2] = h[n] h[n] = α1 δ[n] + α2 δ[n+1] + α3δ[n-1] + α4δ[n-2] = h[n] h[n] = {α2 , α1 , α3 , α4 } b) y[n] = x[n] + (x[n-1] + x[n+1]) /2 h[n] = δ[n] + 0.5 δ[n - 1] + 0.5 δ[n + 1] h[n] = {0.5, 1 , 0.5} , -1 ≤ n ≤ 1

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Linear Time-In Linear Time-Invariant Syst riant Systems ems

Linear Time-Invariant (LTI) System -A system satisfying

both the linearity and the time-invariance properties both the linearity and the time invariance properties

LTI systems are mathematically easy to analyze and

h i d l d i characterize, and consequently, easy to design

There are highly useful signal processing algorithms have

g y g p g g been developed utilizing this class of systems over the last several decades

Linear

time-invariant systems can be completely characterized by their impulse response h[n] In other words characterized by their impulse response h[n]. In other words, the output of a LTI system for any sequence can be computed by knowing its impulse response! We show this in the y g p p following slides.

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Linear Time-In Linear Time-Invariant Syst riant Systems ems

Let h[n] denote the impulse response of a LTI discrete-

time system time system

We compute its output y[n] for the input

0 δ + 2 +1 δ 1 δ 2 + 0 7 δ x[n] = 0.5δ[n + 2] +1.5δ[n −1] − δ[n − 2] + 0.75δ[n − 5]

As the system is linear, we can compute its outputs

for each member of the input separately and add the individual outputs to determine y[n].

Since the system is time-invariant δ[n + 2] → h[n + 2]

[ ] [ ]

δ[n −1] → h[n −1] δ[n − 2] → h[n − 2] δ[n 2] → h[n 2] δ[n − 5] → h[n − 5]

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Linear Time-In Linear Time-Invariant Syst riant Systems ems

Likewise, as the system is linear

0 5δ[n + 2] → 0 5h[n + 2] 0.5δ[n + 2] → 0.5h[n + 2] − δ[n − 2] → −h[n − 2] 1 5δ[ h 1.5δ[n −1] →1.5h[n −1] 0.75δ[n − 5] → 0.75h[n − 5]

Hence because of the linearity property we get

y[n] = 0.5h[n + 2] +1.5h[n −1]− h[n − 2] + 0.75h[n − 5]

Now, any arbitrary input sequence x[n] can be expressed as a

linear combination of delayed and advanced unit sample y p sequences in the form

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Linear Time-In Linear Time-Invariant Syst riant Systems ems

The response of the LTI system to an input x[k]δ[n − k] will

be x[k]h[n − k]. Hence, the response y[n] to an input be x[k]h[n k]. Hence, the response y[n] to an input will be which can be alternately written as

As we discussed in Chapter 2 this summation is called the As we discussed in Chapter 2, this summation is called the

convolution sum of the sequences x[n] and h[n] and represented compactly as represented compactly as

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Linear Time-In Linear Time-Invariant Syst riant Systems ems

Properties of Convolution 1. The convolution between a length-M and length-N 1. The convolution between a length M and length N sequence has a length of M + N – 1 2 The convolution between a sequence defined over N1 ≤ n ≤ 2. The convolution between a sequence defined over N1 ≤ n ≤ M1 and a sequence defined over N2 ≤ n ≤ M2extends over N1 + N2 ≤ n ≤ M1+ M2

1 2 1 2

3. Convolution is commutative 4. Convolution is distributive 5. Convolution is associative

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Linear Time-In Linear Time-Invariant Syst riant Systems ems

Stability of LTI Systems

A LTI system is stable if its impulse response h[n] is A LTI system is stable if its impulse response h[n] is absolutely summable, i.e.

∑

Proof (page 159):

n

S | h[n]|

• ∑

Proof (page 159):

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Linear Time-In Linear Time-Invariant Syst riant Systems ems

Stability of LTI Systems

Example 3.8: Investigate the stability of the system whose impulse response is h[n] = Bn u[ n 1] impulse response is h[n] = -B u[-n -1].

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Linear Time-In Linear Time-Invariant Syst riant Systems ems

Causality of LTI Systems

A LTI system is said to be causal if its impulse response h[n] is A LTI system is said to be causal if its impulse response h[n] is also a causal sequence, i.e. h[n] = 0 , n < 0. Proof (page 161) Proof (page 161)

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Linear Time-In Linear Time-Invariant Syst riant Systems ems

Interconnection of LTI Systems

1.

Cascade Connection: The impulse response h[n] of the

1.

Cascade Connection: The impulse response h[n] of the cascade of two system with impulse response h1[n] and h2[n] is A typical application for cascading systems is the identification of the inverse system h2[n] for some system y y h1[n] such that

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Linear Time-In Linear Time-Invariant Syst riant Systems ems

Interconnection of LTI Systems

Example 3 9: Determine the inverse system for h[n] = u[n] Example 3.9: Determine the inverse system for h[n] = u[n].

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Linear Time-In Linear Time-Invariant Syst riant Systems ems

Interconnection of LTI Systems

2.

Parallel Connection: the impulse response h[n] of two

2.

Parallel Connection: the impulse response h[n] of two systems with impulse response h1[n] and h2[n] that are connected in parallel is

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Linear Time-In Linear Time-Invariant Syst riant Systems ems

Interconnection of LTI Systems

Example 3 10: Determine the impulse response of the Example 3.10: Determine the impulse response of the system shown below if

h [n] = δ[n] + 0 5 δ[n 1]

h3[n]

h1[n] = δ[n] + 0.5 δ[n-1] h2 [n] = 4 δ[n] h [ ] = (0 5)n [ ]

h1[n] h [n] + x[n] y[n]

h3[n] = (0.5)n u[n]

h2[n]

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Linear Time-In Linear Time-Invariant Syst riant Systems ems

LTI systems may have finite impulse response (FIR) such that

h[n] = 0 for n < N1 and n > N2. Such systems usually h[n] 0 for n N1 and n

• N2. Such systems usually

correspond to non-recursive systems and the convolution sum is given by g y

2 1

N k N

y[n] h[k] x[n-k]

• ∑

FIR systems require finite number of multiplications and additions

1

k N

additions Example 3.11: Impulse response of moving average filter

M 1

1 y[n] x[n-k]

• ∑

M 1

1 h[n] [n-k]

• ∑

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k 0

y[ ] [ ] M

• ∑

k 0

[ ] [ ] M

• ∑

Linear Time-In Linear Time-Invariant Syst riant Systems ems

On the other hand, some LTI systems have infinite impulse

response (IIR) such that h[n] is infinite in length. Such response (IIR) such that h[n] is infinite in length. Such systems usually correspond to recursive systems and the convolution sum is given by g y

n

y[n] h[k] x[n-k] ∑

Note how the number of computations are finite but the

k 0

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

• ∑

Note how the number of computations are finite but the complexity increases as n increases.

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Finit Finite-Dimensional L

• Dimensional LTI Discre

I Discrete-Time ime S t S t Sys ystems ems

One popular and important class of LTI systems can be

expressed by a linear constant coefficient difference equation (LCCDE) of the form

• r, after rearranging for y[n]

where dk and pk are constants where dk and pk are constants

The order of the difference equation is max(M,N)

q

Such systems are usually IIR systems in general

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Finit Finite-Dimensional L

• Dimensional LTI Discre

I Discrete-Time ime S t S t Sys ystems ems

Example 3.12:

a) y[n] = 2 x[n] – 3x[n-1] + 4 x[n-2] b) y[n] = 5y[n-1] + x[n] – 3 x[n-3] b) y[n] 5y[n 1] x[n] 3 x[n 3] c) y[n] = 0.5 y[n-3] + x[n] / y[n-1]

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Finit Finite-Dimensional L

• Dimensional LTI Discre

I Discrete-Time ime S t S t Sys ystems ems

For a LTI system described by a LCCDE, we can determine

the output y[n] in response to an arbitrary sequence x[n] recursively. Example 3.13: if x[n] = u[n] and y[n] = y[n-1] + 2 x[n], determine y[n] 0 ≤ n ≤ 3 assuming y[-1] = 0. n = 0 y[0] = y[-1] + 2 x[0] = 2 n = 1 y[1] = y[0] + 2 x[1] = 4 n = 1 y[1] = y[0] + 2 x[1] = 4 n = 2 y[2] = y[1] + 2 x[2] = 6

• n = 3 y[3] = y[2] + 2 x[3] = 8

However, can we find y[n] in a closed form

, y[ ] expression?

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Finit Finite-Dimensional L

• Dimensional LTI Discre

I Discrete-Time ime S t S t Sys ystems ems

The solution of a LCCDE can be expressed as The

complementary (homogeneous natural transient)

The

complementary (homogeneous, natural, transient) solution yc[n] depends on the nature of the system

The particular (non-homogeneous, forced response, steady

t t ) l ti [ ] d d th f f [ ] state) solution yp[n] depends on the form of x[n]

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Finit Finite-Dimensional L

• Dimensional LTI Discre

I Discrete-Time ime S t S t Sys ystems ems

Computation of yc[n] Assumes x[n] = 0 and yc[n] = λn , thus, the LCCDE becomes The polynomial factors into (λ λ )(λ λ )

= 0 with each λ

The polynomial factors into (λ-λ1)(λ-λ2)… = 0, with each λi

satisfying the equation. Also, any combination of the solution also works also works.

Thus, the complementary solution becomes

with αi are constants determined from the initial conditions

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Finit Finite-Dimensional L

• Dimensional LTI Discre

I Discrete-Time ime S t S t Sys ystems ems

In case the root has multiplicity of L, the complementary

solution becomes Example 3.14: Determine the complementary solution of Example 3.14: Determine the complementary solution of y[n] = 2 y[n-1] - y[n-2] – 2 x[n].

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Finit Finite-Dimensional L

• Dimensional LTI Discre

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Computation of yp[n] The particular solution assumes the same form of x[n] scaled

by β

x[n] = A u[n] yp[n] = β u[n] x[n] = An u[n] yp[n] = β An u[n] x[n] = A cos(ωon) yp[n] = β cos(ωon)

If the assumed solution for yp[n] is the same as the

homogeneous solution, the assumed solution is multiplied by n

x[n] = 3n u[n] and yh[n] = 3n +2n yp[n] = β n 3n u[n]

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Finit Finite-Dimensional L

• Dimensional LTI Discre

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Example 3.15: Solve the following LCCDE

y[n] = 3 y[n-1] + 4y[n-2] + x[n] + 2 x[n-1] given that x[n] = 3n u[n] and y[-1] = y[-2] = 0 g y y

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Finit Finite-Dimensional L

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Example 3.16: Solve the following LCCDE

y[n] = 4 y[n-1] - 4y[n-2] + x[n] given that x[n] = 4 u[n] and y[-1] = y[-2] = 1 g y y

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Finit Finite-Dimensional L

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Example 3.17: Solve the following LCCDE

y[n] = -2 y[n-1] + 3 y[n-2] - x[n-1] given that x[n] = 2 (-3)n u[n] and y[-1] = y[-2] = 0 g y y

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Stability of LTI Systems - Revisited

A BIBO stable system has bounded output for every possible bounded input. For a system described by an LCCDE with the output y y p it can be proven that the system is stable if |λk| < 1 it can be proven that the system is stable if |λk| < 1. Proof (page 172)

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Matlab Matlab

• Related functions
• conv
• deconv

deconv

• impz
• stepz
• filter

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