1 Time-Domain Representations of LTI Systems CHAPTER 2.11 Characteristics of Systems Described by Differential and Difference Equations and Difference

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Time-Domain Representations of LTI SystemsTime-Domain Representations of LTI Systems CHAPTER

2.11 Characteristics of Systems Described by Differential2.11 Characteristics of Systems Described by Differential and Difference Equationsand Difference Equations

Complete solution: y = y (n) + y

(f)

y (n) = natural response, y

(f) = forced response

2.11.1 The Natural Response2.11.1 The Natural Response

Example 2.24Example 2.24 RC Circuit (continued): Natural Response

Find the natural response of the this system, assuming that y(0) = 2 V, R = 1 and C = 1 F.

The system In Example 2.17 is described by the differential equation

dy t RC y t x t

dt

<Sol.><Sol.>1. Homogeneous sol.:

1 Vh ty t c e2. I.C.: y(0) = 2 V

y (n) (0) = 2 V c1 = 2

3. Natural Response: 2 Vn ty t e

2


Example 2.25Example 2.25 First-Order Recursive System (Continued): Natural Response

The system in Example 2.21Example 2.21 is described by the difference equation

11

4y n y n x n

Find the natural response of this system.<Sol.><Sol.>1. Homogeneous sol.: 1

1

4

nhy n c

2. I.C.: y[ 1] = 81

1

18

4c

c1 = 2

3. Natural Response:

12 , 1

4

nny n n

2.11.2 The Forced Response2.11.2 The Forced ResponseThe forced response is the system output due to the input signal assuming zero initial conditions.

The forced response is valid only for t 0 or n 0

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The at-rest conditions for a discrete-time system, y[ N] = 0, …, y[ 1] = 0, must be translated forward to times n = 0, 1, …, N 1 before solving for the undetermined coefficients, such as when one is determining the complete solution.Example 2.26Example 2.26 First-Order Recursive System (Continued): Forced ResponseThe system in Example 2.21Example 2.21 is described by the difference equation

11

4y n y n x n

Find the forced response of this system if the input is x[n] = (1/2)n u[n].<Sol.><Sol.>1. Complete solution:

2. I.C.: Translate the at-rest condition y[ 1] to time n = 0

10 0 1

4y x y

1

1 12 , 0

2 4

n n

y n c n

y[0] = 1 + (1/4) 0 =1

3. Finding c1: 0 0

1

1 11 2

2 4c

c1 = 1

4


4. Forced response:

1 12 , 0

2 4

n nfy n n

Example 2.27Example 2.27 RC Circuit (continued): Forced Response

Find the forced response of the this system, assuming that x(t) = cos(t)u(t) V, R = 1 and C = 1 F.

The system In Example 2.17 is described by the differential equation

dy t RC y t x t

dt

<Sol.><Sol.>1. Complete solution: 1 1

cos sin V2 2

ty t ce t t From Example 2.22

2. I.C.:

y(0) = y(0+) = 0 c = 1/23. Forced response:

1 1 1cos sin V

2 2 2f ty t e t t

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2.11.3 The Impulse Response2.11.3 The Impulse Response

Relation between step response and impulse response1. Continuous-time case:

( ) ( )d

h t s tdt

2. Discrete-time case:

[ ] [ 1]h n s n s n

2.11.4 Linearity and Time Invariance2.11.4 Linearity and Time Invariance

Input Forced response

x1 y1(f)

x2 y2(f)

x1 + x2 y1(f) + y2

(f)

Forced response Linearity

Initial Cond. Natural response

I1 y1(n)

I2 y2(n)

I1 + I2 y1(n) + y2

(n)

Natural response Linearity

The complete response of an LTI system is not not time invariant.

Response due to initial condition will not shift with a time shift of the input.

2.11.5 Roots of the Characteristic Equation2.11.5 Roots of the Characteristic Equation

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Roots of characteristic equation

Forced response, natural response, stability, and response time. ★ ★ BIBO Stable:BIBO Stable:

1. Discrete-time case: boundednir 1, for allir i

2. Continuous-time case: boundedir te 0ie r

and1ir 0ie r The system is said to be on the verge of instability.

2.12 Block Diagram Representations2.12 Block Diagram Representations A block diagram is an interconnection of the elementary operations that act on the input signal. Three elementary operations for block diagram:

1. Scalar multiplication: y(t) = cx(t) or y[n] = cx[n], where c is a scalar.2. Addition: y(t) = x(t) + w(t) or y[n] = x[n] + w[n].3. Integration for continuous-time LTI system:

( ) ( )t

y t x d

; and a time shift for discrete-time LTI system: y[n] = x[n 1].

Fig. 2.32.Fig. 2.32.

7

(a)

(b)

( ) ( )t

y t x d

(c)

Figure 2.32 (p. 162)Symbols for elementary operations in block diagram descriptions of systems. (a) Scalar multiplication. (b)

Addition. (c) Integration for continuous-time systems and time shifting for discrete-time systems.


Ex. A discrete-time LTI system: Fig. 2.33.Fig. 2.33.

1. In dashed box:

0 1 2w[n] b x[n] b x[n 1] b x[n 2] (2.49)

2. y[n] in terms of w[n]:

Direct Form I:Direct Form I:

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1 2y[n] w[n] a y[n 1] a y[n 2] (2.50)

3. System output y[n] in terms of input x[n]:

1 2 0 1 2y[n] a y[n 1] a y[n 2] b x[n] b x[n 1] b x[n 2]

1 2 0 1 2y[n] a y[n 1] a y[n 2] b x[n] b x[n 1] b x[n 2] (2.51)

Figure 2.33 (p. 162)Block diagram representation of a discrete-time LTI system described by a second-order difference equation.

Cascade Form(Direct Form I)

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Direct Form II:Direct Form II:

1. Interchange the order of Direct Form I. 2. Denote the output of the new first system as f[n].

1 2f[n] a f[n 1] a f[n 2] x[n] (2.52) Input: x[n]

3. The signal is also the input to the second system. The output of the second system is

0 1 2y[n] b f[n] b f[n 1] b f[n 2] (2.53)

Fig. 2.35.Fig. 2.35.

Figure 2.35 (p. 164)Direct form II representation of an LTI system described by a second-order

difference equation.

)(*)()(*)( 1221 thththth

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Block diagram representation for continuous-time LTI system:

1. Differential Eq.:k kN M

k kk kk 0 k 0

d da y(t) b x(t)

dt dt

(2.54)

2. Let v(0)(t) = v(t) be an arbitrary signal, and set

1 , 1, 2, 3, ...tn nt d n

v(n)(t) is the n-fold integral of v(t) with respect to time 3. Integrator with initial condition:

1 , 0 and 1, 2, 3, ...n ndt t t n

dt

1

00 , 1, 2, 3, ...

tn n nt d n

(N k )N M

(N k)k k

k 0 k 0

a y (t) b x (t)

(2.55)

Integrate N times to eq. (2.54)

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Figure 2.37 (p. 166)Block diagram representations of a continuous-time LTI system described by a

second-order integral equation. (a) Direct form I. (b) Direct form II.

(a)

(b)

Ex. Second-order system:(1) (2) (1) (2)

1 0 2 1 0y(t) a y (t) a y (t) b x(t) b x (t) b x (t)

(2.56)

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2.13 State-Variable Description of 2.13 State-Variable Description of LTILTI Systems Systems The state of a system may be defined as a minimal set of signals that represent the system’s entire memory of the past.

Given initial point ni (or ti) and the input for time n ni (or t ti), we can determine the output for all times n ni (or t ti).

2.13.1 The State-Variable Description2.13.1 The State-Variable Description

1. Direct form II of a second-order LTI system: Fig. 2.39.Fig. 2.39.

2. Choose state variables: q1[n] and q2[n].3. State equation:

1 1 1 2 2q [n 1] a q [n] a q [n] x[n] (2.57)

2 1q [n 1] q [n] (2.58)

4. Output equation:

1 1 1 2 2 2y[n] (b a )q [n] (b a )q [n] x[n] (2.59)

1 11 2

2 2

q [n 1] q [n]a a 1x[n]

q [n 1] q [n]1 0 0

(2.60)

1 1 2 2 1 1 2 2y[n] x[n] a q [n] a q [n] b q [n] b q [n],

5. Matrix Form of state equation:

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Figure 2.39 (p. 167)Direct form II representation of a second-order discrete-time LTI system

depicting state variables q1[n] and q2[n].

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6. Matrix form of output equation:

11 1 2 2

2

q [n]y[n] [b a b a ] [1]x[n]

q [n]

(2.61)

Define state vector as the column vector

1

2

q [n][n]

q [n]q

We can rewrite Eqs. (2.60) and (2.61) as

[n 1] [n] x[n] q Aq b (2.62)

y[n] [n] Dx[n] cq (2.63) where matrix A, vectors b and c, and scalar D are given by

1 2

1 0A

a a

1

0b

1 1 2 2c b a b a 1D

Example 2.28Example 2.28 State-Variable Description of a Second-Order SystemFind the state-variable description corresponding to the system depicted in Fig. 2.40Fig. 2.40 by choosing the state variable to be the outputs of the unit delays.<Sol.><Sol.>

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Figure 2.40 (p. 169)Block diagram of LTI system for Example 2.28.

1. State equation:

1 1 11q n q n x n

2 1 2 21q n q n q n x n

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2. Output equation:

1 1 2 2y n q n q n

3. Define state vector as

1

2

qq n

nq n

In standard form of dynamic equation:

[n 1] [n] x[n]q Aq b

y[n] [n] Dx[n]cq (2.63)

(2.62)

0A

1

2

b

1 2c 2D

State-variable description for continuous-time systems:d

(t) (t) x(t)dt

q Aq b (2.64)

y(t) (t) Dx(t) cq (2.65)

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Example 2.29Example 2.29 State-Variable Description of an Electrical Circuit

Consider the electrical circuit depicted in Fig. 2.42Fig. 2.42. Derive a state-variable description of this system if the input is the applied voltage x(t) and the output is the current y(t) through the resistor.<Sol.><Sol.>

Figure 2.42 (p. 171)Circuit diagram of LTI

system for Example 2.29.

1. State variables: The voltage across each capacitor. 2. KVL Eq. for the loop involving x(t), R1, and C1:

1 1x t y t R q t

11 1

1 1y(t) q (t) x(t)

R R (2.66)

Output equation

3. KVL Eq. for the loop involving C1, R2, and C2:

1 2 2 2( ) ( )q t R i t q t

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2 1 22 2

1 1i (t) q (t) q (t)

R R (2.67)

4. The current i2(t) through R2:

2 2 2( )d

i t C q tdt

2 1 22 2 2 2

1 1( ) ( )

dq t q t q t

dt C R C R (2.68)

Use Eq. (2.67) to eliminate i2(t)

5. KCL Eq. between R1 and R2:

1 2y t i t i t Current through C1 = i1(t)

where 1 1 1

di t C q t

dt

1 1 21 1 2 2 1 2 1 1

1 1 1 1( ) ( ) ( )

dq t q t q t x t

dt C R C R C R C R

(2.69)

◆ Eqs. (2.66), (2.68), and (2.69) = State-Variable Description.

d

(t) (t) x(t)dt

q Aq b (2.64)

y(t) (t) Dx(t)cq (2.65)

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1 1 1 2 1 2

2 2 2 2

1 1 1

,1 1

AC R C R C R

C R C R

1 1

1

0

b C R

1

10 ,c

R

1

1D

Rand

Example 2.30Example 2.30 State-Variable Description from a Block DiagramDetermine the state-variable description corresponding to the block diagram in Fig. 2.44Fig. 2.44. The choice of the state variables is indicated on the diagram.

Figure 2.44 (p. 172)Block diagram of LTI system for Example 2.30.

<Sol.><Sol.>

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1. State equation:

1 1 22dq t q t q t x t

dt

2 1

dq t q t

dt

2. Output equation:

1 23y t q t q t

3. State-variable description:

2 1,

1 0A

1,

0b

3 1 ,c 0D

2.13.2 Transformations of The State2.13.2 Transformations of The State

The transformation is accomplished by defining a new set of state variables that are a weighted sum of the original ones.

The input-output characteristic of the system is not changed.

1. Original state-variable description:

q Aq bx (2.70)

cqy Dx (2.71)

2. Transformation: q’ = Tq

T = state-transformation matrix

q = T1 q’

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3. New state-variable description:

.q TAq Tbx 1) State equation:q Tq

1 .q TAT q Tbx q = T1 q’

2) Output equation: 1 .cT qy Dx 3) If we set

1 1, , , andA TAT b Tb c cT D D then q A q b x and c qy D x

Ex. Consider Example 2.30Example 2.30 again. Let us define new states

2 1 1 2( ) ( ) and ( ) ( )q t q t q t q t Find the state-variable description.

<Sol.><Sol.>1. State equation: 1 1 2 2 2 1( ) 2 ( ) ( ) ( ) ( ) 2 ( ) ( ) ( )

old description New description

d dq t q t q t x t q t q t q t x t

dt dt

22


2 1 1 2( ) ( ) ( ) ( )

Old description New description

d dq t q t q t q t

dt dt

1 2 2 13 ( ) ( ) 3 ( ) ( )Old description New description

y q t q t y q t q t

2. Output equation:

3. State-variable description:

0 1,

1 2A

0,

1b

1 3 ,c 0 .D

Example 2.31Example 2.31 Transforming The State A discrete-time system has the state-variable description

1 41A ,

4 110

2,

4b

11 1 ,

2c 2.D and

Find the state-variable description A, b, c, D corresponding to the new states 1 1

1 1 22 2[ ] [ ] [ ]q n q n q n 1 1

2 1 22 2[ ] [ ] [ ]q n q n q n and

<Sol.><Sol.>1. Transformation: q = Tq, where 1 11

.1 12

T

23


1 1 1.

1 1T

2. New state-variable description:1

2

3

10

0,

0A

1,

3b

0 1 ,c 2.Dand

This choice for T results in A being a diagonal matrix and thus separates the state update into the two decoupled first-order difference equations

1 1

11

2q n q n x n 2 2

31 3

10q n q n x n and

2.14 2.14 Exploring Concepts with MATLABExploring Concepts with MATLAB Two limitations: 1. MATLAB is not easily used in the continuous-time case. 2. Finite memory or storage capacity and nonzero computation times. Both the MATLAB Signal Processing Toolbox and Control System Toolbox are use in this section.

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2.14.1 Convolution2.14.1 Convolution

1. MATLAB command: y = conv(x, h)x and h are signal vectors.

2. The number of elements in y is given by the sum of the number of elements in x and h, minus one.<pf.><pf.>

1) Elements in vector x: from time n = kx to n = lx

2) Elements in vector h: from time n = kh to n = lh

3) Elements in vector y: from time n = ky = kx + kh to n = ly = lx + ly

4) The length of x[n] and h[n] are Lx = lx kx + 1 and Lh = lh kh +1

5) The length of y[n] is Ly = Lx + Lh 1

Ex. Repeat Example 2.1Impulse and Input : From time n = kh = kx = 0 to n = lh = 1 and n = lx =2Convolution sum: From time n = ky = kx + kh = 0 to n = ly = lx + lh = 3 The length of convolution sum: Ly = ly – ky + 1 = 4

MATLAB Program: >> h = [1, 0.5];>> x = [2, 4, -2];>> y = conv(x,h)

y =

2 5 0 -1

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Repeat Example 2.3

Given 1 4 4h n u n u n 10 .x n u n u n and

Impulse response Input

Find the convolution sum x[n] h[n].

1. In this case, kh = 0, lh = 3, kx = 0 and lx = 9<Sol.><Sol.>

2. y starts at time n = ky = 0, ends at time n = ly =12, and has length Ly = 13.

3. Generation for vector h with MATLAB:

>> h = 0.25*ones(1, 4);>> x = ones(1, 10);

4. Output and its plot:

>> n = 0:12;>> y = conv(x, h);>> stem(n, y); xlabel('n'); ylabel('y[n]')

Fig. 2.45.Fig. 2.45.

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Figure 2.45 (p. 177)Convolution sum computed using MATLAB.

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2.14.2 The Step Response2.14.2 The Step Response

1. Step response = the output of a system in response to a step input

2. In general, step response is infinite in duration.

3. We can evaluate the first p values of the step response using the conv function if h[n] = 0 for n < kh by convolving the first p values of h[n] with a finite-duration step of length p.

1) Vector h = the first p nonzero values of the impulse response.2) Define step: u = ones(1, p).3) convolution: s = conv(u, h).

Ex. Repeat Problem 2.12

Determine the first 50 values of the step response of the system with impulse response given by

nh n u n

with = 0.9, by using MATLAB program.<Sol.><Sol.>

1. MATLAB Commands:

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>> h = (-0.9).^[0:49];>> u = ones(1, 50);>> s = conv(u, h);>> stem([0:49], s(1:50))

2. Step response: Fig. 2.47.Fig. 2.47.

Figure 2.47 (p. 178)Step response

computed using MATLAB.

2.14.3 Simulating Difference equations2.14.3 Simulating Difference equations

1. Difference equation:

N M

k kk 0 k 0

a y[n k] b x[n k] (2.36) CommandCommand:: filter

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2. Procedure:

1) Define vectors a = [a0, a1, …, aN] and b =[b0, b1, …, bM] representing the coefficients of Eq. (2.36).2) Input vector: x3) y = filter(b, a, x) results in a vector y representing the output of the system for zero initial conditions.4) y = filter(b, a, x, zi) results in a vector y representing the output of the system for nonzero initial conditions zi.

The initial conditions used by filter are not the past values of the output.

Command zi = filtic(b, a, yi), where yi is a vector containing the initial conditions in the order [y[1], y[2], …, y[N]], generates the initial conditions obtained from the knowledge of the past outputs.Ex. Repeat Example 2.16The system of interest is described by the difference equation

1.143 1 0.4128 2 0.0675 0.1349 1 0.675 2y n y n y n x n x n x n (2.73)Determine the output in response to zero input and initial condition

y[1] = 1 and y[2] = 2.

<Sol.><Sol.>

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1. MATLAB Program:

>> a = [1, -1.143, 0.4128]; b = [0.0675, 0.1349, 0.675];>> x = zeros(1, 50);>> zi = filtic(b, a, [1, 2]);>> y = filter(b, a, x, zi);>> stem(y)

2. Output: Fig. 2.28(b).Fig. 2.28(b).

0 10 20 30 40 50-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

3. System response to an input consisting of the Intel stock price data Intc:

>> load Intc;>> filtintc = filter(b, a, Intc);

We have assume that the Intel stock price data are in the file Intc.mat.

The command [h, t] = impz(b, a, n) evaluates n values of the impulse response of a system described by a different equation.

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2.14.4 State-Variable Descriptions2.14.4 State-Variable Descriptions

MATLAB command: ss1. Input MATLAB arrays: a, b, c, d

Representing the matices A,b,c, and D.

2. Command: sys = ss(a, b, c, d, -1) produces an LTI object sys that represents the discrete-time system in state-variable form.

★ Continuous-time case: sys = ss(a, b, c, d)No 1

System manipulation:

1. sys = sys1 + sys2 Parallel combination of sys1 and sys2.

2. sys = sys1 sys2 Cascade combination of sys1 and sys2.

MATLAB command: lsim1. Command form: y = lsim(sys, x)2. Output = y, input = x.

MATLAB command: impulse

2. This command places the first N values of the impulse response in h.1. Command form: h = impulse(sys, N)

MATLAB routine: ss2ss Perform the state transformation1. Command form: sysT = ss2ss(sys, T), where T = Transformation matrix

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Ex. Repeat Example 2.31.

1. Original state-variable description:

1 41,

4 110A

2,

4b

11 1 ,

2c 2,D and

2. State-transformation matrix:

1 11.

1 12T

3. MATLAB Program:

>> a = [-0.1, 0.4; 0.4, -0.1]; b = [2; 4];>> c = [0.5, 0.5]; d = 2;>> sys = ss(a, b, c, d, -1); % define the state-space object sys>> T = 0.5*[-1, 1; 1, 1];>> sysT = ss2ss(sys, T)

4. Result:

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a = x1 x2 x1 -0.5 0 x2 0 0.3

b = u1 x1 1 x2 3

c = x1 x2 y1 0 1

d = u1 y1 2

Sampling time: unspecifiedDiscrete-time model.

Ex. Verify that the two systems represented by sys and sysT have identical input-output characteristic by comparing their impulse responses .<Sol.><Sol.>

1. MATLAB Program: >> h = impulse(sys, 10); hT = impulse(sysT, 10);>> subplot(2, 1, 1)>> stem([0:9], h)>> title ('Original System Impulse Response');>> xlabel('Time'); ylabel('Amplitude')>> subplot(2, 1, 2)>> stem([0:9], hT)>> title('Transformed System Impulse Response');>> xlabel('Time'); ylabel('Amplitude')

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2. Simulation results: Fig. 2.48.Fig. 2.48.

0 2 4 6 8 100

1

2

3Original System Impulse Response

TimeA

mpl

itude

0 2 4 6 8 100

1

2

3Transformed System Impulse Response

Time

Am

plitu

de

Figure 2.48 (p. 181)Impulse responses associated with the original and transformed state-variable descriptions computer using MATLAB.

We may verify that the original and transformed systems have the (numerically) identical impulse response by computing the error, err = h – hT.

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0 2 4 6 8 10-6

-5

-4

-3

-2

-1

0

1x 10

-17

Time

Am

plitu

de e

rr

Plot for err = h Plot for err = h hT hT

Documents

1 Time-Domain Representations of LTI Systems CHAPTER 2.11 Characteristics of Systems Described by Differential and Difference Equations and Difference