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6.003: Signals and Systems Laplace and Z Transforms October 1, 2009

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Page 1: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

6.003: Signals and Systems

Laplace and Z Transforms

October 1, 2009

Page 2: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Mid-term Examination #1

Wednesday, October 7, 7:30-9:30pm, Walker Memorial.

No recitations on the day of the exam.

Coverage: DT Signals and Systems

Lectures 1–5

Homeworks 1–4

Homework 4 will include practice problems for mid-term 1.

However, it will not collected or graded. Solutions will be posted.

Closed book: 1 page of notes (812 × 11 inches; front and back).

Designed as 1-hour exam; two hours to complete.

Review sessions during open office hours.

Conflict? Contact [email protected] before Friday, October 2, 5pm.

Page 3: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Last Time

Many continuous-time systems can be represented with differential

equations.

Example: leaky tank

r0(t)

r1(t)h1(t)

Differential equation representation:

τ r1(t) = r0(t)− r1(t)

Last time we considered two methods to solve differential equations:

• solving homogeneous and particular equations

• singularity matching

Page 4: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Solving Differential Equations with Laplace Transform

The Laplace transform provides a particularly powerful method of

solving differential equations — it transforms a differential equation

into an algebraic equation.

Method (where L represents the Laplace transform):

differential algebraic

algebraic differential equation

−→

↓ solve

−→

differentialequation

algebraic

equation

algebraicanswer

solution todifferential equation

L−→

↓ solve

L−1−→

Page 5: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Laplace Transform: Definition

Laplace transform maps a function of time t to a function of s.

X(s) =∫x(t)e−stdt

There are two important variants:

Unilateral (18.03)

X(s) =∫ ∞

0x(t)e−stdt

Bilateral (6.003)

X(s) =∫ ∞−∞x(t)e−stdt

Both share important properties — will discuss differences later.

Page 6: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Laplace Transforms

Example: Find the Laplace transform of x1(t):

0t

x1(t)

x1(t) ={Ae−σt if t ≥ 00 otherwise

X1(s) =∫ ∞−∞x1(t)e−stdt =

∫ ∞0Ae−σte−stdt = Ae

−(s+σ)t

−(s+ σ)

∣∣∣∣∣∞

0= A

s+ σ

provided Re{s+ σ} > 0 which implies that Re{s} > −σ.

−σ

s-plane

ROCA

s+ σ; Re{s} > −σ

Page 7: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Check Yourself

0t

x2(t)

x2(t) ={e−t − e−2t if t ≥ 00 otherwise

Which of the following is the Laplace transform of x2(t)?

1. X2(s) = 1(s+1)(s+2) ; Re{s} > −1

2. X2(s) = 1(s+1)(s+2) ; Re{s} > −2

3. X2(s) = s(s+1)(s+2) ; Re{s} > −1

4. X2(s) = s(s+1)(s+2) ; Re{s} > −2

5. none of the above

Page 8: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Check Yourself

X2(s) =∫ ∞

0(e−t − e−2t)e−stdt

=∫ ∞

0e−te−stdt−

∫ ∞0e−2te−stdt

= 1s+ 1

− 1s+ 2

= (s+ 2)− (s+ 1)(s+ 1)(s+ 2)

= 1(s+ 1)(s+ 2)

These equations converge if Re{s + 1} > 0 and Re{s + 2} > 0, thus

Re{s} > −1.

−1−2

s-plane

ROC1(s+ 1)(s+ 2)

; Re{s} > −1

Page 9: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Check Yourself

0t

x2(t)

x2(t) ={e−t − e−2t if t ≥ 00 otherwise

Which of the following is the Laplace transform of x2(t)?

1. X2(s) = 1(s+1)(s+2) ; Re{s} > −1

2. X2(s) = 1(s+1)(s+2) ; Re{s} > −2

3. X2(s) = s(s+1)(s+2) ; Re{s} > −1

4. X2(s) = s(s+1)(s+2) ; Re{s} > −2

5. none of the above

Page 10: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Regions of Convergence

Left-sided signals have left-sided Laplace transforms (bilateral only).

Example:

t

x3(t)

−1x3(t) =

{−e−t if t ≤ 00 otherwise

X3(s) =∫ ∞−∞x3(t)e−stdt =

∫ 0

−∞−e−te−stdt = −e

−(s+1)t

−(s+ 1)

∣∣∣∣∣0

−∞= 1s+ 1

provided Re{s+ 1} < 0 which implies that Re{s} < −1.

−1

s-plane

RO

C1s+ 1

; Re{s} < −1

Page 11: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Left- and Right-Sided Signals

We can concisely express left- and right-sided signals by multiplica-

tion with step functions.

0t

x1(t)x1(t) =

{e−t if t ≥ 00 otherwise

≡ e−tu(t)

t

x3(t)

−1

x3(t) ={−e−t if t ≤ 00 otherwise

≡ −e−tu(−t)

Page 12: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Left- and Right-Sided ROCs

Laplace transforms of left- and right-sided exponentials have the

same form (except −); with left- and right-sided ROCs, respectively.

0t

e−tu(t)time function Laplace transform

−1

s-plane

ROC1s+ 1

t

−e−tu(−t)

−1 −1

s-plane

RO

C

1s+ 1

Page 13: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Check Yourself

Find the Laplace transform of x4(t).

0t

x4(t)

x4(t) = e−|t|

1. X4(s) = 21−s2 ; −∞ < Re{s} <∞

2. X4(s) = 21−s2 ; −1 < Re{s} < 1

3. X4(s) = 21+s2 ; −∞ < Re{s} <∞

4. X4(s) = 21+s2 ; −1 < Re{s} < 1

5. none of the above

Page 14: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Check Yourself

X4(s) =∫ ∞−∞e−|t|e−stdt

=∫ 0

−∞e(1−s)tdt+

∫ ∞0e−(1+s)tdt

= e(1−s)t

(1− s)

∣∣∣∣∣0

−∞+ e−(1+s)t

−(1 + s)

∣∣∣∣∣∞

0

= 11− s︸ ︷︷ ︸

Re{s}<1

+ 11 + s︸ ︷︷ ︸

Re{s}>−1

= 1 + s+ 1− s(1− s)(1 + s)

= 21− s2

; −1 < Re{s} < 1

The ROC is the intersection of Re{s} < 1 and Re{s} > −1.

Page 15: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Check Yourself

The Laplace transform of a signal that is both-sided a vertical strip.

0t

x4(t)

x4(t) = e−|t|

−1 1

s-plane

ROCX4(s) = 21− s2

−1 < Re{s} < 1

Page 16: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Check Yourself

Find the Laplace transform of x4(t). 2

0t

x4(t)

x4(t) = e−|t|

1. X4(s) = 21−s2 ; −∞ < Re{s} <∞

2. X4(s) = 21−s2 ; −1 < Re{s} < 1

3. X4(s) = 21+s2 ; −∞ < Re{s} <∞

4. X4(s) = 21+s2 ; −1 < Re{s} < 1

5. none of the above

Page 17: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Solving Differential Equations with Laplace Transforms

Solve the following differential equation:

y(t) + y(t) = δ(t)

Take the Laplace transform of this equation.

L{y(t) + y(t)} = L{δ(t)}

The Laplace transform of a sum is the sum of the Laplace transforms

(prove this as an exercise).

L{y(t)}+ L{y(t)} = L{δ(t)}

What’s the Laplace transform of a derivative?

Page 18: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Laplace transform of a derivative

Assume that X(s) is the Laplace transform of x(t):

X(s) =∫ ∞−∞x(t)e−stdt

Find the Laplace transform of y(t) = x(t).

Y (s) =∫ ∞−∞y(t)e−stdt

=∫ ∞−∞x(t)e−stdt

= x(t)e−st∣∣∞−∞ −

∫ ∞−∞x(t)(−se−st)dt

The first term must be zero since X(s) converged. Thus

Y (s) = s∫ ∞−∞x(t)e−stdt = sX(s)

Page 19: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Solving Differential Equations with Laplace Transforms

Back to the previous problem:

L{y(t)}+ L{y(t)} = L{δ(t)}

Let Y (s) represent the Laplace transform of y(t).

Then sY (s) is the Laplace transform of y(t).

sY (s) + Y (s) = L{δ(t)}

What’s the Laplace transform of the impulse function?

Page 20: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Laplace transform of the impulse function

Let x(t) = δ(t).

X(s) =∫ ∞−∞δ(t)e−stdt

=∫ ∞−∞δ(t) e−st

∣∣t=0 dt

=∫ ∞−∞δ(t) 1 dt

= 1

Sifting property: δ(t) sifts out the value of the integrand at t = 0.

Page 21: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Solving Differential Equations with Laplace Transforms

Back to the previous problem:

sY (s) + Y (s) = L{δ(t)} = 1

This is a simple algebraic expression. Solve for Y (s):

Y (s) = 1s+ 1

We’ve seen this Laplace transform previously.

y(t) = e−tu(t) (why not y(t) = −e−tu(−t) ?)

Notice that we solved the differential equation y(t) + y(t) = δ(t) with-

out computing homogeneous and particular solutions and without

singularity matching.

Page 22: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Solving Differential Equations with Laplace Transforms

Summary of method.

Start with differential equation:

y(t) + y(t) = δ(t)

Take the Laplace transform of this equation:

sY (s) + Y (s) = 1

Solve for Y (s):

Y (s) = 1s+ 1

Take inverse Laplace transform (by recognizing form of transform):

y(t) = e−tu(t)

Page 23: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Solving Differential Equations with Laplace Transforms

Recognizing the form ...

Is there a more systematic way to take an inverse Laplace transform?

Yes ... and no.

Formally,

x(t) = 12πj

∫ σ+j∞σ−j∞

X(s)estds

but this integral is not generally easy to compute.

This equation can be useful to prove theorems.

We will find better ways (e.g., partial fractions) to compute inverse

transforms for common systems.

Page 24: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Solving Differential Equations with Laplace Transforms

Example 2:

y(t) + 3y(t) + 2y(t) = δ(t)

Laplace transform:

s2Y (s) + 3sY (s) + 2Y (s) = 1

Solve:

Y (s) = 1(s+ 1)(s+ 2)

= 1s+ 1

− 1s+ 2

Inverse Laplace transform:

y(t) =(e−t − e−2t)u(t)

These forward and inverse Laplace transforms are easy if

• differential equation is linear with constant coefficients, and

• the input signal is an impulse function.

Page 25: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Properties of Laplace Transforms

The use of Laplace Transforms to solve differential equations de-

pends on several important properties.

Property x(t) X(s) ROC

Linearity ax1(t) + bx2(t) aX1(s) + bX2(s) ⊃ (R1 ∩R2)

Delay by T x(t− T ) X(s)e−sT R

Multiply by t tx(t) −dX(s)ds

R

Multiply by e−αt x(t)e−αt X(s+ α) shift R by −α

Differentiate in tdx(t)dt

sX(s)

Integrate in t

∫ t−∞x(τ) dτ X(s)

s⊃ (R ∩ {Re{s} > 0})

Convolve in t

∫ ∞−∞x1(τ)x2(t− τ) dτ X1(s)X2(s) ⊃ (R1 ∩R2)

Page 26: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Z Transform

The Z transform is the DT analog of the Laplace transform.

The Z transform converts a difference (recurrence) equation into a

simple algebraic equation.

Method (where Z represents the Z transform):

differential algebraic

algebraic differential equation

−→

↓ solve

−→

differenceequation

algebraic

equation

algebraicanswer

solution todifference equation

Z−→

↓ solve

Z−1−→

Page 27: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Z Transform: Definition

Z transform maps a function of discrete time n to a function of z.

X(z) =∑x(n)z−n

There are two important variants:

Unilateral

X(z) =∞∑n=0x[n]z−n

Bilateral

X(z) =∞∑n=−∞

x[n]z−n

Differences are analogous to those for the Laplace transform.

Page 28: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Z Transforms

Example: Find the Z transform of x1[n]:

−4−3−2−1 0 1 2 3 4n

x1[n]x1[n] =

{(78

)nif n ≥ 0

0 otherwise

≡(

78

)nu[n]

X1(z) =∞∑n=−∞

(78

)nz−nu[n] =

∞∑n=0

(78

)nz−n = 1

1− 78z−1 = z

z − 78

provided |z| > 78 .

−4−3−2−1 0 1 2 3 4n

x1[n] =(

78

)nu[n]

78

z-plane ROC

z

z − 78

Page 29: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Check Yourself

Let X2(z) represent the Z transform of x2[n]:

−4 −3 −2 −1 0 1 2n

x2[n] = −(

78

)nu[−1− n]

What is the relation between X2(z) and X1(z)?

1. X2(z) = −X1(z)2. X2(z) = X1(z)

3. X2(z) = 1X1(z)

4. none of the above

Page 30: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Check Yourself

Let X2(z) represent the Z transform of x2[n].What is the relation between X2(z) and X1(z)?

−4 −3 −2 −1 0 1 2n

x2[n] = −(

78

)nu[−1− n]

X2(z) =∞∑n=−∞

−(

78

)nu[−1− n]z−n = −

−1∑n=−∞

(78

)nz−n

Let l = −1− n:

X2(z) = −∞∑l=0

(78

)−1−lz1+l = −8z

7

∞∑l=0

(8z7

)l= −8z

71

1− 8z7

= z

z − 78

provided∣∣8z

7∣∣ < 1 or |z| < 7

8 .

Page 31: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Z Transforms: Check Yourself

−4 −3 −2 −1 0 1 2n

x2[n] = −(

78

)nu[−1− n]

−(8

7)1−

(87)2−

(87)3

−(8

7)4 7

8

z-plane ROC

z

z − 78

−4−3−2−1 0 1 2 3 4n

x1[n] =(

78

)nu[n]

78

z-plane ROC

z

z − 78

Page 32: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Check Yourself

Let X2(z) represent the Z transform of x2[n]:

−4 −3 −2 −1 0 1 2n

x2[n] = −(

78

)nu[−1− n]

What is the relation between X2(z) and X1(z)? 2

1. X2(z) = −X1(z)2. X2(z) = X1(z)

3. X2(z) = 1X1(z)

4. none of the above

Page 33: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Solving Difference Equations with Z Transforms

Solve the following difference equation:

y[n]− 12y[n− 1] = δ[n]

Take the Z transform of this equation.

Z{y[n]− 1

2y[n− 1]

}= Z {δ[n]}

The Z transform of a sum is the sum of the Z transforms (prove

this as an exercise).

Z {y[n]} − Z{

12y[n− 1]

}= Z {δ[n]}

What’s the Z transform of a delay?

Page 34: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Z transform of a delay

Assume that X(z) is the Z transform of x[n]:

X(z) =∞∑n=−∞

x[n]z−n

Find the Z transform of y[n] = x[n− 1].

Y (z) =∞∑n=−∞

y[n]z−n =∞∑n=−∞

x[n− 1]z−n

Let l = n− 1:

Y (z) =∞∑l=−∞

x[l]z−1−l = z−1∞∑l=−∞

x[l]z−l = z−1X(z)

Page 35: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Solving Difference Equations with Z Transforms

Back to the previous problem:

Z {y[n]} − Z{

12y[n− 1]

}= Z {δ[n]}

Let Y (z) represent the Z transform of y[n]. Then z−1Y (z) is the Z

transform of y[n− 1].

Y (z)− 12z−1Y (z) = Z {δ[n]}

What’s the Z transform of the unit sample function?

Page 36: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Z transform of the unit sample function

Let x[n] = δ[n]. Substituting in the definition of the Z transform:

X(z) =∞∑n=−∞

δ[n]z−n = z−0 = 1

Page 37: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Solving Difference Equations with Z Transforms

Back to the previous problem:

Y (z)− 12z−1Y (z) = Z {δ[n]} = 1

This is a simple algebraic expression. Solve for Y (z):

Y (z) = 11− 1

2z−1

By analogy with the Z transform of x1[n]:

y[n] =(

12

)nu[n]

This method for solving difference equations is analogous to using

the Laplace transform to solve a differential equation.

Page 38: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Solving Difference Equations with Z Transforms

Summary of method.

Start with difference equation:

y[n]− 12y[n− 1] = δ[n]

Take the Z transform of this equation:

Y (z)− 12z−1Y (z) = 1

Solve for Y (z):

Y (z) = 11− 1

2z−1

Take the inverse Z transform (by recognizing the form of the trans-

form):

y[n] =(

12

)nu[n]

Page 39: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Inverse Z transform

The inverse Z transform has defined by an integral that is not par-

ticularly easy to solve.

Formally,

x[n] = 12πj

∫CX(z)zn−1ds

were C represents a closed contour that circles the origin by running

in a counterclockwise direction through the region of convergence.

This integral is not generally easy to compute.

This equation can be useful to prove theorems.

We will find better ways (e.g., partial fractions) to compute inverse

transforms for the kinds of systems that we most frequently en-

counter.

Page 40: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Properties of Z Transforms

The use of Z Transforms to solve differential equations depends on

several important properties.

Property x[n] X(z) ROC

Linearity ax1[n] + bx2[n] aX1(z) + bX2(z) ⊃ (R1 ∩R2)

Delay x[n− 1] z−1X(z) R

Multiply by n nx[n] −z dX(z)dz

R

Convolve in n∞∑

m=−∞x1[m]x2[n−m] X1(z)X2(z) ⊃ (R1 ∩R2)

Page 41: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Concept Map: Discrete-Time Systems

Topics

+

Delay

Delay

X Y

Block Diagram

H = 11−R−R2

System Functional

h : 1, 1, 2, 3, 5, 8, 13, . . .

Unit-Sample Response

y[n] = x[n] + y[n−1] + y[n−2]

Difference Equation

H(z) = z2

z2 − z − 1

System Function

Page 42: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Concept Map: Discrete-Time Systems

Connections

+

Delay

Delay

X Y

Block Diagram

H = 11−R−R2

System Functional

h : 1, 1, 2, 3, 5, 8, 13, . . .

Unit-Sample Response

y[n] = x[n] + y[n−1] + y[n−2]

Difference Equation

H(z) = z2

z2 − z − 1

System Function

Page 43: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Concept Map: Discrete-Time Systems

Example of connections

+

Delay

Delay

X Y

Block Diagram

H = 11−R−R2

System Functional

h : 1, 1, 2, 3, 5, 8, 13, . . .

Unit-Sample Response

y[n] = x[n] + y[n−1] + y[n−2]

Difference Equation

H(z) = z2

z2 − z − 1

System Function

Delay ↔ R

Page 44: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Check Yourself

Where is the Z transform ?

+

Delay

Delay

X Y

Block Diagram

H = 11−R−R2

System Functional

h : 1, 1, 2, 3, 5, 8, 13, . . .

Unit-Sample Response

y[n] = x[n] + y[n−1] + y[n−2]

Difference Equation

H(z) = z2

z2 − z − 1

System Function

1

2

3

4

5 6

7 8

Page 45: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Check Yourself

Where is the Z transform ?

+

Delay

Delay

X Y

Block Diagram

H = 11−R−R2

System Functional

h : 1, 1, 2, 3, 5, 8, 13, . . .

Unit-Sample Response

y[n] = x[n] + y[n−1] + y[n−2]

Difference Equation

H(z) = z2

z2 − z − 1

System Function

Z transform

Page 46: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Concept Map: Discrete-Time Systems

Examples of connections

+

Delay

Delay

X Y

Block Diagram

H = 11−R−R2

System Functional

h : 1, 1, 2, 3, 5, 8, 13, . . .

Unit-Sample Response

y[n] = x[n] + y[n−1] + y[n−2]

Difference Equation

H(z) = z2

z2 − z − 1

System Function

Delay ↔ R

Delay ↔ z−1

Delay ↔ [n− 1]

R ↔ z−1

Step-by-Step Series,

Partial Fractions

Step-by-Step Z transform

Page 47: Laplace and Z Transforms - MITweb.mit.edu/6.003/F09/www/handouts/lec07.pdf · 2009. 12. 21. · Z Transform The Z transform is the DT analog of the Laplace transform. The Z transform

Summary

Introduced Laplace transform

Applied Laplace transform to solving differential equations

Introduced Z transform

Applied Z transform to solving difference equations

Found lots of connections between representations of DT systems