lecture 11: continuous-time frequency response and bode plots · 6.003: signals and systems ct...

6.003: Signals and Systems

CT Frequency Response and Bode Plots

October 18, 2011 1

Mid-term Examination #2

Wednesday, October 26, 7:30-9:30pm,

No recitations on the day of the exam.

Coverage: Lectures 1–12

Recitations 1–12

Homeworks 1–7

Homework 7 will not be collected or graded. Solutions will be posted.

Closed book: 2 pages of notes (812 × 11 inches; front and back).

No calculators, computers, cell phones, music players, or other aids.

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

Review sessions during open office hours.

Conflict? Contact before Friday, Oct. 21, 5pm.

2

mailto:[email protected]

Review: Frequency Response

Complex exponentials are eigenfunctions of LTI systems.

H(s)es0t H(s0) es0t

H(s0) can be determined graphically using vectorial analysis.

H(s0) = K (s0 − z0)(s0 − z1)(s0 − z2) · · · (s0 − p0)(s0 − p1)(s0 − p2) · · ·

z0z0

s0 − z0s0

s-planes0

Response of an LTI system to an eternal cosine is an eternal cosine:

same frequency, but scaled and shifted.

H(s)cos(ω0t) |H(jω0)| cos(ω0t+ ∠H(jω0)

)3

Frequency Response: H(s)|s←jω

s-plane

σ

ω5

−5

5−5

H(s) = s− z1

−5 0 5

5|H(jω)|

−5 5

π/2

−π/2

∠H(jω)

4


s-plane

σ

ω5

−5

5−5

H(s) = 9s− p1

−5 0 5

5|H(jω)|

−5 5

π/2

−π/2

∠H(jω)

5


s-plane

σ

ω5

−5

5−5

H(s) = 3 s− z1s− p1

−5 0 5

5|H(jω)|

−5 5

π/2

−π/2

∠H(jω)

6

Poles and Zeros

Thinking about systems as collections of poles and zeros is an im

portant design concept.

• simple: just a few numbers characterize entire system

• powerful: complete information about frequency response

Today: poles, zeros, frequency responses, and Bode plots.

7

Asymptotic Behavior: Isolated Zero

The magnitude response is simple at low and high frequencies.

σ

ω5

−5

5−5

H(jω) = jω − z1

−5 0 5

5|H(jω)|

|z1|

|ω|

−5 5

π/2

−π/2

∠H(jω)

8


Two asymptotes provide a good approxmation on log-log axes.

H(s) = s − z1

−5 0 5

5|H(jω)|

−2 −1 0 1 2

2

1

0log ω

|z1|

1

log |H(jω)||z1|

lim |H(jω)| = |z1|ω→0

lim |H(jω)| = ω ω→∞

9

Asymptotic Behavior: Isolated Pole

The magnitude response is simple at low and high frequencies.

σ

ω5

−5

5−5

H(s) = 9s− p1

−5 0 5

5|H(jω)|

9|p1|

9|ω|

−5 5

π/2

−π/2

∠H(jω)

10

−5 0 5

5|H(jω)|

−2 −1 0 1 2

0

−1

−2log ω

|p1|

−1

log |H(jω)|9/|p1|


Two asymptotes provide a good approxmation on log-log axes.

9 H(s) =

s − p1

9lim |H(jω)| = ω→0 |p1|

9lim |H(jω)| = ω→∞ ω 11

Check Yourself

Compare log-log plots of the frequency-response magnitudes of

the following system functions:

H1(s) = 1

s + 1 and H2(s) =

1 s + 10

The former can be transformed into the latter by

1. shifting horizontally

2. shifting and scaling horizontally

3. shifting both horizontally and vertically

4. shifting and scaling both horizontally and vertically

5. none of the above

12

Check Yourself

Compare log-log plots of the frequency-response magnitudes of the

following system functions:

1 1 H1(s) =

s + 1 and H2(s) =

s + 10

−2 −1 0 1 2

0

−1

−2logω

−1

|H1(jω)|

|H2(jω)|

log |H(jω)|

13

Check Yourself

Compare log-log plots of the frequency-response magnitudes of

the following system functions:

H1(s) = 1

s + 1 and H2(s) =

1 s + 10

The former can be transformed into the latter by 3

1. shifting horizontally

2. shifting and scaling horizontally

3. shifting both horizontally and vertically

4. shifting and scaling both horizontally and vertically


no scaling in either vertical or horizontal directions !

14

Asymptotic Behavior of More Complicated Systems

Constructing H(s0).

QQ (s0 − zq) ← product of vectors for zeros

q=1 H(s0) = K

PQ (s0 − pp) ← product of vectors for poles

p=1

z1 p1

s0 − z1 s0 − p1

σ

ωs-planes0

15

Asymptotic Behavior of More Complicated Systems

The magnitude of a product is the product of the magnitudes.

Q Q

(s0 − zq) s0 − zq

q=1 q=1P P

p=1 p=1

QQ

QQ|H(s0)| =

K

(s0 − pp)

= |K| s0 − pp

z1 p1

s 0−z 1

s0 −p1

σ

ωs-planes0

16

Bode Plot

The log of the magnitude is a sum of logs.

Q Q

(s0 − zq) s0 − zq q=1 q=1

P P

p=1 p=1

QQ

QQ|H(s0)| = K = |K|

(s0 − pp) s0 − pp

Q PQ log |H(jω)| = log |K| + log jω − zq − log jω − pp

q=1 p=1

Q

17

∣∣∣∣∣∣∣∣∣∣∣∣

∣∣∣∣∣∣∣∣∣∣∣∣

∣∣ ∣∣∣∣ ∣∣

∣∣ ∣∣ ∣∣ ∣∣

s-plane

σ

ω10

−10

10−10

H(s) = s

(s+ 1)(s+ 10)

−2 −1 0 1 2 3

0

−1

−2 logω

log∣∣∣∣ jω

(jω + 1)(jω + 10)

∣∣∣∣log |jω|

−2 −1 0 1 2 3

0

−1 logω

log∣∣∣∣ 1jω + 1

∣∣∣∣

−2 −1 0 1 2 3

−1

−2 logω

log∣∣∣∣ 1jω + 10

∣∣∣∣

Bode Plot: Adding Instead of Multiplying

18

s-plane

σ

ω10

−10

10−10

H(s) = s

(s+ 1)(s+ 10)

−2 −1 0 1 2 3

0

−1

−2 logω

log∣∣∣∣ jω

(jω + 1)(jω + 10)

∣∣∣∣log∣∣∣∣ jω

jω + 1

∣∣∣∣

−2 −1 0 1 2 3

−1

−2 logω

log∣∣∣∣ 1jω + 10

∣∣∣∣


19

s-plane

σ

ω10

−10

10−10

H(s) = s

(s+ 1)(s+ 10)

−2 −1 0 1 2 3

−1

−2

−3 logω

log∣∣∣∣ jω

(jω + 1)(jω + 10)

∣∣∣∣

−2 −1 0 1 2 3

−1

−2 logω

log∣∣∣∣ 1jω + 10

∣∣∣∣


20

s-plane

σ

ω10

−10

10−10

H(s) = s

(s+ 1)(s+ 10)

−2 −1 0 1 2 3

−1

−2

−3 logω

log∣∣∣∣ jω

(jω + 1)(jω + 10)

∣∣∣∣

−2 −1 0 1 2 3

−1

−2 logω

log∣∣∣∣ 1jω + 10

∣∣∣∣


21


The angle response is simple at low and high frequencies.

s-plane

σ

ω5

−5

5−5

H(s) = s− z1

−5 0 5

5|H(jω)|

−5 5

π/2

−π/2

∠H(jω)

22


Three straight lines provide a good approxmation versus log ω.

H(s) = s − z1

−5 5

π/2

−π/2

∠H(jω)

−2 −1 0 1 2

π

2

π

4

0log ω

|z1|

∠H(jω)

lim ∠H(jω) = 0 ω→0

lim ∠H(jω) = π/2 ω→∞

23


The angle response is simple at low and high frequencies.

s-plane

σ

ω5

−5

5−5

H(s) = 9s− p1

−5 0 5

5|H(jω)|

−5 5

π/2

−π/2

∠H(jω)

24

−5 5

π/2

−π/2

∠H(jω)

−2 −1 0 1 2

−π2

−π4

0

log ω

|p1|

∠H(jω)


Three straight lines provide a good approxmation versus log ω.

9 H(s) =

s − p1

lim ∠H(jω) = 0 ω→0

lim ∠H(jω) = −π/2 ω→∞

25

Bode Plot

The angle of a product is the sum of the angles. Q

(s0 − zq) q=1

Q

Pq=1

p=1

z1 p1

∠(s0 − z1) ∠(s0 − p1)σ

ωs-planes0

Q⎞ Q

Q⎛ ⎜⎜⎜⎜⎜⎜⎝

∠H(s0) = ∠ = ∠K + ∠ ∠K s0 − zq − s0 − pp

(s0 − pp)

⎟⎟⎟⎟⎟⎟⎠

PQp=1

The angle of K can be 0 or π for systems described by linear differ

ential equations with constant, real-valued coefficients.

26

s-plane

σ

ω10

−10

10−10

H(s) = s

(s+ 1)(s+ 10)

−2 −1 0 1 2 3

π/2

0

−π/2 logω

log∣∣∣∣ s

(s+ 1)(s+ 10)

∣∣∣∣∠jω

−2 −1 0 1 2 3

0

−π/2 logω

∠1

jω + 1

−2 −1 0 1 2 3

0

−π/2 logω

∠1

jω + 10

Bode Plot

27

s-plane

σ

ω10

−10

10−10

H(s) = s

(s+ 1)(s+ 10)

−2 −1 0 1 2 3

π/2

0

−π/2 logω

log∣∣∣∣ s

(s+ 1)(s+ 10)

∣∣∣∣∠jω

jω + 1

−2 −1 0 1 2 3

0

−π/2 logω

∠1

jω + 10

Bode Plot

28

s-plane

σ

ω10

−10

10−10

H(s) = s

(s+ 1)(s+ 10)

−2 −1 0 1 2 3

π/2

0

−π/2 logω

log∣∣∣∣ s

(s+ 1)(s+ 10)

∣∣∣∣∠jω

(jω + 1)(jω + 10)

−2 −1 0 1 2 3

0

−π/2 logω

∠1

jω + 10

Bode Plot

29

s-plane

σ

ω10

−10

10−10

H(s) = s

(s+ 1)(s+ 10)

−2 −1 0 1 2 3

π/2

0

−π/2 logω

log∣∣∣∣ s

(s+ 1)(s+ 10)

∣∣∣∣∠jω

(jω + 1)(jω + 10)

−2 −1 0 1 2 3

0

−π/2 logω

∠1

jω + 10

Bode Plot

30

From Frequency Response to Bode Plot

The magnitude of H(jω) is a product of magnitudes. Q

jω − zq q=1

P

p=1

QQ|H(jω)| = |K|

jω − pp

The log of the magnitude is a sum of logs. Q PQ

q=1 p=1

Qlog |H(jω)| = log |K| + log jω − zq − log jω − pp

QThe angle of H(jω) is a sum of angles. Q P

q=1 p=1

Q∠H(jω) = ∠K + ∠ jω − zq − ∠ jω − pp

31

∣∣ ∣∣∣∣ ∣∣

∣∣ ∣∣ ∣∣ ∣∣

( ) ( )

Check Yourself

−1 0 1 2 3 4

−2

−3

−4logω

log |H(jω)|

Which corresponds to the Bode approximation above?

1. 1

(s + 1)(s + 10)(s + 100) 2.

s + 1 (s + 10)(s + 100)

3. (s + 10)(s + 100)

s + 1 4.

s + 100 (s + 1)(s + 10)


32

Check Yourself

−1 0 1 2 3 4

−2

−3

−4logω

log |H(jω)|

Which corresponds to the Bode approximation above? 2

1. 1

(s + 1)(s + 10)(s + 100) 2.

s + 1 (s + 10)(s + 100)

3. (s + 10)(s + 100)

s + 1 4.

s + 100 (s + 1)(s + 10)


33

s-plane

σ

ω10

−10

10−10

H(s) = 10s(s+ 1)(s+ 10)

−2 −1 0 1 2 3

0

−1

−2 ω [log scale]logω

−11

log |H(jω)|

−2 −1 0 1 2 3

π/2

0

−π/2 ω [log scale]logω

log∣∣∣∣ s

(s+ 1)(s+ 10)

∣∣∣∣∠H(jω)

Bode Plot: dB

34

s-plane

σ

ω10

−10

10−10

H(s) = 10s(s+ 1)(s+ 10)

0.01 0.1 1 10 100 1000

0

−1

−2 ω [log scale]ω [log scale]

−11

log |H(jω)|

0.01 0.1 1 10 100 1000

π/2

0

−π/2 ω [log scale]ω [log scale]

log∣∣∣∣ s

(s+ 1)(s+ 10)

∣∣∣∣∠H(jω)

Bode Plot: dB

35

s-plane

σ

ω10

−10

10−10

H(s) = 10s(s+ 1)(s+ 10)

0.01 0.1 1 10 100 1000

0

−20


−11

|H(jω)|[dB]= 20 log10 |H(jω)|

0.01 0.1 1 10 100 1000

π/2

0


log∣∣∣∣ s

(s+ 1)(s+ 10)

∣∣∣∣∠H(jω)

Bode Plot: dB

36

s-plane

σ

ω10

−10

10−10

H(s) = 10s(s+ 1)(s+ 10)

0.01 0.1 1 10 100 1000

0

−20


−20 dB/decade20 dB/decade

|H(jω)|[dB]= 20 log10 |H(jω)|

0.01 0.1 1 10 100 1000

π/2

0


log∣∣∣∣ s

(s+ 1)(s+ 10)

∣∣∣∣∠H(jω)

Bode Plot: dB

37

Bode Plot: Accuracy

The straight-line approximations are surprisingly accurate.

0.1 1 10

0

−10

−20 ω [log scale]

|H(jω

)|[d

B]

1 dB3 dB

1 dB

X 20 log10X1 0 dB√2 ≈ 3 dB

2 ≈ 6 dB10 20 dB100 40 dB

H(jω) = 1jω + 1

0.01 0.1 1 10 100

0

−π/4

−π/2 ω [log scale]

∠H

(jω

)

0.1 rad(6◦)

38

Check Yourself

Could the phase plots of any of these systems be equal to

each other? [caution: this is a trick question]

−1

1

−1 1

2

−1( )2

3

−1( )2

4

39

Check Yourself

1.−1

ω

π

−π

2.−1 1

ω

π

−π

3.−1

2 ω

π

−π

4.−1

2 ω

π

−π

40

Check Yourself

1.−1

ω

π

−π

2.−1 1

ω

π

−π if K < 0

3.−1

2 ω

π

−π

4.−1

2 ω

π

−π

41

Check Yourself

Could the phase plots of any of these systems be equal to

each other? [caution: this is a trick question] yes

−1

1

−1 1

2

−1( )2

3

−1( )2

4

phase of 2 could be same as phase of 3: depends on sign of K

42

Frequency Response of a High-Q System

The frequency-response magnitude of a high-Q system is peaked.

s

ω0plane

−1− 1

2Q

√1−

(1

2Q

)2

−√

1−(

12Q

)2

H(s) = 1

1 + 1Q

s

ω0+(s

ω0

)2

−2 −1 0 1 2

0

−1

−2log ω

ω0

log |H(jω)|

43



s

ω0plane

−1− 1

2Q

√1−

(1

2Q

)2

−√

1−(

12Q

)2

H(s) = 1

1 + 1Q

s

ω0+(s

ω0

)2

−2 −1 0 1 2

0

−1

−2log ω

ω0

log |H(jω)|

44



s

ω0plane

−1− 1

2Q

√1−

(1

2Q

)2

−√

1−(

12Q

)2

H(s) = 1

1 + 1Q

s

ω0+(s

ω0

)2

−2 −1 0 1 2

0

−1

−2log ω

ω0

log |H(jω)|

45



s

ω0plane

−1− 1

2Q

√1−

(1

2Q

)2

−√

1−(

12Q

)2

H(s) = 1

1 + 1Q

s

ω0+(s

ω0

)2

−2 −1 0 1 2

0

−1

−2log ω

ω0

log |H(jω)|

46



s

ω0plane

−1− 1

2Q

√1−

(1

2Q

)2

−√

1−(

12Q

)2

H(s) = 1

1 + 1Q

s

ω0+(s

ω0

)2

−2 −1 0 1 2

0

−1

−2log ω

ω0

log |H(jω)|

47

Check Yourself

Find dependence of peak magnitude on Q (assume Q > 3).

s

ω0plane

−1− 1

2Q

√1−

(1

2Q

)2

−√

1−(

12Q

)2

H(s) = 1

1 + 1Q

s

ω0+(s

ω0

)2

−2 −1 0 1 2

0

−1

−2log ω

ω0

log |H(jω)|

48

Check Yourself

Find dependence of peak magnitude on Q (assume Q > 3).

Analyze with vectors.

low frequencies

σ/ω0

ω/ω0

−1− 1

2Q

1× 1 = 1

high frequencies

σ/ω0

ω/ω0

−1− 1

2Q1

2Q × 2 = 1Q

Peak magnitude increases with Q ! 49


As Q increases, the width of the peak narrows.

s

ω0plane

−1− 1

2Q

√1−

(1

2Q

)2

−√

1−(

12Q

)2

H(s) = 1

1 + 1Q

s

ω0+(s

ω0

)2

−2 −1 0 1 2

0

−1

−2log ω

ω0

log |H(jω)|

50



s

ω0plane

−1− 1

2Q

√1−

(1

2Q

)2

−√

1−(

12Q

)2

H(s) = 1

1 + 1Q

s

ω0+(s

ω0

)2

−2 −1 0 1 2

0

−1

−2log ω

ω0

log |H(jω)|

51



s

ω0plane

−1− 1

2Q

√1−

(1

2Q

)2

−√

1−(

12Q

)2

H(s) = 1

1 + 1Q

s

ω0+(s

ω0

)2

−2 −1 0 1 2

0

−1

−2log ω

ω0

log |H(jω)|

52



s

ω0plane

−1− 1

2Q

√1−

(1

2Q

)2

−√

1−(

12Q

)2

H(s) = 1

1 + 1Q

s

ω0+(s

ω0

)2

−2 −1 0 1 2

0

−1

−2log ω

ω0

log |H(jω)|

53



s

ω0plane

−1− 1

2Q

√1−

(1

2Q

)2

−√

1−(

12Q

)2

H(s) = 1

1 + 1Q

s

ω0+(s

ω0

)2

−2 −1 0 1 2

0

−1

−2log ω

ω0

log |H(jω)|

54

Check Yourself

Estimate the “3dB bandwidth” of the peak (assume Q > 3).

Let ωl (or ωh) represent the lowest (or highest) frequency for

which the magnitude is greater than the peak value divided by√2. The 3dB bandwidth is then ωh − ωl.

s

ω0plane

−1− 1

2Q

√1−

(1

2Q

)2

−√

1−(

12Q

)2

−2 −1 0 1 2

0

−1

−2log ω

ω0

log |H(jω)|

55

Check Yourself

Estimate the “3dB bandwidth” of the peak (assume Q > 3).


low frequencies

σ/ω0

ω/ω0

−1− 1

2Q√

2 12Q × 2 =

√2Q

1− 12Q

high frequencies

σ/ω0

ω/ω0

−1− 1

2Q√

2 12Q × 2 =

√2Q

1 + 12Q

Bandwidth approximately 1 Q

56


As Q increases, the phase changes more abruptly with ω.

s

ω0plane

−1

H(s) = 1

1 + 1Q

s

ω0+(s

ω0

)2

−2 −1 0 1 2

0

−π/2

−πlog ω

ω0

∠H(jω)

57



s

ω0plane

−1− 1

2Q

√1−

(1

2Q

)2

−√

1−(

12Q

)2

H(s) = 1

1 + 1Q

s

ω0+(s

ω0

)2

−2 −1 0 1 2

0

−π/2

−πlog ω

ω0

∠H(jω)

58



s

ω0plane

−1− 1

2Q

√1−

(1

2Q

)2

−√

1−(

12Q

)2

H(s) = 1

1 + 1Q

s

ω0+(s

ω0

)2

−2 −1 0 1 2

0

−π/2

−πlog ω

ω0

∠H(jω)

59



s

ω0plane

−1− 1

2Q

√1−

(1

2Q

)2

−√

1−(

12Q

)2

H(s) = 1

1 + 1Q

s

ω0+(s

ω0

)2

−2 −1 0 1 2

0

−π/2

−πlog ω

ω0

∠H(jω)

60



s

ω0plane

−1− 1

2Q

√1−

(1

2Q

)2

−√

1−(

12Q

)2

H(s) = 1

1 + 1Q

s

ω0+(s

ω0

)2

−2 −1 0 1 2

0

−π/2

−πlog ω

ω0

∠H(jω)

61

Check Yourself

Estimate change in phase that occurs over the 3dB bandwidth.

s

ω0plane

−1− 1

2Q

√1−

(1

2Q

)2

−√

1−(

12Q

)2

H(s) = 1

1 + 1Q

s

ω0+(s

ω0

)2

−2 −1 0 1 2

0

−π/2

−πlog ω

ω0

∠H(jω)

62

Check Yourself

Estimate change in phase that occurs over the 3dB bandwidth.


low frequencies

σ/ω0

ω/ω0

−1− 1

2Qπ

2 −π

4 = π

4

1− 12Q

high frequencies

σ/ω0

ω/ω0

−1− 1

2Qπ

2 + π

4 = 3π4

1 + 12Q

Change in phase approximately π 2 .

63

Summary

The frequency response of a system can be quickly determined using

Bode plots.

Bode plots are constructed from sections that correspond to single

poles and single zeros.

Responses for each section simply sum when plotted on logarithmic

coordinates.

64

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6.003 Signals and SystemsFall 2011

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lecture 11: continuous-time frequency response and bode plots · 6.003: signals and systems ct...

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