Where we’ve been

Attenuate,Amplify,Linearize,Filter

Timing limitations to making measurements

Gain-BandwidthProduct, etc.

ThermalMass, ForExample

Where we’re going

Speed,StorageIssues

Frequency Space

Some signals are like this...

time

voltage

Some signals are like this...

time

voltage

But many signals are like this...

time

voltage

Introducing imaginary numbers

6or 636

366 2

366 2

36 ? ? ?

Introducing “i” (sometimes called ‘j’)

i 1 12 i

Any number that is a multiple of i is an imaginary number:

i

i

i

i

01.0

4

2

cos

4

222

i

ix

i

i

Multiplication rules

A real number times a real number is a real number

A real number times an imaginary number is an imaginary number

An imaginary number times an imaginary number is a real number

5.45.13

ii 2.16.02

335.12 2 iii

Powers of i

1

1

1

4

3

2

1

0

i

ii

i

ii

i anything to the “0” power = 1

anything to the “1” power = itself

Introducing complex numbers

Any number that is a sum of a real number and an imaginarynumber is a complex number:

i23

imaginarypart

realpart

323Re i 223Im i

NOT “2i”

Real versus imaginary parts

All of the real components of a complex number, takentogether, are the real part. The same holds for the imaginarypart:

i

iii

24

42314321

imaginarypart

realpart

424Re i 224Im i

Adding/subtracting complex numbers

When adding (or subtracting) complex numbers, add(or subtract) the real and imaginary parts separately:

i

i

i

75

32

43

i

i

i

53

34

21

i

i

i

21

42

23

Multiplying complex numbers

When multiplying a complex number by any other number,multiply both the real and imaginary parts

i

ii

1215

4353453

Multiplying complex numbers (continued)

When multiplying two complex numbers, an easy method is the “FOIL” method:

i

ii

iiiiii

223

8101215

425243534523

First Outer Inner Last

Example

Perform the following addition. Identify the real and complexparts of the answer:

ii 2352

Example

Perform the following subtraction. Identify the real and complex parts of the answer:

ii 342

Example

Perform the following multiplication. Identify the real and complex parts of the answer:

ii 2321

Complex numbers as vectors

R e

I m

+ 2

- 3 i

3 + 4 i

real axis

imaginary axis

complex plane

Magnitude of a complex number

22ImRe AAA

R e

I m

3 + 4 i

3 2 + 42 = 52

5251694343 22 i

Example

Sketch the following complex numbers as vectors. Whatare their magnitudes?

i42 i23 i 4

Direction of a complex number

R e

I m

3 + 4 i

3 2 + 42 = 52

A

A

Re

Imtan 1

A

Example

What is the phase angle of each of these complex numbers?

i42 i23 i 4

063)2

4arctan( 034)

3

2arctan(

0166)4

1arctan(

Definition of the complex exponential

sincos iei

Re

Im

0

13543

Magnitude of the complex exponential

Re

Im

0

13543

11sincos 22 ie

For any

1

1

Magnitude of the complex exponential

Re

Im

0wt

owt 13543

tA

tiAtAAe ti

cos

sincosReRe

Amplitude A

A

AAAeAAe ii 1sincos 22

Simple harmonic motion!

A·cos(135)

The real part is what we observe.

= t

Aeit is in effect a spinning complex vector that generates

- a cosine function on the real axis and - a sine function on the imaginary axis

re

im

x=Acos(t)

y=A

sin(

t)F(t)= Aeit =

A[cos(t)+ i sin(t)] =

Acos(t) + iA sin(t)A

A

A AA

A

t=0t=900

t=1800

Real part describesmotion of mass on spring

Using a complex amplitude

tieA 5

tieiB 43

Re

Im0t

Same magnitude, different phase

tieA 5

tieiB 43

Re

Im

0

011

011

53

05

0tan

Re

Imtan

533

4tan

Re

Imtan

AB

A

B

isdifferencephase

A

A

B

B

t = 0 positions(Aeit = e0 = 1)

52543 22 B

B

A

Using a complex amplitude

25ieA

243ieiB

Re

Im

)90(2

0 ttovectorsrotatenow

Phase differenceis still the same sinceboth vectors rotated by900!

Real amplitude yields pure cosine wave in real space

-4

-3

-2

-1

0

1

2

3

4

0 0.5 1 1.5 2 2.5

time

t

tite

eExampleti

ti

cos3

sin3cos3Re3Re

3:

Imaginary amplitude yields pure sine wave in real space

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5

time

t

titiie

ieExampleti

ti

sin2

sin2cos2Re2Re

2:2

Complex Real amplitude = sine/cosine mixture

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5

time

tt

titi

titei ti

sincos2

sinsin2

coscos2Re2Re

2

Sine waves can be mixed with DC signals, or with other sine

waves to produce new waveforms. Here is one example of a

complex waveform:

V(t) = Ao + A1sin1t + A2sin 2t + A3sin 3t + … + Ansin nt--- in this case---V(t) = Ao + A1sin1t

Ao

A1

Fourier Analysis

Just an AC component superimposed on aDC component

More dramatic results are obtained by mixing a sine wave of a particular frequency

with exact multiples of the same frequency. We are adding harmonics

to the fundamental frequency. For example, take the fundamental frequency and add 3rd

harmonic (3 times the fundamental frequency) at reduced amplitude, and subsequently add

its 5th, 7th and 9th harmonics:

Fourier Analysis, cont’d

the waveform begins to look more and more like a square wave.

This result illustrates a general principle first formulated by the

French mathematician Joseph Fourier, namely that any complex waveform

can be built up from a pure sine waves plus particular harmonics of the

fundamental frequency. Square waves, triangular waves and sawtooth waves

can all be produced in this way.

...)7sin(7

1)5sin(

5

1)3sin(

3

1)sin(

1

1)(

,

tttttf

thatshownbecanitwavesquarethefor

oooo

(try plotting this using Excel)

Fourier Analysis, cont’d