where we’ve been attenuate, amplify, linearize, filter
Post on 03-Jan-2016
219 Views
Preview:
TRANSCRIPT
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
top related