射頻電子 - [第一章] 知識回顧與通訊系統簡介

高頻電子電路第一章知識回顧與通訊系統簡介

李健榮助理教授

Department of Electronic EngineeringNational Taipei University of Technology

大綱

• dB的定義

• 相量(Phasor)

• 調變• 線性調變與線性發射機• 線性解調變與線性接收機• 調變訊號譜• 複數波包

Department of Electronic Engineering, NTUT2/40

dB的定義

• , where

• Power gain

• Voltage gain

• Power (dBW)

• Power (dBm)

• Voltage (dBV)

• Voltage (dBuV)

( )dB 10 log G= ⋅ ( )aG b=

2

110 log P

P = ⋅

2

120 log V

V = ⋅

( )10 log 1-WP= ⋅

( )10 log 1-mWP= ⋅

( )20 log 1-VoltV= ⋅

( )20 log 1- VV

µ= ⋅

相對的相對的相對的相對的(Relative )(比值比值比值比值, 無單位無單位無單位無單位, dB)

絕對的絕對的絕對的絕對的(Absolute )(單位單位單位單位, dBW, dBm, dBV…)


In some textbooks, phasor may be

represented as

尤拉公式

• Euler’s Formula states that: cos sinjxe x j x= +

( ) ( ) ( ){ } { }cos Re Rej t j j tp p pv t V t V e V e eω φ φ ωω φ += ⋅ + = ⋅ = ⋅

( )cos sindef

jp p pV V e V V jφ φ φ φ= ⋅ = ∠ = +• Phasor :

� Don’t be confused withVector which is commonly denoted as .A�

phasor

A real signal can be represented as:

V

V

( ) ( )cospv t V tω φ= ⋅ +


Euler’s Trick on the Definition of e

2 3

lim 1 11! 2! 3!

nx

n

x x x xe

n→∞

= + = + + + +

…

x jx=

( ) ( )2 3 2 4 3 5

1 11! 2! 3! 2! 4! 3! 5!

jx jx jxjx x x x xe j x

= + + + + = − + − + + − + − +

… … …

• Euler played a trick : Let , where 1j = −

1lim 1

n

ne

n→∞

= +

6/33

2 4

cos 12! 4!

x xx = − + − +…

3 5

sin3! 5!

x xx x= − + − +…

cos sinjxe x j x= +

cos sinjxe x j x− = −

cos2

jx jxe ex

−+=

sin2

jx jxe ex

j

−−=

• Use and

we have


座標系統

x-axis

y-axis

x-axis

y-axis

P(r,θ)

θ

rP(x,y)

2 2r x y= +1tan

y

xθ −=

cosx r θ=siny r θ=

� Cartesian Coordinate System � Polar Coordinate System

(x,0)

(0,y)

( )cos ,0r θ

( )0, sinr θ

Projectionon x-axis

Projectionon y-axis


正弦波形

x-axis

y-axis

P(x,y)

x

yr

θ θθ

y

θ0 π/2 π 3π/2 2π

� Go along the circle, the projection on y-axis results in a sine wave.


x

θ

0

π/2

π

3π/2

餘弦波形

x-axis

y-axis

θ

� Go along the circle, the projectionon x-axis results in a cosine wave.

� Sinusoidal waves relate to aCirclevery closely.

� Complete going along the circle tofinish a cycle, and the angleθrotates with 2π rads and you areback to the original starting-pointand. Complete another cycle again,sinusoidal waveformin one periodrepeats again. Keep going along thecircle, the waveform willperiodically appear.


複數平面(I)

It seems to be the same thing with x-y plan, right?

• Carl Friedrich Gauss (1777-1855) defined the complex plan.He defined the unit length on Im -axis is equal to “j”.A complex Z = x + jy can be denoted as (x, yj) on the complex plan.(sometimes, ‘j’ may be written as ‘i’ which represent imaginary)

Re-axis

Im-axis

Re-axis

Im-axis

P(r,θ)

θ

rP(x,yj)

2 2r x y= +1tan

y

xθ −=

cosx r θ=siny r θ=

(x,0j)

(0,yj)

( )cos ,0r θ

( )0, sinr θ

( )1j = −


複數平面(II)

Re-axis

Im-axis

1

Every time you multiply something by j, that thing will rotate 90 degrees.

1j = − 2 1j = − 3 1j = − − 4 1j =

1*j=jj

j* j=-1

-1

-j-1*j=-j -j* j=1

(0.5,0.2j)(-0.2, 0.5j)

(-0.5, -0.2j)(0.2, -0.5j)

• Multiplying j by j and so on:


正弦波

Re-axis

Im-axis

P(x,y)

x

yr

θ θθ

y = rsinθ

θ0 π/2 π 3π/2 2π

To see the cosine waveform, the same operation can be applied to trace out the projection on Re-axis.


相量表示法 (I) –以sine為基底

( ) ( ) { } { }sin Im Imj j t j jsv t A t Ae e Ae eφ ω φ θω φ= + = =

Re-axis

Im-axis

P(A,ϕ)

y = Asinϕ

θ0 π/2 π 3π/2 2π

ϕ

tθ ω=

Given the phasor denoted as a point on the complex-plan, you should know itrepresents a sinusoidal signal. Keep this in mind, it is veryimportant!

time-domain waveform


相量表示法 (II) –以cosine為基底

( ) ( ) { } { }cos Re Rej j t j jsv t A t Ae e Ae eφ ω φ θω φ= + = =

Re-axis

Im-axis

P(A, ϕ)

y = Acosϕ

θ0 π/2 π 3π/2 2π

ϕ

tθ ω=

time-domain waveform


相量表示法 (III)

( ) ( ) { }11 1 1 1sin Im j j tv t A t Ae eφ ωω φ= + =

Re-axis

Im-axis

P(A1, ϕ1)

ϕ1

P(A2, ϕ2)

P(A3, ϕ3)

θ0 π/2 π 3π/2 2π

tθ ω=

A1sinϕ1

( ) ( ) { }22 2 2 2sin Im j j tv t A t A e eφ ωω φ= + =

( ) ( ) { }33 3 3 3sin Im j j tv t A t A e eφ ωω φ= + =

A2sinϕ2

A3sinϕ3


到處都是相量

• Circuit Analysis, Microelectronics:Phasor is often constant.

• Field and Wave Electromagnetics, Microwave Engineering:Phasor varies with the propagation distance.

• Communication System:Phasor varies with time (complex envelope, envelope, orequivalent lowpass signal of the bandpass signal).

( ) ( )5cos 1000 30sv t t= + � 5 30sV = ∠ �

( ) ( ) ( ) ( ) ( ){ }, cos cos Re j x t j x tv x t A x t B x t Ae Beβ ω β ωβ ω β ω − − += − + + = +

( ) j x j xV x Ae Beβ β−= +

( ){ }Re j tV x eω=


調變(調制)

• Why modulation?

� Communication

� Bandwidth

� Antenna Size

� Security, avoid Interferes, etc.

Voice

Electric signal

AudioEquipment

AudioEquipmentModulator Demodulator

Electric signal

Voice


振幅調變(Amplitude Modulation)

( ) ( ) cos 2m BB cs t s t A f tπ= ⋅

Baseband real signal

Voice

Electric signal

AudioEquipment


Electric signal

Voice

( )BBs tcos2 cA f tπ

Carrier (or local)

High-frequency sinusoid

Amplitude-modulated signal(AM signal)


頻率調變(Frequency Modulation)

( ) ( ){ }cos 2m c f BBs t A f K s t tπ = + ⋅

Voice

Electric signal

AudioEquipment


Electric signal

Voice



Carrier (or local)


Frequency-modulated signal(FM signal)


相位調變(Phase Modulation)

Voice

Electric signal

AudioEquipment


Electric signal

Voice

( ) ( )cos 2m c p BBs t A f t K s tπ = + ( )cos 2 c BBA f t tπ φ= +



Carrier (or local)


Phase-modulated signal(PM signal)


線性調變(Linear Modulation)

( ) ( ) ( )cos 2m BB c BBs t A t f t tπ φ= ⋅ +

Voice

Electric signal

AudioEquipment


Electric signal

Voice



Carrier (or local)


Linear-modulated signal

( )BBs t ( ) ( ), ?BB BBA t tφ


線性調變之數學推導

• Consider a modulated signal

( ) ( ) ( ) ( ) ( ){ }2cos 2 Re c BBj f t t

m BB c BB BBs t A t f t t A t e π φπ φ + = ⋅ + = ⋅

( ) ( ) ( ) ( ) ( ){ }2 2Re Re cos sinBB c cj t j f t j f tBB BB BB BBA t e e A t t j t eφ π πφ φ = ⋅ = ⋅ +

( ) ( ) ( ) ( ) ( ) ( )cos sinBBj tl BB BB BB BBs t A t e A t t j tφ φ φ= ⋅ = ⋅ +

( ) ( ) ( ) ( ) ( ) ( )cos sinBB BB BB BBA t t jA t t I t jQ tφ φ= ⋅ + ⋅ = +

( ) ( ) ( ) ( ){ }Re cos2 sin 2m c cs t I t jQ t f t j f tπ π= + ⋅ +

( ) ( )cos 2 sin 2c cI t f t Q t f tπ π= −

Time-varying phasor (information in both amplitude and phase)

( )BBs t : real

( )ls t : complex

Modulated signal is the linear combination of I(t), Q(t), and the carrier. Thus the linear modulator is also called “I/Q Modulator,” and it is an universal modulator.


線性調變器

• The modulator accomplishes the mathematical operation.

( ) ( ) ( ) ( ) ( ){ }Re cos sin cos 2 sin 2m BB BB BB c cs t A t t j t f t j f tφ φ π π= ⋅ + +

( ) ( ) ( ) ( )cos cos2 sin sin 2BB BB c BB BB cA t t f t A t t f tφ π φ π= −

( ) ( )cos 2 sin 2c cI t f t Q t f tπ π= −

( )I t

cos ctωsin ctω−

( )Q t

( )ms t

( )I t

cos ctωsin ctω

( )Q t

( )ms t

+

− 90�

( )I t

cos ctω

( )Q t

( )ms t

Department of Electronic Engineering, NTUT

I component Q component

I channel Q channel

22/40

線性發射機架構

• Linear Transmitter

90�

( )I t

cos ctω

( )Q t

( )ms t

Power Amplifier(PA)

Antenna

Bas

eban

d

Pro

cess

or

90�cos ctω

( )ms t

Power Amplifier(PA)

Antenna

Matching /BPF

Matching

( )I t

( )Q t

Bas

eban

d

Pro

cess

or


線性解調變

( ) ( ) ( ) ( ) ( )cos 2 cos2 sin 2m BB c BB c cs t A t f t t I t f t Q t f tπ φ π π= ⋅ + = −

( ) ( ) ( ) ( ) ( ) ( ) ( )2 1 1cos2 cos 2 sin 2 cos2 cos4 1 sin 4 sin0

2 2m c c c c c cs t f t I t f t Q t f t f t I t f t Q t f tπ π π π π π= − ⋅ = ⋅ + − ⋅ +

( ) ( ) ( ) ( ) ( ) ( ) ( )2 1 1sin 2 cos2 sin 2 sin 2 sin 4 sin0 1 cos4

2 2m c c c c c cs t f t I t f t f t Q t f t I t f t Q t f tπ π π π π π− = − + = − ⋅ + + ⋅ −

( ) ( ) ( )cos4 sin 4

2 2 2c c

I t I t Q tf t f tπ π

= + −

( ) ( ) ( )sin 4 cos4

2 2 2c c

Q t I t Q tf t f tπ π

= − +

?Receiver

( )ms t ( )BBs t

Received modulated signal:

Multiplied by “cosine”:

Multiplied by “ −−−− sine”:

High-frequency components (should be filtered out)

High-frequency components (should be filtered out)


線性解調器

( )I t

cos ctωsin ctω−

( )Q t

( )ms t

LPF

LPF

( )I t

( )Q t

( )ms t

LPF

LPF

90�cos ctω

( ) ( ) ( ) ( ) ( )BBj tl BBs t A t e I t jQ tφ= ⋅ = +

( ) ( ) ( )2 2BBA t I t Q t= +

( ) ( )( )

1tanBB

Q tt

I tφ −=

BasebandProcessing

Original Information (or data)( )I t

( )Q t


線性接收機架構

• Linear Receiver (direct conversion)

90�

( )I t

cos ctω

( )Q t

( )ms t

Low Noise Amplifier(LNA)

Bas

eban

d

Pro

cess

orLPF

LPF

Matching /BPF

90�

( )I t

cos ctω

( )Q t

( )ms t

Low Noise Amplifier(LNA)

Bas

eban

d

Pro

cess

orLPF

LPF

Matching


調變訊號的頻譜

• Fourier Series Representations

• Non-periodic Waveform and Fourier Transform

• Spectrum of a Real Signal

• AM, PM, and Linear Modulated Signal

• Concept of Complex Envelope


傅立葉級數

• There are three forms to represent the Fourier Series of aperiodic signal :

� Sine-cosine form

� Amplitude-phase form

� Complex exponential form

( ) ( )0 1 11

cos sinn nn

x t A A n t B n tω ω∞

=

= + +∑

( ) ( )0 11

cosn nn

x t C C n tω φ∞

=

= + +∑

( ) 1jn tn

n

x t X e ω∞

=−∞

= ∑

( )x t


t

x(t)

t

t

t

( )X jω

ω1f 13 f 15 f

.etc

T1

1 1C φ∠

2 2C φ∠

3 3C φ∠

28/40

Sine-Cosine Form

( )0 0

area under curve in one cycle

period T

1 TA x t dt

T= =∫

( ) 10

2cos , for 1 but not for 0

T

nA x t n tdt n nT

ω= ≥ =∫

( ) 10

2sin , for 1

T

nB x t n tdt nT

ω= ≥∫

is the DC term(average value over one cycle)

• Other than DC, there are two components appearing at a givenharmonic frequency in the most general case: a cosine termwith an amplitudeAn and a sine termwith an amplitudeBn.

(A complete cycle can also be noted from )~2 2

T T−


Amplitude-Phase Form

( ) ( )0 11

cosn nn

x t C C n tω φ∞

=

= + +∑

( ) ( )0 11

sinn nn

x t C C n tω θ∞

=

= + +∑

2 2n n nC A B= +

• The sumof two or more sinusoids of a given frequency isequivalent to a single sinusoid at the same frequency.

• The amplitude-phase formof the Fourier series can beexpressed as either

or

0 0C A= is the DC term

is the net amplitude of a given component at frequencynf1,since sine and cosine phasor forms are alwaysperpendicular to each other.

where


Complex Exponential Form (I)

11 1cos sinjn te n t j n tω ω ω= +

11 1cos sinjn te n t j n tω ω ω− = −

1 1

1cos2

jn t jn te en t

ω ω

ω−+=

1 1

1sin2

jn t jn te en t

j

ω ω

ω−−=

cos sinjxe x j x= +

cos sinjxe x j x− = −

cos2

jx jxe ex

−+=

sin2

jx jxe ex

j

−−=

Recall that• Euler’s formula

1nω is called thepositive frequency, and 1

nω− thenegative frequency

� From Euler’s formula, we know that both positive-frequency and negative-frequency terms are required to completely describe the sine or cosinefunction with complex exponential form.

Here

1jn te ω

1jn te ω−


Complex Exponential Form (II)

1 1jk t jk tk kX e X eω ω−

−+ �( )where kkX X− =

( ) 1jn tn

n

x t X e ω∞

=−∞

= ∑

( ) 1

0

1 T jn tnX x t e dt

Tω−= ∫

• The general formof the complex exponential formof theFourier series can be expressed as

whereXn is a complex value

• At a given real frequencykf1, (k>0), that spectral representationconsists of

The first term is thought of as the “positive frequency” contribution, whereas the second is thecorresponding “negative frequency” contribution.Although either one of the two terms is acomplex quantity, they add together in such a manner as to create a real function, and thisis why both terms are required to make the mathematical form complete.


當週期趨近無限大

T 2T 3T 4T 5T

( )x t

f

nX

T 2T

T

T

f

nX

f

nX

f

nX

Single pulse T → ∞


傅立葉轉換

( ) ( ) X f F x t= F ( ) ( )1 x t F X f−= F

( ) ( ) j tX f x t e dtω∞ −

−∞= ∫

( ) ( ) j tx t X f e dfω∞

−∞= ∫

• Fourier transformation and its inverse operation :

• The actual mathematical processes involved in these operationsare as follows:

2 fω π=

• The Fourier transform is, in general, a complex functionand has both a magnitude and an angle:

( )X f

( ) ( ) ( ) ( ) ( )j fX f X f e X f fφ φ= = ∠( )X f

fFor the nonperiodic signal, its spectrum iscontinuous, and, ingeneral, it consists of components atall frequenciesin therange over which the spectrum is present.


調變譜 (I)

• From Euler’s Formula :

• AM signal (DSB-SC)

cos2

jx jxe ex

−+=

A “ real signal” is composed of positive and negative frequency components.

( ) ( )cos2m cs t A t f tπ=

Two-sided amplitude frequency spectrum

( ) ( )2 1000 2 1000150cos 2 1000

2j t j tt e eπ ππ × − ×× = +

2525

0 Hz 1 kHz1 kHz−f

One-sided amplitude frequency spectrum

50

0 Hz 1 kHz

( )50cos 2 1000tπ ×

f

t( ) ( )BBs t A t=

ff

cf0 Hzcf−0 Hz

USBLSBUSBLSBLSBUSB

cos2 cf tπ


“ real signal”

35/40

PhaseModulator

調變譜 (II)

t( )BBs t

f0 Hz

USBLSBcos2 cf tπ

( ) ( )2 2

2 2c cj t j tj f t j f tA A

e e e eφ φπ π− −= +

( ) ( )( )cos 2m cs t A f t tπ φ= +

( ){ } ( ){ }2 2Re Rec cj f t t j t j f tA e A e eπ φ φ π+ = ⋅ = ⋅


“ real signal”

fcf0 Hzcf−

USBLSBLSBUSB

“complex”“complex” “ real”

• PM signal

Complex conjugate

36/40

調變譜 (III)

I/QModulator

t( )BBs t

f0 Hz

USBLSBcos2 cf tπ

( ) ( ) ( ) ( )2 2

2 2c cj t j tj f t j f tA t A t

e e e eφ φπ π− −= +

( ) ( ) ( )( )cos 2m cs t A t f t tπ φ= +

( ) ( ){ }2Re cj t j f tA t e eφ π= ⋅

“ real signal”

• I/Q modulated signal

( )I t

( )Q t

fcf0 Hzcf−

USBLSBLSBUSB

“complex”“complex” “ real”


Complex conjugate

37/40

複數波包的概念 (I)

• Bandpass real signal :

( ) ( ) ( )( ) ( ) ( ) ( ) ( )2 2cos 22 2

c cj t j tj f t j f tm c

A t A ts t A t f t t e e e eφ φπ ππ φ − −= + = +

( ) ( ) ( ) ( )2 21 1

2 2c cj t j tj f t j f tA t e e A t e eφ φπ π− −= +

( )ls t ( )ls t∗

( )lS f∗( )lS f

Complex timed value

Spectrum

( ) ( ) ( ) ( )2 21 1

2 2c cj t j tj f t j f tA t e e A t e eφ φπ π− −= +

( ) 2 cj f tls t e π⋅ ( ) 2 cj f t

ls t e π−∗ ⋅

( )l cS f f∗ − −( )l cS f f−

Complex timed value

Spectrum( ) ( ) ( )1

2m l c l cS f S f f S f f∗ = − + − −

fcf0 Hzcf−

USBLSBLSBUSB

( )1

2 l cS f f−( )1

2 l cS f f∗ − −

Spectrum of the bandpass signal


複數波包的概念 (II)

• Equivalent low-pass signal (complex envelope):

f0 Hz

( )lS f

cfcf−

( ) 21

2cj f t

ls t e π⋅( ) 21

2cj f t

ls t e π−∗ ⋅

( ) ( ) ( ) ( ) ( )j tls t A t e I t jQ tφ= = +

( ) ( ) ( )1

2m l c l cS f S f f S f f∗ = − + − −

fcf0 Hzcf−

USBLSBLSBUSB

( ) ( )1

2I t jQ t+

Spectrum of the bandpass signal

( ) ( )1

2I t jQ t−

( )ms t

( ) ( ) ( ) ( ) ( )BBj tls t A t e I t jQ tφ= = +

complex envelope

( ) ( ) ( ) ( ) ( ) 2cos 2 Re cj t j f tm cs t A t f t t A t e eφ ππ φ = ⋅ + = ⋅

( ) ( ){ }2Re cj f tI t jQ t e π= +

complex envelopecarriercarrier2 cj f te π

carrier


本章總結

• In this chapter, the phasor was introduced to manifest itself inthe mathematical operation for communication engineering.

• A modulated signal is a linear combination ofI(t), Q(t), andthe carrier. This mathematical combination can be realizedwith a practical circuitry, say, “modulator.”

• The demodulation is the decomposition of the modulatedsignal, which is the reverse process to recover the basebandsignalI(t) andQ(t).

• The modulated signal can be viewed as a complex envelopecarried by a sinusoidal carrier. With this equivalent lowpassform to represent a bandpass system, the mathematicalanalysis can be easily simplified.


射頻電子 - [第一章] 知識回顧與通訊系統簡介

Engineering

Transcript of 射頻電子 - [第一章] 知識回顧與通訊系統簡介