multiband transceivers - [chapter 1]

Multiband RF Transceiver SystemChapter 1

From Basics to RF Transceivers李健榮助理教授

Department of Electronic EngineeringNational Taipei University of Technology

Outline

• Definition of dB

• Phasor

• Modulation

• Transmitter Architecture

• Demodulation

• Receiver Architecture

• From Fourier Transform to Modulation Spectrum

Department of Electronic Engineering, NTUT2/44

Definition of dB

• , 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= ⋅

Relative (Ratio, unitless, dB)

Absolute (Have unit, dBW, dBm, dBV…)


In some textbooks, phasor may be

represented as

Euler’s Formula

• 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


Coordinate Systems

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


Sine Waveform

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

Cosine Waveform

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.


Complex Plan (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 = −


Complex Plan (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:


Sine Waveform

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.


Phasor Representation (I) – Sine Basis

( ) ( ) { } { }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


Phasor Representation (II) – Cosine Basis

( ) ( ) { } { }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


Phasor Representation (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


Phasor Everywhere

• 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=


Modulation

• Why modulation?� Communication

� Bandwidth

� Antenna Size

� Security, avoid Interferes, etc.

Voice

Electric signal

AudioEquipment

AudioEquipmentModulator Demodulator

Electric signal

Voice


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)


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)


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)


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


Linear Modulation

• 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.


Linear Modulator

• 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

22/44

Transmitter Architecture (I)

• 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


Transmitter Architecture (II)

• Polar Transmitter

( ) ( ) ( ) ( ){ } ( ) ( ){ }22cos 2 Re Re c BBcj f t tj f t

m BB c BB l BBs t A t f t t s t e A t e π φππ φ + = ⋅ + = ⋅ = ⋅

( )BBA t

cos ctω

( )ms t

Switching-mode PA

Antenna

PhaseModulator

Matching

( )BBA t

( )BB tφ

Bas

eban

d

Pro

cess

or

AmplitudeModulator

• Linear regulator• PWM modulator• Class-S modulator

• Linear modulator to generate PM signal• Frequency synthesizer or PLL-based PM modulator

• Analog scheme: EER

( ){ }2Re c BBj f t te

π φ+


Linear Demodulation

( ) ( ) ( ) ( ) ( )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)


Linear Demodulator

( )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


Receiver Architecture

• 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


Digital Modulation

• I(t) and Q(t) for digital transmission?

• Assume thatI(t) and Q(t) are pulses at TX,I(t) and Q(t)waveforms will be recovered ideally after the demodulationprocess, of course, pulses.


( ) ( ) ( )cos 2 sin 2m c cs t I t f t Q t f tπ π= −

( )I t

cos ctωsin ctω−

( )Q t

( )ms t

LPF

LPF

( )I t

cos ctωsin ctω−

( )Q t

TX RX

Assume the LPF has asufficiently wide bandwidth torecover the pulse waveform.

28/44

• A Quadrature Phase Shift Keying (QPSK) signal is a goodexample.

Quadrature Phase Shift Keying

( )I t

( )Q t

( )1,1

( )1, 1−( )1, 1− −

( )1,1−

cA+

cA+cA−

cA−

Constellation

( )I t

cos ctωsin ctω−

( )Q t

S/PConverter

BinaryBaseband

Data( )I t

( )Q t

BinaryBaseband

Data

bT

2 bTt

S/P Converter

� The linear combination shows that the QPSK signalhas 4 different phase states(1 symbol = 2 bits = 4states).


Symbol

29/44

• Phase transition in QPSKsignal due to simultaneous transitionof I(t) andQ(t).

Phase Transition

( )I t

( )Q t

S/PConverter

BinaryBaseband

Data

( )I t

( )Q t

t

( )I t

( )Q t

( )1,1

( )1, 1−( )1, 1− −

( )1,1−

cA+

cA+cA−

cA−

Constellation

Constant envelope


Bandwidth Consideration (I)

• A rectangular waveformhas many frequency componentscovering within a wide bandwidth. For many reasons, themodulation spectrumoccupying such a wide bandwidth is notpreferable.

( ) ( ) ( )cos 2 sin 2m c cs t I t f t Q t f tπ π= −

( )I t

cos ctωsin ctω−

( )Q t

TX

f

f

f

( )mS f

How to limit the bandwidth?

cf

cff

0

0


Bandwidth Consideration (II)

• Is that good to limit the bandwidth at passband?

( )I t

cos ctωsin ctω−

( )Q t

TX

BPF

cff

channel

f

( )I t

cos ctωsin ctω−

( )Q t

BPF2

BPF1

BPFnchannel

f

channelchannel

Requiring many BPFs for each channel is impractical

selector

cf


Bandwidth Consideration (III)

• Limit the bandwidth at baseband –Pulse Shaping

( )I t

cos ctωsin ctω−

( )Q t

LPF

LPF

f

t

0

f

t

0

f0

f0

cff

The low-pass filter is use to shape thewaveform, thus called “pulse shapingfilter,” or “shaping filter.”

Constant envelope (before shaping)

180� 180�90�

Time-varying envelope (after shaping)

Smooth the sharp transition

( )1,1( )1,1−( )Q t

( )I t

( )1, 1−( )1, 1− −


Band-limiting and Inter Symbol Interference (I)

• Nyquist Filter :Produce pulse shapes with no ISI at each sampling instant

Brick-wall LPF

IFT

Sinc shape

Raised Cosine Filter


Band-limiting and Inter Symbol Interference (II)

• Gaussian Filter :Time domain response is Gaussian as well, it exhibits no overshot or ringing in thetime domain. This smooth well-behaved impulse response results in very littleISI

� Reduce bandwidth : Smaller BT causes even faster spectral roll-off , but this has a price, smaller BT cause more ISI.

� Relative bandwidth BT = (filter BW) / (Bite rate)

Power spectra of MSK and GMSK Signals for varying BT

BT=0.2

BT=0.25

BT=0.3

Impulse response

MSK : BT is infinite(no filter)


Optimum Receiver

( )I t

cos ctωsin ctω−

( )Q t

( )ms t

LPF

LPF

( )I t

cos ctωsin ctω−

( )Q t

LPF

LPF

TX RX

Matched Filter


Picture of a Practical Transceiver

( )I t

cos ctωsin ctω−

( )Q t

LPFFilter

Filter

DAC

DAC LPF

Digital Processor

Digital pulse shaping filter

Waveform recovery filterfor DAC

Data (bits)Waveform

(digital, M-bits)Quantized Waveform (analog)Recovered Waveform (analog)

( )I t

cos ctωsin ctω−

( )Q t

( )ms t

LPF

LPF

Filter

Filter

Digital Processor

Matched filter orcorrelator

ADC

ADC

Demodulated Waveform (analog)

Sampled Waveform (digital)

Decision

Decision

Data (bits)

Mixing spurs remover

Encoder and decoder are notincluded here for simple.


Modulation in Frequency Domain

• Spectrum of a Real Signal

• AM, PM, and Linear Modulated Signal

• Concept of Complex Envelope


Modulation Spectrum (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”

39/44

PhaseModulator

Modulation Spectrum (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

40/44

Modulation Spectrum (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

41/44

Concept of the Complex Envelope (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−

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


( )l cS f f∗ − −

Concept of the Complex Envelope (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


Summary

• 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.
