wireless access systems: introduction and course outline

Wireless Access Systems:Introduction and Course Outline

2Communication Technology LaboratoryWireless Communication Group

Some Wireless Access Systems

• Wireless Access Systems provide short to medium range tetherless access to a backhaul network, a central unit or peer nodes

• Examples include

– Bluetooth– WLAN– Vehicular Networks– WiMax– RFID– WBAN– WPAN


Wireless Access Systems are Ubiquitous

Internet


Some More Applications

intelligent homeambient intelligencesecurity wearable computing

shopping

defencesurveillance

supply chain management

logisticsindustrial

information exchangepervasive computingvirtual reality

health carehome care

communicationsinstant messagingenterprise communication

environment

trafficsecuritysurveillanceaccess control

Internet


Characteristics of Wireless Access Systems

Heterogeneous standards• IEEE 802.11 WLAN• IEEE 802.15 WPAN• IEEE 802.16 WMAN• (Hiperlan)• Bluetooth• DECT• various RFID standards

• RFID tags, readers• sensors, actors• communication appliances• information access• information processing• backhaul access points

Heterogeneous nodes

Lots of spectrum (approx.)• [email protected] (ISM)• [email protected] (ISM)• [email protected] (ISM)• [email protected] (ISM)• >3GHz@5GHz (UWB)

WLAN

Internetbackhaul

Sensornetwork

RFID

cellular:GSMUMTS

BluetoothWPAN

WMAN

Pervasive wirelessaccess


The Throughput - Range Tradeoff

RFID

Bo

dy A

rea Netw

orks

100M

10M

1M

100k

10k

1k

1 3 10 30 100 range [m]

link throughput[bps]

11b

11a 11g

15.4

15.3

15.1

Sensor Netw

orks

15.3a

WLAN

WPAN

UWB

Bluetooth

ZigBee

UWB region(conceptional)


Outline of Course

Fundamentals

1. Fundamentals of short/medium range wireless communication 1

– digital transmission systems– equivalent baseband model– digital modulation and ML-detection


– fading channels– diversity– MIMO wireless


– Multicarrier modulation and OFDM

Systems I: OFDM based broadband access

4. WLAN 1: IEEE 802.11g, a

5. WLAN 2: IEEE 802.11n

6. WMAN: (mobile) WiMAX

7. Vehicular Networks

Systems II: Wireless short range access technolgies and systems

8. UWB 1: Promises and challenges of Ultra Wideband Systems

9. UWB 2: Physical Layer options

10. Wireless Body Area Network case study: UWB based human motion tracking

11. The IEEE 802.15x family of Wireless Personal Area Networks (WPAN): • Bluetooth, • ZigBee, • UWB

Systems III: RF identification (RFID) and sensor networks

12. RFID 1

13. RFID 2


Exercises: Motivation

• Simulate, practice, verify, learn and have fun

• We will simulate the theoretical ideas/methods/techniques that we learn throughout the lecture.

• MATLAB (matrix laboratory) will be used for simulations.

• In general we will simulate

– Single carrier transmission– Multi-carrier transmission– Wireless Channel– Channel coding– Simple UWB transceiver


Exercises: Organization

• Students organize in 2 or 4 groups

• There will be three exercises with two tasks each during the semester.

• Each group will perform one of the two tasks and then present the results.

• The general schedule of tasks:

– Introduction of tasks.

– Working period (2 weeks). Present afterwards.• Each group will work individually.

– Combining period (1 week). Present afterwards.• Two groups will work in collaboration.

• For further details will be presented in the first exercise lecture next week 8:15


Schedule:

8:15-9:00 9:15-10:00 10:15-11:00 11:15-12:00

Week 1 Fundamentals of wireless

communications. 1 Fundamentals of wireless


communications. 1

Week 2Introduction –First Exercise

Fundamentals of wireless communications. 2



Week 3 Fundamentals of wireless

communications. 3Fundamentals of wireless


communications. 3

Week 4 Presentation of Ex 1/ 1 Presentation of Ex 1/2 WLAN - 1 WLAN - 1

Week 5optional: wrap up of

simulation basicsoptional: revised solutions of Ex

1/1 and EX 1/2 WLAN - 2 WLAN - 2

Week 6Introduction-

Second ExercisePresentation of Ex 1 -

Combination step WiMAX 1 WiMAX 1

Week 7 Vehicular Networks Vehicular Networks

Week 8 Presentation of Ex 2/1 Presentation of Ex 2/2 UWB 1 UWB 1

Week 9Introduction –

Third ExercisePresentation of Ex 2 -

Combination step UWB 2 UWB 2

Week 10 WBAN WBAN

Week 11 Presentation of Ex 3/1 Presentation of Ex 3/2 WPAN WPAN

Week 12 Presentation of Ex 3 -

Combination step RFID 1 RFID 1

Week 13 RFID 2 RFID 2

Wireless Access Systems:Fundamentals of Short Range Wireless Communications


Fundamentals of Short Range Wireless: Outline

1. Digital transmission and detection on the AWGN channel– digital transmission systems– equivalent baseband model– digital modulation and ML-detection

2. Fading channels– fading channels– diversity– MIMO wireless

3. Modulation schemes for frequency selective channels– multicarrier modulation – Orthogonal Frequency Division Multiplexing (OFDM)


Equivalent Baseband Representation

5.14Communication Technology LaboratoryWireless Communication Group

Narrowband Case: Equivalent Baseband Model with Bandpass Channel, Different TX and RX Lowpasses and Frequency/Phase Offset

Notes • f0 and are called the reference

frequency and phase of the BB model

• for f0 = f1 the BB model is time-invariant (a filter)

1 12 cos t

1 12 sin t

trI

tsRX

trQ tsTX

tsQ

0 02 cos t

0 02 sin t

+

tsI

( )BP

H f

I Q

s t

s t js t

I Qr t r t jr t

( )TX

H f ( )RX

H f

)( 10 je

( )TX

H f ( )RX

H f

0real AWGN; / 2N

0complex AWGN; N

TXH f 0BPH f f RXH f

Narrowband case:

0 1( )e j t

00

0

TX

RX

H f f f

H f f B

Notation:

for 0

0 elsewhereBP

BP

H f fH f

0

0 0 1with max 0,B f f f

5.15Communication Technology LaboratoryWireless Communication Group

Narrowband Case: Relation of Physical Signals and Their Complex Baseband Representation

• The spectrum of the analytic signal in terms of the physical signal is given by

Re{} Re{}

0( )BPH f f

0 1( )e j t

)( 10 je

0 0( )2 j te

)(ts )(tr

Re{} Im{}

)(tsI )(tsQ ( )TXs t )(tsRX

Re{} Im{}

)(trI )(trQ

( )TX

H f ( )RX

H f

0complex AWGN; N

( ) : complex envelope of ( )

( ): analytic signal (pre-envelope of ( ))

( ): physical passband signal

TX TX

TX TX

TX

s t s t

s t s t

s t

( )TXs t

( )TXs t

Names and Notation:

2 0

0 elsewhereTX

TX

S f fS f


Transmission of Digital Information I: Generation of Finite Signal Sets (Modulation)


General Block Diagram of a Digital Modulator

• The information bit vector contains N bit • It is mapped onto a message index (i)• We use a look-up table with 2N transmit waveforms• The transmit signal is selected according to the message index• The process of selecting a transmit signal according to an information bit

vector is called modulation• For finite N this structure models a block transmission

(1)TXs t

Mapper( )i

i

TXs t

(2)TXs t

(2 )N

TXs t

TXs t


Signal Space Representation of Digital Modulator

• The signal space is defined by a set of orthonormal basepulses

• The basepulses are stacked to form the basepulse vector – orthonormality implies

• The signal space representation of the transmit signals is obtained by the projection

– we refer to as transmit symbol vector

(1)TXs

Mapper( )i

i

TXs

(2)TXs

(2 )N

TXs

H t

2NM

t

( ) ( )i iTX TXs t s t dt

Ht t dt I

TXs t( )i

TXs

TXs

Look-up table


Linear Modulation

• For linear modulation schemes the transmit symbol vector is obtained by a linear transformation of the input symbol vector– precoding matrix GTX

• Dramatically reduces the size of the look-up table– general modulation: exponential growth with the number N of information

symbols– linear modulation: linear growth

TXG

TXs

H t

TXs t


Some Popular Linear Modulation Schemes

name symbol alphabet 3TX SG N

2-PAM1 0 0

0 1 0

0 0 1

1 1

4-QAM(QPSK)

1 0 0 0 0

0 0 1 0 0

0 0 0 0 1

j

j

j

2 1 2 0 0 0 0 0 0 0 0

0 0 0 0 2 1 2 0 0 0 0

0 0 0 0 0 0 0 0 2 1 2

j j

j j

j j

16 - QAM


Filter Implementation of Linear Modulator: Nyquist Basepulses

• Nyquist basepulses (orthonormal)

• Nyquist criterion

TXG

TXs

H t

TXs t

g t

t

H t

g t mT g t nT dt m n

T

2/ 1/

k

G f k T T

t

TXs t

T


Transmission of Digital Information II: Transmission and Detection


• Channel is modelled as additive noise source

• In many cases of practical interest the noise can be characterized as zero mean stationary Gaussian random process w(t)

– any set of samples is jointy normally distributed

– autocorrelation function

– power density spectrum

Additive White Gaussian Noise (AWGN) Channel

• For analytical tractability usually a white noise process is assumed

– for physical system models (real-valued signals) we have

– for complex baseband representations as used herein we have

TXs t

w t

r t

*wR E w t w t

wD f R

0 0

2 2w

N ND f R

0 0wD f N R N


Sufficient Statistic and Symbol Discrete System Model

• Bank of correlators generates the decision vector

• the decision vector is a sufficient statistics (for additive white Gaussian noise; AWGN)

– contains all information for the optimal estimation of the transmit symbol vector

• With the impulse correlation matrix

we obtain the symbol discrete system model

– the elements of the noise vector are statistically independent and identically distributed Gaussian random variables

TXs t

H t

AWGN

w t

r t dTXs

t

Bank of correlators

dTXs

0,w CN I

1/ 2

Ht t dt

d

Symbol discrete system model

Continuous time system model

w


Frequency Selective Channel

w t

TXG TXs t

H t

h t

channel

TXs t

h t r t

t

d

• The channel is represented by a filter h(t) and AWGN

• A channel matched filter is required prior to the correlator bank in order to obtain a sufficient statistics

• These filters may affect the resulting impulse correlation matrix– intersymbol interference

(ISI)channelmatchedfilter


P

Form-Invariant Basepulses

TXs t

w t

r t d

TXs

Continuous time system model

impulsemodulator

g(t) h(t) h*(-t) g*(-t)S

kT

d

TXs

0, ppn CN

pp

Symbol discrete system model

0 1 2 3

1 0 1 2

2 1 0 12 2

3 2 1 0

with

and

k

pp

p p p pp p kTp p p p

p p p p

p p p p p t G f H f


Transmission of Digital Information III: Decoding


Maximum Likelihood Decoder and Decision Regions

• With orthonormal basepulse vector the impulse correlation matrix becomes the identity matrix

• The decoder observes the decision vector and generates an estimate of the transmit symbol vector

• To minimize the probability of error the decoder selects the hypothesis, which has the minimum Euclidean distance to the decision vector (Maximum Likelihood (ML) decoder)

– decision regions in the signal space

d

TXs

20, ww CN I

I decoder

T̂Xs

2

ˆ

ˆ arg min

ˆ

kTX

k

k

TX TX

k d s

s s

d

TXs T̂Xs


Example: Error Performance of QPSK

Decision regions

bE

2 22

Bit error probability Symbol error probability

2 1 1 2b

b s b b bw

EP Q P P P P

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Q-Function

Q(x

)x

(1,1)

(1,0)

(0,1)

(0,0)

Gray mapping(bit 1, bit 2)


Fading I: Time Selective (Narrowband) Fading Channels

Communication Technology LaboratoryWireless Communication Group

Path Loss and Short Term Fading

TX RX

power [dB]

distance(log(x))

free space20 dB/dec

urban40 dB/dec

rural30 dB/dec

distance (log(x))

0( ) expTXs t j t 0( ) ( ) expr t r x j t

210log( ( ) )r x

32


Doppler Shift I: 1800 Angle of Arrival

• Received signal in complex passband notation

• For (small scale effects) we obtain the complex envelope of the receive signal

– depends only on the displacement . In the spectral domain we obtain the (spatial) Doppler shift

e j t0 x0 x

0 0 0

0

( , ) ( ) exp 2 / exp 2 / expcr x t a x x j x j x j t

c

0( ) exp 2 /r x c j x

0( ) ( ) ( 1/ )x xr x R f c f

0( , ) ( ) exp 2 / expcr x t a x j x j t

x

0x x

x1/

33


• For a linear movement of the receiver the spatial variations translate linearly into equivalent temporal variations

– the corresponding frequency shift follows as

Doppler Shift II: Arbitrary Angle of Arrival

• Complex envelope of received signal

– in the spectral domain we obtain the spatial Doppler shift

e j t0

x

xx cos( )

1( ) exp 2 (cos / )r x c j x

cos HzD xD

vf f v

1 1( ) ( ) exp 2 (cos / )r t r x vt c j vt

cosxDf

x v t

cosx

34


Multipath Propagation and Fading

• Complex envelope of the received signal

• Due to the different frequency shifts of the components, the magnitude of the received signal varies with the displacement: fading

• Example:

– note the spaced zero crossingse j t0

xx0 x

,1

( ) exp 2N

n xD nn

r x c j f x

( ) cos 2 /r x x

/x

( )r x

0.5 1

/ 2

,1 ,2 1 21/ ; .5xD xDf f c c

35


Doppler Spectrum: Power Spectral Density of Fading Process

• infinite number of scatterers under average receive power constraint: continuous PSD of fading process

• Scattering coefficients cn modelled as uncorrelated random variables with variance – fading described as random

process• Power spectral density (PSD) of

fading process for

– note the relation between Doppler shift fxD,n and the angle of arrival

,1

( ) exp 2N

n xD nn

r x c j f x

2, ,

1

N

r x c n x xD nn

D f f f

,

cos nxD nf

fxD

PSD

2,c n

n

,1 , xD xD Nf f 0n

E c

36


Jake's Doppler Spectrum

• Cumulative power distribution versus frequency

• Power spectral density

– "Jake's Spectrum"

cP

arccos xf

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

fx

Pc

arccosc x xP f f

2

1

1x c x

xx

dD f P f

df f

uniform continuousscattering around receiver

cumulative power distribution

• Relation of angle of arrival and Dopper shift


Jake's Channel Model for Linear Movement

• Multiplicative fading• Speed of movement: v

TXs t

z t

AWGN

w t

r t

1/ 421 /D DH f f v

complex whiteGaussian noiseprocess

fD


Fading II: Frequency Selective Fading


Broadband Channel Measurement

• Channel measurement with a short impulse h(t) (broadband)

• All scatterers, which lead to a given path delay

are located on an ellipse• Typical received signal:( )h t x

x0x

1,

1,,

0

1 1 1 , ,, const

' ''1 1 1

0 1 2 3

40


Scattering Function

• The scattering function describes the average power spectral density of the received signal as a function of Doppler shift fx and delay

x

S2S1

S3

S4

321

2 31

S1

S2

S3

S4

Dopplershift fx

, xS f

41


Doppler Spectrum of Narrowband System

• rms Doppler spread

with the mean Doppler shift

Doppler Spectrum2 31

S1S3

S4Dopplershift

, xS f

,r x xD f S f d

1/ 22

x x r x x

f

r x x

f f D f df

D f df

x r x x

x

r x x

f D f dff

D f df

• Scattering function– 2nd order statistics of the spatio-

temporal fading process • a narrow band system can not

resolve the multiple paths– narrowband fading with Doppler

spectrum

42


Delay Power Spectrum of Broadband System

• Scattering function– 2nd order statistics of the spatio-temporal fading process

• Delay power spectrum

– rms delay spread

with the mean delay

2 31

S1S3

S4Dopplershift

2 31

Delaypowerspectrum

, x xP S f df

, xS f

1/ 22P d

P d

P d

P d

43


Classification of Multipath Channels

• The signal bandwidth and the duration of the transmit burst determine the fading model

– flat: no significant variation over the interval of interest

– selective: varies significantly over the interval of interest

• Narrowband systems experience frequency-flat fading

• Broadband systems experience frequency-selective fading

• A block fading model is suitable in the time-flat regime

– may be either frequency-flat or frequency selective

• Systems below the red curve are not physically implementable

• Note the role of Doppler spread and delay spread

signal bandwidth B

burst duration TBURST

time-selectivefrequency-selective

time-flatfrequency-selective

time-selectivefrequency-flat

time-flatfrequency-flat

1/

1/ f

TBurst=1/B


Typical Time-Selective/Frequency Selective Channel Model

• A discrete delay power spectrum is specified

– paths (delays) are usually clustered

• For each path (k) (i.e. delay ) a Doppler spectrum is specified

– default: Jake's spectrum– if specified in terms of spatial frequency fx,

substitute f x = f/v for linear movement with velocity v

• The filter coefficients are filtered complex normal random processes

– in line of sight (LoS) situations: nonzero mean

1

( )s t

(1) ( )z t

2

(2) ( )z t

3

(3) ( )z t

+

( )r t

delay power spectrum

delay

k

kD f

white complex normal random process

kD f

( ) ( )kz t( ) ( )kw t

Structure

Generation of fading processes

kP

Specification

45


Special Cases

1

( )s t

(1)z

2

(2)z

3

(3)z

+

( )r t

Block fading channel

• Note that the coefficients are random variables (not processes)

• For each channel realization a new set of random variables is generated

–

– non LoS:

,kk kz CN m P

0km

Frequency-flat fading

( )s t

( )z t ( )r t

white complex normal random process D f

( )z t( )w t

• Generation of fading processes

46


Fading III: Impact on Error Performance

47


Frequency-Flat Fading Channel

w t

TXG

H t

fadingchannel

t

d

channelmatched"filter"

z t *z t

1 /d d z

TXs

20, ww CN I

1/ 2

Symbol discrete system model with block fading

z

• block fading: fading variable instead of fading process

• note multiplication with magnitude of fading variable due to

– channel matched "filter"– normalization of decision vector

1 /d d z


Error Performance of QPSK in Frequency-Flat Block Fading

• In frequency-flat block fading the error performance of QPSK is determined by the instantaneous value of the fading variable

• We can define various figures of merit. Frequently used are– outage probability: probability, that

the instantaneous bit error probability is above a target value

– fading averaged bit error probability

• Clearly these figures of merit depend on the probability density function (pdf) of the fading amplitude

z

2

2 bb

w

EP z Q z

0 0Prout bP P P z P

b bzP E P z

• here is a chi2 random variable with 2 degrees of freedom

2y z

2 20 / dBb wE z E

fadi

ng a

vera

ged

bit e

rror

pro

babi

lity

0P

z


• Special case L=1– the fading variable z is complex

normally distributed; – is the sum of two

statistically independent squared real-valued normal random variable

– If , is Rayleigh distributed; Rayleigh fading

– if , is Rician distributed. K-factor:

• General case L=N: N-fold diversity– For , is the sum of 2L

squared real-valued Gaussian random variables

– chi2-distribution with 2L degrees of freedom

– e.g. achievable with L receive antennas

• Approximation: BER (c/SNR)L

Diversity

•

2 2 21 2y z x x

2 20 / dBb wE z E

fadi

ng a

vera

ged

bit e

rror

pro

babi

lity

z

22 2

1

L

kk

y z x

2y z

z 0E z

2 2zK E z

0E z

0E z

Vector/Matrix Channels

Single Input/Single Output (SISO)

• channel coefficient

Single Input/Multiple Output (SIMO)

• channel vector

Multiple Input/Multiple Output (MIMO)

• channel matrix

h

h

H H

h

h

52

53

Diversity Techniques

• Wireless channel varies in time, frequency and space Time, frequency and space diversity available

• Examples: – Time diversity: repeat same codeword after channel varied (Repetition Code)

– Frequency diversity: transmit same symbol over two or more OFDM sub-carriers (if fading of the sub-carriers is uncorrelated)

– Space diversity: use more than one antenna at RX or TX or on both sides

– (But usually pure repetition is not an efficient way to code: repetition of the same information in time or frequency sacrifices bandwidth space diversity seems promising)

Diversity, MIMO

54

• Receive diversity: using NRX receive antennas

(spatial dimension)

• High diversity factors available for high carrier frequencies and large bandwidths

• Transmit diversity: in addition a temporal coding needed

Space-Time Codes

Diversity, MIMO

Spatial (or Antenna) Diversity

TX RX


System Model with RX Diversity and Maximum Ratio Combining

w t

TXG

H t

block fadingvector channel

t

d

channelmatched"filter"

h

Hh

1 /d d h

TXs

20, ww CN I

1/ 2

Scalar symbol discrete system model

h

• note multiplication with magnitude of fading vector due to– channel matched "filter"– normalization of decision vector

1 /d d h

RXN 11TXs

56

• Receive diversity:hi : channel gain between the TX

antenna and the RX antenna i

Diversity, MIMO

Probability of Error

TX RX

1 2, , ,RX

T

Nh h h h

new argument of Q-Function:

2

22 b

Sw

EP Q h

2 2

2 2

1=b b

RXw w RX

E Eh N h

N

Array Gain and Diversity Gain

2

22 b

w

EQ h

Array (Power) gain Diversity gain

• Expression converges to constant for increasing NRX

i.e. fading is eliminated in the limit

• 3 dB gain per doubling of the number of RX antenna

57

Multiple Input/ Multiple Output

Single Input/Single Output (SISO)

• channel coefficient

Single Input/Multiple Output (SIMO)

• channel vector

Multiple Input/Multiple Output (MIMO)

• channel matrix

h

h

H H

h

h

58

Free Space vs. Multipath Propagation

/ 2

scattering

fading

h

h

h

h

59

Multiple Antennas and Spatial Multiplexing

h h

1 1 1

1 1 1

1 1 1

H

0.8 0.4 0.6

0.9 1.3 1.5

0.2 1.0 1.4

H

Channel Matrix

Singular Value Decomposition H H U S V

3 0 0

0 0 0

0 0 0

S

2.9 0 0

0 0.7 0

0 0 0.1

Srank 1• full rank• 3 spatial subchannels• spatial multiplexing

unitary3 3;H H U U I V V I

60

61

MIMO Wireless Capacity (1)

• MIMO channel capacity grows nearly linearly with N = min(NTX, NRX)

[Foschini, Gans, 1998] [Telatar, 1999]

TX RX

• N decoupled spatial sub-channels available (Spatial Multiplexing)

• Higher data rate without need of higher bandwidth spectral efficiency

Diversity, MIMO

K-f

acto

r of

Ric

ian

fadi

ng

62

MIMO Wireless Capacity (2)

TX RX

Diversity, MIMO

20

log detRX

HSN

TX

EC E

N N

I HH

Telatar, Foschini:

NTX: number of TX antennas, NRX: number of RX antennas.

K-f

acto

r of

Ric

ian

fadi

ng

63

MIMO Systems: Spatial Subchannels

Subchannels

0 0

0 0

0 0

S

RXN

TXN

A priori transmit channel state information (CSIT) necessary !

TX Diversity:Take only the “best“subchannel !

Spatial Multiplexing:Take all !

, unitary matricesHH USV U VSVD of MIMO channel matrix:

H H H U H V U U S V V S

Diversity, MIMO

64

MIMO Systems without CSIT: Spatial Subchannels

if no a priori CSIT: TX Combining not possible;Spatial multiplexing leads to ISI

Receiver has to compensate ISI due to V (cf. BLAST);

Diversity, MIMO

65

BLAST Architecture

[Gesbert, et al.: From Theory to Practice: An Overview of MIMO Space–Time Coded Wireless Systems]

Diversity, MIMO

66

BLAST (2)

TX RX

H

•Antenna array at TX and RX

•Spatial Data Pipes in rich scattering (MIMO channel H of high rank) without

increasing the bandwidth

•Spatial Multiplexing achieves ergodic MIMO capacity

•ISI compensation at RX necessary, because no CSIT used in BLAST system

•BLAST: no combination of diversity techniques and spatial multiplexing

Diversity, MIMO


Multicarrier Modulation I: Continuous Time Implementation


P

Discrete Channel Impulse Response

2 2* ** * *

k

p t g t g t h t h t G f H f

p p kT

w td

ka t kT g(t) h(t) h*(-t) g*(-t)

S

kT

-T -T T T

p0p-1 p1

+

ka kn

kd


Low and High Data Rate Systems

* ** **g t g tt h t tp h

gT

2 gT

hT

2 hT

t t

t

2 g hT T

low data rate:

high data rate:

g hT T T

g hT T T

p t

no ISI

severe ISI

transmit basepulse channel impulse response

**g t g t **h t h t


Multicarrier Modulation

• Transmit in N subbands, for each of which the channel transfer function is approximately constant– minimizes ISI in each subband– subband center frequency fk

– for non-overlapping subbands, there is no inter-subband (inter-carrier) interference (ICI)

• One multicarrier (MC) symbol consists of N transmit symbols:• Subbands implemented by letting

• The symbol rate on each subcarrier is

f

H(f)

TXs

H t

TXp t

exp

sinc( / )

k k

TX MC

t j t

p t t T

1/MC MCf T

1f Nf

TXs


Transmit and Receive Window

• For sinc-windows there is strictly no interference between adjacent subcarriers (non-overlapping spectra)– the ISI matrix in the discrete system model is diagonal

• Due to their infinite duration sinc-windows are not implementable.• Is it possible to design finite length windows without introducing

interference?

H t

TXp t w t

h(t)

t

RXp t

d

pp

TXs

transmit window

receive window


Eigenfunction of the Convolution

• We consider the response of the channel h(t) to a step function with center frequency fk

• We observe a transient with duration Th

• After the transient the response is a scaled version of the input signal– scaling factor H(fk)

– complex exponentials are eigenfunctions of convolution

• After an appropriate window, which blanks the transient, we obtain the input-output relation

• This observation is the key to the design of a finite window for MC

hT

t

t

t

exp kt j t

h t

expk kH f j t

transient exp exph tk h k kt j t t T H f j t

t


Equivalent Diagonal Channel Matrix

expHkt j t

TXp t w t

h(t)

t

RXp t

d

MCT

CT

hT

H t

rect / ct T w t t

RXp t

d

1 0 0

0 0

0 0 N

H f

H f

Equivalent model:

TX and RX window:

equivalent channel matrixTXs

TXs


Receive Window

• As the equivalent channel matrix is diagonal, it does not affect orthogonality any more

• What is the impact of the receive window?

• Pulse correlation matrix

– the Fourier transform PRX(f) of the receive window pRX(t) determines the pulse correlation matrix

exp

HRX

mn n m RX

RX n m

t t p t dt

j t p t dt

P f f f

• For a uniform subcarrier spacing we have– the Fourier transform of the

receive window needs to have

spaced zeros• In this case the received window

has to fullfil the following condition (temporal dual to the spectral Nyquist condition)

– the most compact window thus is given by

– this implies • A temporal roll off improves the

robustness to frequency offsets

n mf f n m f

/RXk

p t k f const

f

rectRXp t t f 1/MCT f


Transmit Window

• With we obtain

• Example for N=21; pTX(t)=rect(t/(Tc+Th)

MCT

CT

hT

• The transmit window has to be constant for at least

• This window determines the power spectral density of the transmit signal

h cT T

TXD f

2

1

N

TX TX kk

D f P f f

TX TXP f p t

-50 -40 -30 -20 -10 0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency/f

pow

er s

pect

ral d

ensi

ty


Multicarrier Modulation II: Discrete Implementation (Orthogonal Frequency Division Multiplexing; OFDM)


P

Discrete Channel Impulse Response

2 2* ** * *

k

p t g t g t h t h t G f H f

p p kT

w td

ka t kT g(t) h(t) h*(-t) g*(-t)

S

kT

-T -T T T

p0p-1 p1

+

kakn

kdd

a

n

Vector signal model


Response to Periodic Input Sequence

T T

p0 p1

+

ka kn

kd

d1 ..................................... dNd1 ..................................... dN

1T

Na a

• Discrete channel impulse response with L taps

• for illustration assumed causal

Idea: use periodic input sequence to generate periodic response

1 Na a1 Na a

d

a

n

pp0 1

1 0

1 0

1 0

0 0

0 0

0 0

0 0

pp

p p

p p

p p

p p

Equivalent channel matrix is circulant

1T

Nd d d


Cyclic Prefix and Circulant Channel Matrix

• Circulant channel matrix due to cyclic prefix of length larger or equal to L-1

T T

p0 p1

+

ka kn

kd

d1 ..................................... dNcyclic prefix

1 Na a

d

a

n

pp0 1

1 0

1 0

1 0

0 0

0 0

0 0

0 0

pp

p p

p p

p p

p p

NCP

• Discrete channel impulse response with L taps

• for illustration assumed causal

1CPN N Na a


Diagonalization of Circulant Matrix: Orthogonal Frequency Division Multiplexing (OFDM)

• All (NxN) circulant matrices are diagonalized by the Fourier matrix FN

1

1

1 with

DFT C :,1

H H HN N N N N N N N N N

T

N

F C F diag c c F DFT I F F F F IN

c c

w t

d

g(t) h(t) a(t)

kT; k=1.. N+NCP

insertCP

removeCP

HNF NF

a S

P

P

S

d

a

n

diagonal channel matrix CD

diag DFT C :,1CD

circulant channelmatrix C

kT; k=1.. N+NCP IFFTN a

1/ FFTN r


Comparison of Discrete and Continuous Time Implementation of Multicarrier Modulation

d

LowPass

LowPass

kT; k=1.. N+NCP

insertCP

removeCP

HNF NF

a S

P

P

S

kT; k=1.. N+NCP

expHmt j t

TXp t

t

RXp t

d

a

CPN N T N T

Multicarrier Transmitter Multicarrier Receiver

Note:

[ , ] exp( 2 )N

m

mF m n j n

NT

, 0,..., 1m n N

wireless access systems: introduction and course outline

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