Post on 02-Apr-2018
EECE5984CE5984Orthogonal Frequency Division Multiplexing and Related Orthogonal Frequency Division Multiplexing and Related
TechnologiesTechnologiesFall 2007Fall 2007
Mohamed Essam Khedr
Modulation (Mapping) in OFDM
SyllabusSyllabus• Wireless channels characteristics (7.5%) 1
• OFDM Basics (10%) 1
• Modulation and Coding (10%) 2– Linear and nonlinear modulation – Interleavin\g and channel coding – Optimal bit and power allocation – Adaptive modulation
P/SQAM
decoder
invert channel
=frequencydomain
equalizer
S/P
quadratureamplitude
modulation (QAM)
encoder
N-IFFTadd
cyclic prefix
P/SD/A +
transmit filter
N-FFT S/Premove cyclic prefix
TRANSMITTER
RECEIVER
N subchannels N complex samples
N complex samplesN subchannels
Receive filter
+A/D
multipath channel
An OFDM Modem An OFDM Modem
Bits
00110
OFDM Mathematics1
2
0
( ) k
Nj f t
kk
s t X e π−
=
= t ≡ [ 0,Τos]
Orthogonality Condition
*1 2
0
( ). ( ) 0T
g t g t dt =In our case
2 2
0
. 0p q
Tj f t j f te e dtπ π− =
For p ≠ q Where fk=k/T
os
os
os
Spectrum ShapingSpectrum Shaping
• FCC manages spectrum• Specifies power spectral
density mask– Adjacent channel interference– Roll-off requirements
• Implications to OFDM– Zero tones on edge of band– Time domain windowing
‘smoothes’ adjacent symbols
IEEE 802.11a
Inband
Adjacent channel
Reference: Std 802.11a
frequency
Zero tones
Ofdm symbol
Mitigating Mitigating MultipathMultipath effectseffects• Channel estimation required • Training based methods
– Tradeoffs in overhead, complexity, and delays
• Linear equalizers can completely eliminate ISI, but this may enhance noise.
• Decision feedback (nonlinear) equalizers can improve performance.
EqualizationEqualization• Digital Equalizer
• Criterion for coefficient choice– Minimize BER (Hard to solve for ws)– Eliminate ISI (Zero forcing, enhances noise)– Minimize MSE between dn and dn
nneq zwzwwzH −− +++= ...)( 1
10
Discrete Random Variables and ProbabilityDiscrete Random Variables and ProbabilityQuick RevisionQuick Revision
Random variable X assumes a value as a function from outcomes of a process which can not be determined in advance.
Sample space S of a random variable is the set of all possible values of thevariable X.
ΩΩΩΩ: set of all outcomes and divide it into elementary events, or states
1)(
=x
xp 0)(1 ≥≥ xp
a5
a5
1
Continuous Random VariablesContinuous Random Variables
1
)()( αα
α XX Ff∂∂=
)Pr()( αα ≤= XFX
Cummulative distribution function(CDF)
Probability density function (PDF)
1)( =∂∞
∞−
ααXf
?)(
?)(
=−∞=∞
X
X
F
F
Ensemble Average Ensemble Average
• Mean:– Continuous
– Discrete
• Variance
∞
∞−
∂== ααα )( E XfXX
( )222 XX −=σ
==k
kk XX )Pr( αα
( ) ( ) ( ) ααασ ∂−=−= ∞
∞−Xx fXXX
222 E
Correlation & CovarianceCorrelation & Covariance• Crosscorrelation • Covariance
• If <X> or <Y> equal zero, correlation equals covariance
• Note that X* means the complex conjugate of X
*XYE=XYr
( )( ) ***XY YXXYYYXX-c −=−= EE
irir jXXjXXX −=+= ** )(
1−=j
Random ProcessRandom Process
• X, Y need not be separate events• X, Y can be samples of process observed at different instants t1, t2
( ) ( ) 2*
121 E),( tZtZttRZ =
( )( ) *2211
*21 )()()(t)(E),( tZtZZtZttCZ −−=
( )1tZX =
( )2tZY =
*XY XYr E=
( )( ) *XY YYXX-c −= E
Independence vs. UncorrelatednessIndependence vs. Uncorrelatedness
• R.V.s independent
• Uncorrelated (weaker condition), when
• R.V. X, Y uncorrelated if covariance is zero.
• Independent R.V. always uncorrelated.• Uncorrelated R.V. may not be independent!
)()(),( βαβα YXXY fff =
*** EEE YXYXXYrXY ===
( )( ) *EE YXXYYYXX-c **XY −=−=
Major Channel EffectsMajor Channel Effects• Propagation Loss: attenuation, also called path loss
• Gaussian Noise and Interference: Time variant nature due to mobility of objects in an environment
• Time Dispersion: multiple reflections due to obstacles leading to multipaths
• Doppler Effects: Time variant nature due to mobility of objects in an environment
2
4
=d
GGPP
RTT
R
πλ
TransmittersTransmitters
Multicarrier System –Wireless (Complex Transmission)
Map Block into N complex
IFFT BandpassFilterRate 1/T
N symbols
Rate N/T
Rate N/T
cos(wct)
sin(wct)
ReceiversReceiversMulticarrier System –Wireless (Complex Transmission)
Demap
Block
FFT
BandpassFilter
cos(wct)
j sin(wct)
SampleRate N/T
Sync
Frequency Domain EqualizationFrequency Domain Equalization
• For the kth carrier:xk = Hk sk + vk
where Hk = n hk(nTs) exp(j2π k n / N) where n = 0, …,. N-1
• Frequency domain equalizer xk
Hk-1
ssk
• Noise enhancement factor
k
|Hk|2
|Hk-1|2
kgood
bad
Example: IEEE 802.11aExample: IEEE 802.11a
• IEEE 802.11 employs adaptive modulation– Code rate & modulation depends on distance from base station– Overall data rate varies from 6 Mbps to 54 Mbps
Reference: IEEE Std 802.11a-1999
Ideal Channel EstimationIdeal Channel Estimation• Wireless channels change frequently ~ 10 ms• Require frequent channel estimation• Many systems use pilot tones – known symbols
– Given sk, for k = k1, k2, k3, … solve xk = l=0L hl e-j2π k l/N sk for hl
– Find Hk = l=0L hl e-j2π k l / N (significant computation)
• More pilot tones– Better noise resilience– Lower throughput (pilots are not informative)
frequency
mag
nitu
de Pilot tones
Channel Estimation Via InterpolationChannel Estimation Via Interpolation
• More efficient approach is interpolation• Algorithm
– For each pilot ki find Hki = xki / ski
– Interpolate unknown values using interpolation filter– Hm = αm,1 Hk1 + αm,2 Hk2 + …
• Comments– Longer interpolation filter: more computation, timing sensitivity– Typical 1dB loss in performance in practical implementation
frequency
mag
nitu
de
Vector modulator & complex baseband
• Independently modulate cos(2πfCt) & sin(2πfCt) and sum.• Coherent demodulator for ‘cos’ transmission blind to ‘sin’ trans. and vice-versa.
Mult
Mult
ADD
Cos(2ππππfCt)
Sin(2ππππfCt)
bR(t)
bI(t)
• “2 channels for price of 1”
• Still single carrier
• Complex baseband:
b(t) = bI(t) + jbR(t)
• More about this later
Vector demodulator
Mult
Mult
Cos(2ππππfCt)
Sin(2ππππfCt)
bR(t)
bI(t)
Derive local carrier(cos & sin)
Lowpassfilter
Lowpassfilter
bR(t)cos(2ππππfCt)+
bI(t)sin(2ππππfCt)
Mapping bit stream to baseMapping bit stream to base--bandband
Pulse-shaping filter
..1 1 0 1 0 ... Generate impulses
t
V V
t
‘Map to base-band’
• Stream of impulses produced according to bits & approach
e.g. for unipolar: unit impulse for ‘1’ & zero for ‘0’.
• Pass impulse stream through pulse shaping filter.
• Impulses & filter may be analogue or digital (generally digital)
Techniques for digital transmission
• Can modulate amplitude, frequency &/or phase of cos(2πƒπƒπƒπƒCt).
• These 3 forms of modulation when used independently give us
(a) amplitude shift keying (ASK) (b) frequency shift keying (FSK)(c) phase shift keying (PSK).
• There are many versions of each of these.
• Possible to use a combination of more than one form.
• Consider simplest binary forms first.
Constellation diagramsShow “in phase” and “quadrature” components as a graph as illustrated below for two examples:
Binary ASK with symbols 0 & Acos(..)
In phase with carrier
Quadrature to carrier
Q
I
4-ary ASK with symbols 0, Acos(..), 2Acos(..), 3Acos(..)
0 A 2A 3A
Complex baseband & vector-modulator/demodulator
Vector modulator:
..11010.. Map
sin(2ππππfCt)
cos(2ππππfCt)
bI(t)
bR(t)
bR(t)cos(2ππππfCt)+
bI(t)sin(2ππππfCt)Map..10010..
Vector demodulator
Mult
Mult ThresholdDetector
ThresholdDetector
Cos(2ππππfCt)
Sin(2ππππfCt)
bR(t)
bI(t)Lowpass
Lowpass
..11010..
Derive localcarrier(cos & sin)
Receivedsignal r(t)
..10010..
Constellation diags for ASK with complex baseband
In phase with carrier
Quadrature to
carrier
0 A 2A 3A
Binary ASK for bR(t) & bI(t) 4-ary ASK
for bR(t) & bI(t)
In quadrature
Inphas
A
A 3A
A
0
ReReal pt
8-PSK16-PSK
Imag pt
QPSK is 4-PSK. What about 8-PSK & 16-PSK?
Can have 8-PSK (3 bits/symbol) & 16-PSK (4 bits/symbol). Constellation diagrams for shown below.
Differential forms of QPSK & M-PSK often used where changes in phase signify the data. Principle similar to DPSK .
‘Single carrier’ receiver
• Receiver must demodulate to obtain base-band b(t) .• Pulse shapes distorted & affected by noise. • Sample & detect for rectangular pulses discussed in last lecture.• May work for low bit-rates over channels with little distortion or noise• Performance can be improved by introduction of
– a matched filter optimally tuned to shape of transmitted pulses to minimize effect of noise (AWGN).
– a channel equalizer to cancel out distortion introduced by channel.
..1100..Matchedfilter
Demodulator Channelequaliser
Sample&
detectb(t)
Channelsignal + AWGN
Recall: Attenuation, Dispersion Effects: ISI!Recall: Attenuation, Dispersion Effects: ISI!
Source: Prof. Raj Jain, WUSTL
Inter-symbol interference (ISI)
Channel equaliser
• Channel equaliser’ is an ‘adaptive filter’ • Programmed to correct any differences between pulses seen at output
of matched filter & ideal RC pulses required by detector. • Aims to cancel out effect of the channel, • In particular the effects of frequency selective fading. • Received amplitude reduced at some frequencies & reinforced at
others.• Equalizer must do opposite of this. • Must adapt to changes in fading channel characteristics.• A demanding filtering task, and it cannot always be successful.• If there is a very deep fade, it will just not be possible to reverse it.• Trying to do so will just emphasize noise at frequency of deep fade.• Single carrier sine-wave modulation still widely used.
Spectra of Spectra of rectrect & & sincsinc pulsespulses
T.sinc1/T(f)
t
T/2-T/2
1
rectT(t)
f
T
1/T-1/T
2/T-2/T
3/T-3/T
4/T-4/T
Fourier transform
Real part shownImag part = 0
f
1/(2T)-1/(2T)
T
T.rect1/T(f)sincT(t)
t
1
T-T
2T-2T
3T-3T
4T-4T
Fourier transform
Real pt shownImag pt = 0
Spectra of 50% RC pulses & spectraSpectra of 50% RC pulses & spectra
RC(f)
tT/2-T/2
1
rc(t)
f
T
1/T-1/T
2/T-2/T
3/T-3/T
4/T-4/T
Fourier transform
Real part shownImag part = 0
-3T/43T/4
f
1/(2T)-1/(2T)
T
RC(f)rc(t)
t
1
T-T
2T-2T
3T-3T
4T-4T
Fourier transform
Real pt shownImag pt = 0
3/(4T)
-3/(4T)
Modulation of sub-carriers
• With IEEE802.11, each OFDM sub-carrier modulated by choice of:– binary-PSK, (1 bit per pulse)– QPSK, (2 bits per pulse)– 16-QAM (4 bits per pulse)– 64-QAM (6 bits per pulse)
• 16-QAM & 64-QAM are multi-level schemes.• Implement by vector-modulator according to ‘constellations’.• Illustrate for QPSK & 16-QAM• ‘Gray coding’ for 16-QAM makes nearest dots differ in just 1 bit.• Differential PSK, QPSK & QAM used where the difference between the current &
previous pulse specifies the bit pattern.
Constellation for QPSK
modulating cos
0,0
0,1
1,0
1,1
Bit1 Bit2 bR bI0 0 A A0 1 A -A1 0 -A A1 1 -A -A
Modulating sin
‘16_QAM’ constellation
A
3A
-A
-3A
A 3AReal
Imag
(0000)
-A
(0001)
(0010)
(0011)
(0100)(1000)
(1001)
(1010)
(1011)
(1100)
(1101)
(1110)
(1111)
(0110)
(0101)
(0111)
(modulates cos)
(modulates sin)
MulticarrierMulticarrier vsvs EqualizersEqualizers
• Equalizers use signal processing in receiver to eliminate ISI. • Linear equalizers can completely eliminate ISI (ZF), but this may enhance
noise. MMSE better tradeoff.
• Equalizer design involves tradeoffs in complexity, overhead, andperformance (ISI vs. noise).– Number of filter taps, linear versus nonlinear, complexity and overhead of training
and tracking
• Multicarrier is an alternative to equalization– Divides signal bandwidth to create flat-fading subchannels.
Introduction
-60 -40 -20 0 20 40 60-50
-40
-30
-20
-10
0
10
f [MHz]
pow
er s
pect
rum
mag
nitu
de [
dB] OFDM spectrum for N
FFT = 128, N
w in = 12, N
guard = 24, oversampling = 1
0 20 40 60 80 100 120 140 160 180 200-0.2
-0.1
0
0.1
0.2time domain signal (baseband)
sample nr.
imaginaryreal
OFDM Block DiagramOFDM Block Diagram
OFDM modulation
(IFFT)
Channel coding /
interleaving
Guard interval
I/Q I/QSymbol mapping
(modulation)
Transmitter
N symbols
OFDM demod. (FFT)
Decoding / deinter-leaving
Guard interval removal
Time sync.
I/Q I/Q
symbol de-mapping
(detection)
Channel est.
ReceiverFFT-part
time
1 OFDM symbol
Channel impulse response:
0101010010110
P/S
QAM demod
decoder
invert channel
=frequencydomain
equalizer
S/P
quadratureamplitude
modulation (QAM)
encoder
N-IFFTadd
cyclic prefix
P/SD/A +
transmit filter
N-FFT S/Premove cyclic prefix
TRANSMITTER
RECEIVER
N subchannels 2N real samples
2N real samplesN subchannels
Receive filter
+A/D
multipath channel
Summary: An OFDM Modem Summary: An OFDM Modem
Bits
00110