OFDM Basics 1

Overview on OFDM Design

Ho Chin Keong

Communication Systems & Signal Processing

Centre for Wireless Communications

National University of Singapore

November 9, 2001

Topics

• OFDM Transmitter Design Considerations

• Design Considerations in Transmission through Multipath Channel

• Additional OFDM Receiver Design Considerations

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1 Ho Chin Keong: Overview on OFDM Design

OFDM Transmitter Design Considerations

• OFDM modulation in IF vs OFDM modulation using IFFT

• OFDM symbol windowing

• OFDM spectrum

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OFDM modulation using IF circuitry

+

××

××

cos(2πnfot)

-sin(2πnfot)

XR (k)

XI (k)

(a) One sub-carrier at IF.

up-convert to frequency kf0

up-convert to frequency 0

up-convert to frequency (N-1)f0

:

:

XR (k)

XI (k)

XR (N-1)

XI (N-1)

XR (0)

XI (0)

+

x' (t)

(b) OFDM modulation implemented at

IF.

x′(t) =

N−1∑

k=0

xR(k) cos(2πkfot)− xI(t) sin(2πkfot) (1)

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OFDM modulation at baseband

××

exp(j2πnfot)

X (k) = XR (k)+ jXR

(k)

(a) One sub-carrier at baseband.

up-convert to frequency kf0

up-convert to frequency 0

up-convert to frequency (N-1)f0

:

:

X (0)

+

x (t)

X (k)

X (N-1)

(b) OFDM modulation implemented at

baseband.

x(t) =

N−1∑

k=0

X(k) exp(j2πkfot) (2)

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OFDM modulation at baseband using IFFT

IFFT

P/S

S/P :

:

:

:

DAC

x (t)

X (k)

x (n)

• obtain discrete values at t = nT , select fo = 1NT

x(n) = x(t = nT ) =

N−1∑

k=0

X(k) exp(j2πnk/N) (3)

• becomes IDFT of {X(k)}k mod N

• implemented using IFFT, then use DAC

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OFDM symbol windowing - allows other OFDM symbols transmission

0 0.5 1 1.5 2 2.5 3−2

−1

0

1

2

3

4Time domain OFDM signal for 5 subcarriers modulated with {1,1,1,1,1}.

0 0.5 1 1.5 2 2.5 3−2

−1

0

1

2

3

4Time domain rectangular windowed OFDM signal

time / ( NT)

(a) Time domain before/after windowing.

−6 −4 −2 0 2 4 60

0.2

0.4

0.6

0.8

1Frequency domain OFDM signal for 5 subcarriers modulated with {1,1,1,1,1}.

subcarrier k

Spe

ctru

m

−6 −4 −2 0 2 4 6−0.5

0

0.5

1

1.5Frequency domain rectangular windowed OFDM signal

subcarrier k

Spe

ctru

m

(b) Freq. domain before/after windowing.

• before windowing: X(f ) = X(k)δ(f − kfo)

• after windowing:X(f )W (f ) = X(k)δ(f − kfo)⊗ sinc(f/fo) = X(k)sinc(f/fo − k)

Note: X(f) from here on will be assumed to be windowed with rectangular window.

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Spectrum efficiency η for single carrier vs OFDM

assume R is data rate of each (sub)carrier, bandwidth defined from

null-to-null, N subcarriers

• for single carrier : η = R2fo

• for OFDM : η = NR(N+1)fo

• for OFDM : as N →∞, η → Rfo

• but - windowing possible for single carrier to improve η

• windowing not possible on a subcarrier to exploit implementation

simplicity of IFFT

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Digital to Analogue Converter (DAC)

Need to perform upsampling for interpolation.

Solutions:

• null subcarriers to ease filter design

• windowing - spectrum decays at 1/|f | for rectangular window

• digital filtering - usually FIR

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−30

−25

−20

−15

−10

−5

0

5

Frequency

Pow

er S

pect

rum

Mag

nitu

de (

dB)

Effects of raised cosine windowing (with 16 null subcarrriers)

3 overlapping window points2 overlapping window points1 overlapping window points

Effects of windowing and null subcarriers.

Spectrum of OFDM symbol before DAC in KRONDOR.

High peak-to-average power ratio (PAPR)

• x(n) is Gaussian distributed by law of large number - leads to high PAPR

• effect is more prominent as N increases

• requires expensive and high power amplifier with high dynamic range, or

else

– distorts time domain signal

– non-linearity or clipping introduces out-of-band spectral regrowth

• solutions

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Design Considerations in Transmission through Multipath Channel

• multi-dimension interference (MDI)

– inter-[OFDM] symbol interference (ISI)

– inter-[sub]carrier interference (ICI)

• use of guard interval to remove ISI

• use of cyclic prefix to remove ICI

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Time selective channel

• usually valid to treat channel as time invariant over each OFDM symbol

• channel changes after every OFDM symbol

Frequency selective channel

• use discrete time baseband representation

• y(n) = x(n)⊗ h(n) + v(n) where v(n) is AWGN, ⊗ is linear convolution

• ISI occurs which spills to other OFDM symbols

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Mitigation of ISI

0 1 2−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5Time domain rectangular windowed OFDM signal with ISI

time / ( NT)

Transmitted OFDM symbolReceived OFDM symbol

Guard space ≥ maximum delay spread

• append guard interval (GI) at transmitter

• GI longer than maximum delay spread

• remove GI at receiver

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Introduce cyclic prefix (CP)

• how to remove ISI within OFDM symbol? we don’t

• fill appended GI with cyclic data to form CP

• define c© to be circular convolution modulo N ,

y(n) = x(n)⊗ h(n)

→ ycp(n) = xcp(n)⊗ h(n)

→ y(n) = x(n) c©h(n) (4)

Note: All discrete index from now on will be assumed to be modulo N .

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ICI removed through cyclic prefix (CP)

• removing cyclic prefix, received time domain signal is

y(n) =

N−1∑

k=0

H(k)X(k) exp(j2πnk/N), (5)

where H(k) =∑N−1

n=0 h(n) exp(−j2πnk/N).

• demodulation using FFT,

Y (k) =

N−1∑

n=0

y(n) exp(−j2πnk/N)

= X(k)H(k) + V (k) (6)

when orthogonality is satisfied, i.e.∑N−1n=0 exp(−j2πnk/N) exp(j2πnk′/N) = δ(k − k′)

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Additional OFDM Receiver Design Considerations

• Time/frequency duality

• Timing Synchronization

• Frequency Synchronization

• Channel Estimation

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Time/frequency duality

• single carrier in time selective channel vs OFDM in frequency selective

channel

y(n) = x(n)h(n) + v(n) Ã Y (k) = X(k)H(k) + V (k)

• single carrier in frequency selective channel vs OFDM with time selective

channel

y(n) = Ax(n)h(n) + ISI + v(n) Ã Y (k) = BX(k)H(k) + ICI + V (k)

where A,B are complex attenuation factors

• ISI occurs for single carrier satisfying Nyquist pulse-shaping criterion when

sampling offset occurs

• ICI occurs for OFDM when frequency offset occurs

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Synchronization in general

• timing synchronization errors- correctable in a pilot based system

y(n + ∆t) ® Y (k) exp(j2πk∆t/N) (7)

• frequency synchronization errors when

– carrier frequency offset

– phase jitter

– Doppler spread

• frequency synchronization errors- introduces ICI

y(n) exp(j2πn∆/N) ® Y (k + ∆) (8)

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Windowing at receiver

• consider windowing after removing cyclic prefix, before FFT

yw(n) = y(n)w(n)

= [x(n) c©h(n)]w(n) (9)

• the DFT is

Yw(k) = [X(k)H(k)] c©W (k)

=

N−1∑

l=0

X(l)H(l)W (k − l) (10)

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Effects of Frequency Offset

• presence of normalized frequency offset ∆ between tx and rx, (5) becomes

y(n) =

N−1∑

k=0

H(k)X(k) exp(j2πnk/N) exp(j2πn∆/N), (11)

• define w(n; ∆) = exp(j2πn∆/N) and thus W (k; ∆) is the dirichlet kernel

• using (9) and (10), we can re-write the DFT of (11) as

Y (k) = H(k)X(k)W (0; ∆) + I(k) (12)

where the ICI is

I(k) =

N−1∑

l=0,l 6=k

H(l)X(l)W (k − l; ∆) (13)

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ICI due to frequency offset illustrated

-6 -4 -2 0 2 4 6

0

0.5

1

1.5

2

subcarrier k

Sp

ec

tru

m

Spec trum of subcarriers for OFDM

-6 -4 -2 0 2 4 6

0

0.5

1

1.5

2

subcarrier k

Sp

ec

tru

m

Spec trum of des ired subcarrier and ICI

ICI des ired subcarrier

ICI occurs when sampling at frequency k + ∆ instead of k .

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Remove ICI caused by frequency offset

• time domain methods:

– estimate ∆ using averaging of phase difference of repeated samples,

multiply complex term to de-rotate signal with increasing phase

– use windowing to reduce ICI

• frequency domain method:

– estimate ∆, then

– express as matrix form

y = W(∆)diag{x}h + v (14)

= W(∆)diag{h}x + v (15)

– solve for h or x in the frequency domain using least squares by

minimizing 2-norm of v

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OFDM model with respect channel estimation methodology

LegendPI: pilot insertion, FEQ: frequency domain equaliserS/P: serial to parrallel, P/S: parrallel to serial

Removecyclicprefix

Channel Estimation

PSAM/APSB

Time Domain signalFrequency Domain signal

IDFT DFTS/P S/PAddCP

Channel withTime andFrequencySelectivity

Frequency Domain signal

SymbolRate 1/T

SymbolRate 1/LT

OFDM SymbolRate 1/PT

.

.

.P/S

x � (n) y � (n)x(n)4 y(n)4y(n)x(n) h(n)

noise,

b(n)

PI:PSAM/APSB

pilot,

p � (n)

data,

d � (n)

Blind Scheme

FEQ...

.

.

.

.

.

.P/S

or

.

Note: Some notations here are not inconsistent with this presentation.

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Channel estimation in OFDM systems

• (semi-)blind methods based on cyclic prefix or cyclostationarity properties

• decision feedback - not popular, probably due to large processing delay in

FFT

• training - substantial initialization, usually for fixed line, not for packet

based system

• pilots - most common method for wireless applications

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0 200 400 600 800 100010

−1

100

101

time n

MS

E

Cyclostationarity based with λo = 0.995

Subspace based with λo = 0.98

Switch based with λo = 0.97

Slow convergence found in some blind schemes.

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Pilot based channel estimation

• useful to think of OFDM symbols in a 2-dimensional (2D) time/frequency

grid

• usually time correlation is higher than frequency correlation

• pilots used to sample 2D channel - need to satisfy sampling theorem (2

times of coherence bandwidth/time)

• 2D filters used for interpolation

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OFDM Basics 1

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