OHIO STATE - Eurecom · Phil Schniter The Ohio State University ... freq selective) chans. Some...

Phil Schniter The Ohio State University'

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Efficient Multi-Carrier Communication

over Mobile Wideband Channels

Phil Schniter

OHIOSTATE

T . H . E

UNIVERSITY

July 2008(Joint work with Mr. Sungjun Hwang and Dr. Sib Das)

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Multipath, Mobility, and Bandwidth:

Trends:

• As the signal bandwidth increases, there are more gain variations across

the signal bandwidth, and hence more channel parameters.

Increased BW ⇒ longer channel impulse response.

• As the mobilities of the transmitter, receiver, and reflectors increase,

there are more channel gain variations per unit time.

Increased mobility ⇒ faster impulse response variation.

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Multipath Fading (cont.):

• Implication:

As bandwidth & mobility increase, time-domain receivers need to

implement longer filters and adapt them at faster rates.

• Example — North American Digital TV:

Typical receivers use decision feedback equalization (DFE) with 1000

taps in the feedforward path and 500 in the feedback path.

+FF

FB

decisions

DFE

Even with a fixed transmitter and receiver, it is very difficult to adapt

such long filters quickly enough!

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Orthogonal Frequency Division Multiplexing (OFDM):

Main Ideas:

1. Transmit data in parallel over non-interfering narrowband subchannels.

• Flat frequency response across each subchannel ⇒ 1 channel coefficientsubchannel

.

• Equalization = one complex gain adjustment per subchannel!

2. Implement subchannelization using the Fast Fourier Transform (FFT).

• No calibration or drift issues like with analog modulators.

• Minimal frequency spacing (⇒ good spectral efficiency).

• Fast implementation: N log2 N multiplications per N -FFT.

• Requires a guard interval of length-L−1.

time? ?? ?

L−1L−1 L−1L−1 NNN

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Equalization Complexity:

Single-carrier:

.+

3L

2L

L

decisions

decisions

Viterbi

DFE:

Linear:

MLSD:

3L mults per QAM symbol

3L mults per QAM symbol

|S|L mults per QAM symbol

Complexity scales linearly or exponentially in the channel length L.

Multi carrier:

N + N log2 N mults per block, where typically N ≈ 4L.

⇒ 1 + log2 N = 3 + log2 L mults per QAM symbol.

Complexity scales logarithmically in the channel length L.

Example: When L = 512, we have 3L = 1536 and 3 + log2 L = 12.

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CP-OFDM:

Principal advantage:

• Low-complexity demod with delay-spreading (i.e., freq selective) chans.

Some disadvantages:

• Sensitive to Doppler-spreading (i.e., time selective) channels.

• Loss of spectral efficiency due to the insertion of guards.

What if we increased N relative to L (i.e., P , NL≫ 4)?

– Complexity increases to 1 + log2 P + log2 L multsQAM symbol

...not bad.

– Reduced subcarrier spacing ⇒ more sensitive to Doppler spread!

• Slow spectral roll-off causes interference to adjacent-band systems.

Improves with raised-cosine pulse, but at further loss in efficiency:

timeL−1 L−1L−1 N N N

• High peak-to-average power ratio (PAPR).

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The Big Question:

Can we fix CP-OFDM’s

• sensitivity to Doppler spread

• loss in spectral efficiency, and

• slow spectral roll-off,

without spoiling its O(log2 L) multsQAM symbol

complexity scaling?

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The Big Question:

Can we fix CP-OFDM’s

• sensitivity to Doppler spread

• loss in spectral efficiency, and

• slow spectral roll-off,

without spoiling its O(log2 L) multsQAM symbol

complexity scaling?

Yes!

Re-think the role of “pulse shaping” in multi-carrier modulation. . .

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Rectangular Pulses:

A standard CP-OFDM block can be recognized as a sum of N infinite-length

complex exponentials windowed by a rectangular pulse of width N+L−1.

=+

+

t

t

t

tN+L−1

N+L−1

N+L−1

N+L−1N+L−1N+L−1

⇒ Dirichlet sinc in DTFT domain, whose slow side-lobe decay causes:

• strong interference to adjacent-band communication systems, and

• high sensitivity to Doppler spreading:

Doppler(ejω) |Dsinc(ejω)|2

∗

−fD fD −2π . . . . . . . . . . . . −4πN

−2πN

0 2πN

4πN

. . .. . . . . .. . . 2π

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Smooth Overlapping Pulses:

What if we applied a smooth window instead?

=+

+

t

t

t

t Ns

Ns

NsNs

The main-lobe is wider but the sidelobes decay more quickly, implying:

• reduced adjacent-band interference, and

• strong interference from adjacent subcarriers, but very little interference

from all other subcarriers, even under large Doppler spreads:

Doppler(ejω) |A(ejω)|2

∗

−fD fD −2π . . . . . . . . . . . . −4πN

−2πN

0 2πN

4πN

. . .. . .. . . . . . 2π

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Smooth Overlapping Pulses:

Challenge: The use of smooth overlapping pulses potentially causes

inter-carrier interference (ICI) and inter-block interference (IBI):

x(i) =∞∑

q=−∞

H(i, q)︸︷︷︸

subcarriercoupling matrix

s(i − q) + z(i). Difficult to equalize!

Solution: Design the pulse shapes with the goal of. . .

1. Completely suppressing IBI: H(i, q)∣∣q 6=0

= 0.

2. Allowing ICI only within a radius of D ≪ N subcarriers. (Often D = 1.)

= + Not difficult to equalize.D

x(i) H(i, 0) s(i) z(i)

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Receiver Pulse-Shaping:

Though so far we’ve considered a non-rectangular transmission pulse {an},

tn =∞∑

i=−∞

an−iNs

N−1∑

k=0

sk(i)ej 2π

Nkn, n = −∞ . . .∞,

we can use, in addition, a non-rectangular reception pulse {bn}:

xk(i) =∞∑

n=−∞

rn−iNsbn e−j 2π

Nkn, k = 0 . . . N − 1.

Ns specifies the interval between the start of one OFDM block and the next.

• Modulation efficiency η , NNs

symbols

sec Hz

• For OFDM, Ns = N+L−1, but now there is no constraint on Ns!

We focus on Ns = N ⇔ no guard interval ⇔ η = 1.

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Max-SINR Pulse Design:

Writing the received signal energy components due to

Es =∑

(q,k,l)∈¥

E{|Hk,l(·, q)|2} “signal” and

Ei =∑

(q,k,l)∈¥

E{|Hk,l(·, q)|2} “interference” (IBI+ICI)

· · ·· · ·interferencedon’t caresignal

where {H(·, q)} =

we can write SINR =Es

Ei + En

=aHP 1(b)a

aHP 2(b)a=

bHP 3(a)b

bHP 4(a)bwhere

a = transmission pulse coefficientsb = reception pulse coefficients

P 1(·),P 2(·),P 3(·),P 4(·) = matrix fxns dependant on Doppler & SNR.

⇒ SINR-maximizing pulses are generalized eigenvectors.

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Max-SINR Pulse Examples:

0 100 200 300

0

0.5

1

1.5

TxRx

0 100 200 300 400 500

0

0.5

1

1.5

0 100 200 300 400 500

0

0.5

1

1.5

0 100 200 300 400 500

0

0.5

1

1.5

ZP-OFDM (η = 0.803) Optimized receiver (η = 1)

Optimized transmitter (η = 1) Jointly optimized (η = 1)

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IBI/ICI Energy Profiles (same for each subcarrier):

0 5 10 15−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

0

10

0 5 10 15−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

0

10

0 5 10 15−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

0

10CP−OFDM η=0.803

ZP−OFDM η=0.803

ROMS η=1

TOMS η=1

JOMS η=1

previous block same block next block

dB

subcarrier separationsubcarrier separationsubcarrier separation

D = 1, SNR = 15dB, L = 64, fDTc = 7.6× 10−4, Jakes Doppler spectrum.

(For example, fc = 20GHz, BW=3MHz, Th = 5.4µs, v = 120km/hr.)

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Non-Orthogonal FDM:

To summarize, Orthogonal FDM is possible only when

1. the channel is time-invariant, and

2. an adequate-length guard is included.

With a properly-designed Non-Orthogonal FDM, we can

• eliminate the guard, and

• tolerate large delay and Doppler spreads,

at the cost of

• a short span of intercarrier-interference,

which can be properly handled via low-complexity equalization.

Thus, we advocate shaping ISI/ICI rather than suppressing ISI/ICI.

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Outage Capacity vs NfDTs for various ICI-radii D:

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

1.4

1.6

1.8

2

2.2

2.4

2.6

0.00

01−

outa

ge c

apac

ity [b

its/u

se]

GP−FDMN=16M=4Nh=8Ng=0

SNR=10

D=0D=1D=2D=3D=4D=5

NfDTs

• The outage-capacity optimal D obeys D ≈ ⌊NfDTs⌉!

• ICI shaping is better than ICI suppression when 2fDTs ≥1N

.

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ICI Equalization:

Coherent approaches (i.e., known channel): multsQAM symbol

1. Viterbi [Matheus/Kammeyer GLOBE 97] O(|S|DD)

2. Iterative Soft [Das/Schniter Asilomar 04] O(D2)

3. Linear MMSE [Rugini/Banelli/Leus SPL 05] O(D2)

4. MMSE DFE [Rugini/Banelli/Leus SPAWC 05] O(D2)

5. Tree Search [Hwang/Schniter SPAWC 06] O(D2)

Non-coherent approaches (i.e., unknown channel):

1. Hard Tree Search [Hwang/Schniter WUWNet 07] O(D2L2)

2. Soft Tree Search [Hwang/Schniter Asilomar 07] O(D2L2)

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1) Coherent Tree Search:

Two-step procedure:

1. MMSE-GDFE pre-processing [Damen/ElGamal/Caire TIT 03]:

DD + 1 2D + 1 2D

N

Ã O(D2N) algorithm [Rugini/Banelli/Leus SPAWC 05].

2. Near-optimal yet efficient tree search. Options include:

• Depth-first search (e.g., Schnor-Euchner sphere decoder),

• Best-first search (e.g., Fano alg, stack alg),

• Breadth-first search (e.g., M-alg, T-alg, Pohst sphere decoder).

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Performance:

10 15 20 2510

−4

10−3

10−2

10−1

100

SNR in dB

fram

e er

ror

rate

DFEFanoM−algT−algadaptive T−algSE−SpDML

10 15 20 2510

−4

10−3

10−2

10−1

100

SNR in dB

fram

e er

ror

rate

DFEFanoM−algT−algadaptive T−algSE−SpDML

fDTc = 0.001 fDTc = 0.003

Suboptimal tree search is almost indistinguishable from ML!

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Average Complexity (MACs/frame):

10 15 20 25 302.2

2.4

2.6

2.8

3

3.2

3.4

SNR in dB

log

(ope

ratio

ns)

Viterbi

Fano

SE−SpD

M−alg

T−alg

Adaptive T−alg

DFE

10 15 20 25 302.2

2.4

2.6

2.8

3

3.2

3.4

SNR in dB

log

(ope

ratio

ns)

Viterbi

Fano

SE−SpD

M−alg

T−alg

Adaptive T−alg

DFE

fDTc = 0.001 fDTc = 0.003

NN

Breadth-first & DFE stay cheap, while depth-first & Fano explode!

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Error Masking due to V-shaped Channel Matrix:

After MMSE-GDFE pre-processing, we get the following system:

= +

0 2D + 1 2D

N − 2D − 1N − 2D − 1N − 4D − 2

Key point: The blue symbol does not affect any of the red observations.

Error-masking explains the complexity explosion of the depth-first

and Fano searches!

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2) Iterative Soft ICI Cancellation:

+=

xk

Hk

hksk

sk

zk

xk = Hksk + zk

• Soft interference cancellation using mean of sk.

• Assuming Gaussian residual interference and us-

ing the covariance of sk, compute LLRs(sk).

• Using LLRs(sk), update mean/covariance of sk.

• k → 〈k + 1〉N .

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Iterative Soft ICI Cancellation (BPSK example):

updatepriors SIC

updateLLR

LLR(i)k

s(i)k , v

(i)k g

(i)k LLR

(i+1)k

s(i)k , E{sk|sk} = tanh(LLR

(i)k /2)

v(i)k , var(sk|sk) = 1 − (s

(i)k )2

y(i)k = xk − Hks

(i)k

g(i)k = y

(i)Hk

(Rz + Hk D(v

(i)k )HH

k

)−1hk

LLR(i+1)k = LLR

(i)k + 2 Re(g

(i)k )

Complexity: Miters

× O(D2)mtx inv

per BPSK symbol.

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Turbo Equalization:

Soft bit information (i.e., LLRs) can be exchanged with a soft-input

soft-output (SISO) decoder:

x(i) IterativeEqualizer

Lo

eq

Li

eq

Le

eq Li

dc

Le

dc Lo

dc

Π−1

Π

SISODecoder

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Turbo Equalization Performance:

0 5 10

10−4

10−3

10−2

10−1

Eb/N

o (dB)

BE

R

0 5 10

10−4

10−3

10−2

10−1

Eb/N

o (dB)

0 5 10

10−4

10−3

10−2

10−1

Eb/N

o (dB)

LINML1MS1MS8ML8PLICPGIC



TOMS JOMS ROMS rate-12

conv code

QPSK

N = 64

D = 2

L = 16

fDTc = 0.003

for example:

fc = 20GHz

BW= 3MHz

Th = 5.4µs

v = 3×160km/hr

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3) Non-Coherent Equalization:

• So far we have assumed that the equalizer is provided with channel

estimates.

But with long and quickly varying channels, accurate channel

estimation requires a high proportion of pilots!

• Instead, one might consider joint estimation of channel and symbols.

Actually, we are not interested in explicitly estimating the channel,

but rather doing non-coherent equalization: inferring transmitted

bits using knowledge of channel statistics but not channel state.

For this it helps to re-parameterize the system model. . .

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Basis-Expansion Model:

From the pulse-shaped FDM model:

x(n) = H(n)s(n) + z(n) H(n) is banded with band radius D

= S(n)h(n) + z(n) h(n) ∈ C(2D+1)N ICI coefs

we can use a basis-expansion model (BEM) for the ICI coefs:

h(n) = Bθ(n) θ(n) ∈ C(2D+1)L delay/Doppler coefs

B =

(FL ...

FL

)

F L ∈ CN×L truncated DFT matrix

to rewrite the observation as

x(n) = S(n)Bθ(n) + z(n).

Note: if we knew the locations of K active sparse taps, we would only

include those columns of the DFT matrix, giving θ(n) ∈ C(2D+1)K .

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Non-coherent MLSD:

Treating the delay/Doppler coefs θ as nuisance parameters,

sML = arg maxs

p(x|s)

= arg maxs

∫

θ

p(x|s, θ)p(θ)dθ

Assuming θ ∼ CN (0, Rθ),

sML = arg maxs

{xHSBΣ

−1s

BHSHx − σ2 log |Σs|}

Σs , BHSHSB + σ2R−1θ

Since θMMSE|s = Σ−1s

BHSHx, we can rewrite

sML = arg maxs

{

‖x − SBθMMSE|s‖2 − σ2 log |Σs|

}

But how do we avoid an exhaustive search over s?

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Fast Tree Search:

By turning off the first and last D subcarri-

ers, H becomes upper-triangular, facilitat-

ing the use of tree-search.

= +

x H s w

s0 = [s0]T

s1 = [s0, s1]T

s2 = [s0, s1, s2]T

s3 = [s0, s1, s2, s3]T

(for BPSK)

The important thing here is that the partial ML metric

µML(sk) = ‖xk − SkBkθMMSE|sk‖2 − σ2 log |Σk|

can be computed recursively. Thus, total search complexity via the

M -algorithm is only

2M |S|(2D + 1)2L2 mults per QAM-symbol!

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Complexity Reduction via Pilots:

• With non-coherent decoding, we require only a single pilot subcarrier.

• But, with more pilots, the initial channel estimate θMMSE improves,

allowing more aggressive branch pruning in the tree search.

Example: M-algorithm

(BPSK, 25% pilots,

M=8) compared to

coherent MLSD with

genie-aided θMMSE:

5 10 15 20 25 30

10−5

10−4

10−3

10−2

10−1

100

BE

R

SNR (dB)

joint est/det

genie-chan+MLSD

joint est/det, genie-sparse

genie-chan+MLSD, genie-sparse

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Non-coherent Turbo Equalization:

softnon-coherent

equalizer

softdecoder

Π

Π−1

pilots

xbits {bk}

QN−1k=0

{Le(bk|x)}QN−1k=0

{La(bk)}QN−1k=0

Le(bk|x) = ln

∑

s: bk=1 exp µMAP(s)∑

s: bk=0 exp µMAP(s)− La(bk) ”extrinsic LLR”

µMAP(s) = ln p(x|s

)+

∑

k: bk=1

La(bk) ”MAP metric”

Need O(2QN) evaluations of µ(s) Ã Computationally infeasible!

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Simplified LLR Evaluation:

The “max-log” approximation:

Le(bk|x) ≈ maxs∈L∩{s:bk=1}

µMAP(s) − maxs∈L∩{s:bk=0}

µMAP(s) − La(bk)

L : set containing the M most probable s,

requires only a few evaluations of µMAP(s).

To find L, the set of most probable s, and the MAP metrics {µMAP(s)}s∈L,

we use a tree search, as in the uncoded case.

Here again, there exists a fast metric update such that the total search

complexity, with the M-algorithm, is only

2M |S|(2D+1)2L2 mults per QAM-symbol!

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Non-coherent Turbo Performance:

4 6 8 10 12 1410

−5

10−4

10−3

10−2

10−1

Eb/No(dB)

BE

R

Soft Kalman Estim Np=9Kalman−LVA Np=9NC (Proposed) Np=9Genie−Estim CHPerfect CH Knowledge rate-1

2LDPC

QPSK

N = 64

D = 2

L = 16

K = 3

fDTc = 0.003

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Related Work:

• Channel estimation techniques (for coherent equalization).

• Single-carrier frequency-domain equalization.

• Pilot pattern designs:

– MMSE optimal (under CE-BEM assumption).

– Achievable-rate optimal (under CE-BEM assumption).

• Theoretical analysis of pulse-shaped multicarrier modulation:

– Achievable-rate characterized.

• Analysis of doubly selective channel:

– Capacity characterized (under CE-BEM assumption).

(See http://www.ece.osu.edu/∼schniter/pubs by topic.html)

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Conclusions:

Single Carrier:

• O(L) multsQAM symbol

equalization of delay-spread channels.

• Challenging to track quickly time-varying channels.

Orthogonal FDM:

• O(log2 L) multsQAM symbol

equalization of delay-spread channels.

• Loss in spectral efficiency due to guard interval.

• Sensitive to Doppler spread.

• Slow spectral roll-off ⇒ high adjacent-band interference.

Non-Orthogonal FDM:

• No need for a guard interval; high spectral efficiency.

• Large simultaneous delay & Doppler spreads ⇒ no IBI and short ICI.

• Fast spectral roll-off ⇒ low adjacent-band interference.

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Equalization/Decoding of Short ICI Span:

• Uncoded Coherent:

Tree search gives ML-like performance with DFE-like complexity.

• Coded Coherent:

Iterative soft ICI cancellation in turbo configuration performs close to

perfect-interference-cancellation bound.

• Uncoded Non-Coherent:

Tree-search with fast metric update gives ML-like performance with

O(D2L2) complexity.

• Coded Non-Coherent:

Tree-search with fast metric update performs close to genie-aided bound

with O(D2L2) complexity, or O(D2K2) if sparseness leveraged.

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Thanks for listening!

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OHIO STATE - Eurecom · Phil Schniter The Ohio State University ... freq selective) chans. Some...

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