faster-than-nyquist transmission for wireless and optical

Lutz LampeUniversity of British Columbia, Canada

Faster-than-Nyquist transmission for wireless and optical fibre communication

(FTN) OR (Faster than Nyquist) OR (time-frequency packing) OR (super Nyquist)

0

0.5

1

1.5

2

2.5

3

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

2003

2005

2007

2009

2011

2013

2015

2017

What is faster than Nyquist?What is faster than Nyquist?

More bits per second per Hertz, i.e., higher spectral efficiency

Why fast (-er than Nyquist)?©

Sca

ling

Opt

ical F

iber

Net

wor

ks: C

halle

nges

and

Sol

utio

nsPe

ter J

. Winz

er,

Opt

ics a

nd P

hoto

nics N

ews V

ol. 2

6, Iss

ue 3

, pp.

28-3

5 (2

015)

http

s://d

oi.o

rg/1

0.136

4/O

PN.26

.3.00

0028

https://doi.org/10.1364/OPN.26.3.000028

Why fast (-er than Nyquist)?

Cisco Visual Networking Index: Forecast and Methodology, 2016–2021

0

50000

100000

150000

200000

250000

300000

2016 2017 2018 2019 2020 2021

Global IP traffic in PB per month

© Scaling Optical Fiber Networks: Challenges and SolutionsPeter J. Winzer, Optics and Photonics News Vol. 26, Issue 3, pp. 28-35 (2015)https://doi.org/10.1364/OPN.26.3.000028

CAGR 24%

https://doi.org/10.1364/OPN.26.3.000028


VNI Mobile Forecast Highlights, 2016-2021 IEEE Signal Processing Magazine, 2014


© Ericsson, “Microwave towards 2020,” White paper, September 2015

. © Huawei, “From today to tomorrow,” White paper, February 2016.


I. Concept of FTN

II. Pragmatic approaches

III. Two case studies: some numerical results

Nyquist Signalling

s(t) =Pna[n]h(t� nT )

0 1 2 3 4 5 6-2

-1.5

-1

-0.5

0

0.5

1

1.5

t/T �!

+1 +1-1 -1 -1 +1 +1

AWGN, matched filterr(t) =

Pna[n]g(t� nT ) + w(t)

r[n] = a[n] + w[n]

Sampling Nyquist pulse

Achievable rate*:

*complex baseband signal

CN =

1T log2

⇣1 +

PN0/T

⌘

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

fT �!

|H(f)|2

/T�!

Observation: rate isindependent of pulse shape

Nyquist Signalling

Achievable rate*:

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

fT �!

C =

1R�1

log2

⇣1 +

PN0

|H(f)|2⌘df

=

W/2R

�W/2

log2

⇣1 +

PN0

|H(f)|2⌘df

W� 1T

> 1T log2

⇣1 +

PN0/T

⌘= CN

*complex baseband signal

|H(f)|2

/T�!

Not Nyquist Signalling

s(t) =Pna[n]h(t� n⌧T ) 0 < ⌧ 1

Rate increases by a factor ⌧

P = �2a

⌧TPower is scaled:

Faster-than-Nyquist Signalling

0 5 10 15 20 25 30 35 400

2

4

6

8

10

12

14

16

18

bit/T

s�!

SNR = P · T/N0 �!

CN

C ⌧ = {0.7, 0.8, 0.9, 1}

CFTN

CFTN =

1/(2⌧T )R

�1/(2⌧T )

log2

1 +

PN0

1Pk=�1

��H�f +

k⌧T

��2!df

⌧=1/WT= C

Observation: capacity increases with decreasing 𝜏

⌧=1/WT> CN

Faster-than-Nyquist Capacity

CFTN =

1/(2⌧T )R

�1/(2⌧T )

log2

1 +

PN0

1Pk=�1

��H�f +

k⌧T

��2!df

⌧=1/WT= C

⌧ = {0.7, 0.8, 0.9, 1}

bit/T

s�!

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

18

SNR = Eb/N0 �!

Observation: capacity increases with decreasing 𝜏

CN

C

⌧=1/WT> CN

Faster-than-Nyquist Capacity

⌧ = {0.7, 0.8, 0.9, 1}

CFTN

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

-1.5

-1

-0.5

0

0.5

1

1.5

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

-1.5

-1

-0.5

0

0.5

1

1.5

t/T �!

+1 +1-1 -1 -1

Observation: loss in eye opening …

Nyquist

FTN

t/T �!

Faster-than-Nyquist Distance

t/T �!

… but minimum distance retained

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

-1.5

-1

-0.5

0

0.5

1

1.5

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

-1.5

-1

-0.5

0

0.5

1

1.5

t/T �!

+1 +1-1 -1 -1

Faster-than-Nyquist Distance

1

2

3

4

5

6

7

8

SNR = Eb/N0 �!

bit/(s·H

z)�!

�⌧

2 dB

*absolute numbers need to be verified

2PAM/4QAM

4PAM/16QAM

8PAM/64QAMSince dmin does not change, spectral efficiency increases for fixed Eb/N0 *

beyond Mazo limit, trade-off between spectral and power efficiency

Faster-than-Nyquist Mazo Limit


I. Concept of FTN



Faster-than-Nyquist ISI

-5 -4 -3 -2 -1 0 1 2 3 4 5

-0.2

0

0.2

0.4

0.6

0.8

1

Intersymbol-interference (ISI) channel

n-k ⟶

g(n-

k) ⟶

r[n] =X

k

a[k]g(n� k) + w[n]

Faster-than-Nyquist Equalization a[n]

r[n]

E.g. binary transmission, L=3

g1 g2 gL�1 gL

Faster-than-Nyquist Equalization Method Pro Con

MLSE performance optimal complexity polynomial in constellation size and exponential in L

RSSEM-algorithm

close to optimal performance with reduced complexity relatively high complexity

DFE complexity independent of constellation size and about

linear in L

performance loss

Linear

FDE reduce complexity further through filtering via FFT possibly use of cyclic prefix

Relaxed combinatorial optimization

complexity independent of constellation size and

polynomial in L

block based and complexity polynomial in block size,

requires random trials

Faster-than-Nyquist Coding

FEC encoder

FTNtransmission

FTNequalization

FEC decoder

Turbo equalization

Channelsoft output

Faster-than-Nyquist Prequalization LDPC

Enc

oder

Inte

rleav

er

QA

M

Map

per RRC

2𝜏𝑇

Opt

ical

Fro

nt-e

nd

DA

C

SSMF

Coh

eren

t Rx.

2x2

MIM

O B

utte

rfly

PM

D E

q.

Dem

appe

r

𝑣′ LDPC

Dec

oder

𝑎

Car

rier R

ecov

ery

FTN

Pr

e-eq

ualiz

er

𝑟

QAMMapper

FTNPre-equalizer

DACRRC Pulse shape

ℎ(𝑡)

QA

M

Map

per

FTN

Pr

e-eq

ualiz

er

DA

C AD

CA

DC

WM

F+

CD

Com

p.

Dem

appe

r De-

inte

rleav

er

𝑣′

𝑎 𝑟

DataIn

DataOutTransmitter Receiver

WF𝐹(𝑧)

WM

F+

CD

Com

p.

SoftDemapper

Rx Matched Filterℎ∗(−𝑡)

AWGN𝜏𝑇-Sampling

Transmitter

Receiver

𝑎 𝑠𝑟

𝑣′

RRC2𝜏𝑇

Interleaver

De-interleaver

FECEncoder

FECDecoder

Data In

Data Out

Tomlison-Harashima Precoding

Linear Precoding

© M. Jana, A. Mehdra, L. Lampe, J. Mitra "Pre-equalized Faster-than-Nyquist Transmission," IEEE Transactions on Communications, vol. 65, no. 10, pp. 4406-4418, October 2017.

Faster-than-Nyquist Equalization Method Pro Con

MLSE performance optimal complexity polynomial in constellation size and exponential in L

RSSEM-algorithm

close to optimal performance with reduced complexity relatively high complexity

DFE complexity independent of constellation size and about

linear in L

performance loss Linear

FDE reduce complexity further through filtering via FFT possibly use of cyclic prefix

Relaxed combinatorial optimization

complexity independent of constellation size and

polynomial in L

block based and complexity polynomial in block size,

requires random trials

Precoding low complexity, high performance

application to limited FTN acceleration range

Faster-than-Nyquist Complications

SynchronizationNo excess bandwidth for timing synchronization

Varying signal amplitude complicates carrier-phase estimation

Channel estimation

Effective constellation changes due to FTN signalling is a challenge for pilot-based and blind channel

estimation

Peak-to-average power ratio Generally increases


I. Concept of FTN



Optical Fibre CommunicationLD

PC E

ncod

er

Inte

rleav

er

QA

M

Map

per RRC

2"#

Opt

ical

Fro

nt-e

nd

DA

C

SSMF

Coh

eren

t Rx.

2x2

MIM

O B

utte

rfly

PMD

Eq.

LDPC

Dec

oder

Car

rier

Rec

over

y

QA

M

Map

per

FTN

Eq

ualiz

er

DA

C AD

CA

DC

WM

F+

CD

Com

p.

De-

inte

rleav

er


WM

F+

CD

Com

p.RRC2"#

FTN

Eq

ualiz

er

LDPC

Enc

oder

Inte

rleav

er

QA

M

Map

per RRC2"#

Opt

ical

Fro

nt-e

nd

DA

CSSMF

Coh

eren

t Rx.

2x2

MIM

O B

utte

rfly

PMD

Eq.

Dem

appe

r

LDPC

Dec

oder

Car

rier

Rec

over

y

FTN

Pr

e-eq

ualiz

er

QA

M

Map

per

FTN

Pr

e-eq

ualiz

er

DA

C AD

CA

DC

WM

F+

CD

Com

p.

Dem

appe

r De-

inte

rleav

er

DataIn


WM

F+

CD

Com

p.RRC2"#

© M. Jana, A. Mehdra, L. Lampe, J. Mitra "Pre-equalized Faster-than-Nyquist Transmission," IEEE Transactions on Communications, vol. 65, no. 10, pp. 4406-4418, October 2017.

BER

�!

OSNR �!

QPSK, roll-off 0.3, optical comms setting, fixed baud rate𝜏=0.85

8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 1310-6

10-5

10-4

10-3

10-2

10-1

Nyquist (τ = 1)THP FTNTHP FTN demapperMAP, 10 iterations

BER

�!

OSNR �!

QPSK, roll-off 0.3, optical comms setting, fixed baud rate𝜏=0.80

8 9 10 11 12 13 14 1510-6

10-5

10-4

10-3

10-2

10-1

Nyquist (τ = 1)THP FTNTHP FTN demapperMAP, 10 iterationsLinear pre-filtering

Microwave Communication

Ana

log

Fron

t-end

&U

p-co

nver

sion

H-Antenna

V-Antenna

InterleaverFECEncoder

QAMMapper

DACRRC Pulse shape

!(#)%&

InterleaverFECEncoder

QAMMapper

DACRRC Pulse shape

!(#)%'

Sync

hron

ized

R

ecei

ver

Har

d D

ecod

ing

2-D

Ada

ptiv

e D

FE(I

SI &

XPI

)+

Phas

e no

ise

trac

king

Rx Matched Filter!∗(−%)

'(-Sampling

Rx Matched Filter!∗(−%)

'(-Sampling

QAMDemapper

De-interleaver+

FEC Decoder

QAMDemapper

De-interleaver+

FEC Decoder

Rx DSP

)*

)+

M. Jana, L. Lampe, J. Mitra "Dual-polarized Faster-than-Nyquist Transmission Using Higher-order Modulation Schemes,” IEEE SPAWC 2018.

24 26 28 30 32 34 36 38 4010

−6

10−5

10−4

10−3

10−2

10−1

100

SNR in dB

BE

R

Nyq (τ=1), w/o PN

Nyq (τ=1), IPNT

Nyq (τ=1), CPNT

FTN, w/o PN

FTN, IPNT

FTN, CPNT

CPNT vs IPNT 3.2 dB

3.3 dB

1024−QAM Nyquist

256−QAM FTN

256−QAM Nyquist

CPNT vs IPNT 0.55 dB

Dual-polarized FTN transmission over microwave channel with phase noise, roll-off 0.4, 𝜏=0.80


Dual-polarized FTN transmission over microwave channel with phase noise, spectral efficiency (SE)

22 24 26 28 30 32 334

6

8

10

12

SNR in dB

SE (b

its/s

/Hz/

pol.)

Shannon bound, w/o PN and XPINyquist (τ = 1)DFE−FTNLPE−FTN

1024Qβ = 0.4

1024Qβ = 0.3

512Qβ = 0.25256Q

β = 0.4,τ = 0.8

β = 0

β = 0.3

β = 0.4

256Q β = 0.3, τ = 0.8

β = 0.25

256Q β = 0.25, τ = 0.89

256Qβ = 0.25

256Qβ = 0.4


Faster-than-Nyquist Extension2D (time-frequency packing)

𝜏T

𝜍𝛥f

a good number of works …

Faster-Than-Nyquist Signal Design for Multiuser Multicell Indoor Visible Light Communications, IEEE Photonics Journal, 2016

Faster-Than-Nyquist Precoded CAP Modulation Visible Light Communication System Based on Nonlinear Weighted Look-Up Table Predistortion, IEEE Photonics Journal, 2018

Polar coding for faster-than-Nyquist signaling, IEEE/CIC International Conference on Communications in China (ICCC), 2017

Application domain

Makes use of excess bandwidth to transmit dataImproves rate in non-(modulation-interference) limited systemsAlternative/complement to larger constellation sizes

Requires equalization, changes to synchronization, channel estimation etcPre-filtering is akin to pulse-shaping and partial-response transmission

Faster-than-Nyquist Transmission

faster-than-nyquist transmission for wireless and optical

Documents