LUND UNIVERSITY
PO Box 117
221 00 Lund
+46 46-222 00 00
Multistream faster than Nyquist signaling
Rusek, Fredrik; Anderson, John B
Published in:
IEEE Transactions on Communications
DOI:
10.1109/TCOMM.2009.05.070224
2009
Link to publication
Citation for published version (APA):
Rusek, F., & Anderson, J. B. (2009). Multistream faster than Nyquist signaling.
IEEE Transactions on
Communications
,
57
(5), 1329-1340. https://doi.org/10.1109/TCOMM.2009.05.070224
Total number of authors:
2
General rights
Unless other specific re-use rights are stated the following general rights apply:
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors
and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the
legal requirements associated with these rights.
• Users may download and print one copy of any publication from the public portal for the purpose of private study
or research.
• You may not further distribute the material or use it for any profit-making activity or commercial gain
• You may freely distribute the URL identifying the publication in the public portal
Read more about Creative commons licenses: https://creativecommons.org/licenses/
Take down policy
If you believe that this document breaches copyright please contact us providing details, and we will remove
access to the work immediately and investigate your claim.
1
Multistream Faster than Nyquist Signaling
Fredrik Rusek and John B. Anderson
Electrical and Information Technology Dept. and
Swedish Strategic Center for High Speed Wireless Communication
Lund University, Box 118, 221 00 Lund, Sweden
Abstract— We extend Mazo’s concept of faster-than-Nyquist
(FTN) signaling to pulse trains that modulate a bank of
subcarriers, a method called two dimensional FTN signaling.
The signal processing is similar to orthogonal frequency division
multiplex (OFDM) transmission but the subchannels are not
orthogonal. Despite nonorthogonal pulses and subcarriers, the
method achieves the isolated-pulse error performance; it does so
in as little as half the bandwidth of ordinary OFDM. Euclidean
distance properties are investigated for schemes based on
several basic pulses. The best have Gaussian shape. An efficient
distance calculation is given. Concatenations of ordinary codes
and FTN are introduced. The combination achieves the outer
code gain in as little as half the bandwidth. Receivers must work
in two dimensions, and several iterative designs are proposed
for FTN with outer convolutional coding.
I. INTRODUCTION
Consider baseband signals of the form
s(t)=
2E
s
/T
n
a
n
h(t − nT ), (1)
in which a
n
are data values over an M-ary alphabet and
h(t) is a unit-energy baseband pulse. This simple form
underlies QAM, TCM, and the subcarriers in orthogonal
frequency division multiplex (OFDM), as well as many
other transmission systems. In these schemes, h(t) is a
T -orthogonal pulse, meaning that the correlation
h(t −
nT )h
∗
(t − mT )dt is zero, m = n. In 1975 Mazo [1]
pointed out that binary sinc(t/T ) pulses in (1) could be
sent every T
Δ
seconds, T
Δ
<T, without loss in asymptotic
error probability. This he called faster than Nyquist (FTN)
signaling, because the pulses appear faster than allowed by
Nyquist’s limit for orthogonal pulses. FTN signaling has
since been generalized in a number of ways.
This paper extends the FTN concept to the frequency
dimension. This extension to two dimensions opens up a
number of attractive possibilities. Many signals of type (1)
are stacked in frequency through modulation by a set of
carriers at frequencies f
0
+ {f
k
} to form the in-phase and
quadrature (I/Q) signal given by the real part of
s(t)=
2E
s
T
K−1
k=0
N−1
n=0
[a
I
k,n
+ja
Q
k,n
]h(t−nT ) e
j2π(f
0
+f
k
)t
.
(2)
This is a superposition of 2K linear carrier modulations, and
it carries 2NK data values. The K ×N matrix A = {a
k,n
}
is called the data matrix and consists of the complex data
values a
k,n
= a
I
k,n
+ ja
Q
k,n
. The K rows in this matrix
correspond to subcarriers, and the N columns to pulse
positions. If f
k
= kf
Δ
, k =0, 1,...K− 1 and f
Δ
is equal
twice the single-sided bandwidth of h(t), the 2K carrier
signals are orthogonal; if h(t) is T -orthogonal, all NK
pulses are mutually orthogonal. In OFDM, h is ordinarily
the width-T square pulse, the subcarriers are 1/T -spaced
frequency orthogonal, and all pulses are again mutually
orthogonal.
The signal design in the sinc and OFDM cases is thus
based on orthogonality. According to theory, there exist
about 2W T orthogonal signals in W positive Hertz and T
seconds. Data values that modulate the amplitude of each
can be maximum-likelihood detected independently, and
therefore about 2W T symbols can be transmitted this way.
Take for example h(t)=
1/T sinc(t/T ), and measure
time and bandwidth of (2) by some reasonable method
(such as 99% power bandwidth). Then as K and N grow,
T/N → T , W = Kf
Δ
= K/T, and the product 2W T
tends in ratio to 2(K/T )(NT)=2KN; thus Eq. (2) carries
as many data values for large NK as any scheme based on
orthogonality can carry. A similar outcome occurs when h is
a root RC pulse. For a given number of symbols carried by
(2), T may be varied, which trades off W against T. N may
also be traded against K. The time–bandwidth product is
unaffected, and (2) always carries about twice W T symbols.
If the aim is to achieve the error rate of a stacked
orthogonal-signal system (2), without necessarily using or-
thogonal signals, the story is more complex, and more can
be achieved. By error rate is meant the error probability
of the maximum likelihood sequence estimation (MLSE)
receiver when h is employed in (1) with additive white
Gaussian noise (AWGN) of density N
0
/2 in the channel.
As the signal-to-noise ratio E
b
/N
0
grows, the probability
of incorrect detection of an a
n
is asymptotically P
e
∼
Q(
d
2
min
E
b
/N
0
), where d
min
is the minimum distance of
the set of signals and d
min
≤ d
MF
. Here E
b
= E
s
/ log
2
M,
E
s
is the average symbol energy, and d
MF
is the matched-
filter bound distance for the data alphabet. d
MF
measures
the performance of simple orthogonal-pulse signaling with
the same data values. The paper will concentrate on the
binary case, for which d
2
MF
=2, so the target orthogonal-
pulse error rate is Q(
2E
b
/N
0
).IftheK I/Q signals in
(2) do not overlap in frequency, the same asymptotic error
rate applies there.
Achieving more at the same E
b
and error probability
means that FTN signals need to consume less bandwidth.
We need to define it carefully. With independent and iden-
tically distributed (IID) data symbols the power spectral
density (PSD) of the kth subcarrier S
k
(f) is proportional to
|H(f−kf
Δ
−f
0
)|
2
+|H(f+kf
Δ
+f
0
)|
2
, k =0,...,K−1,
where H(f) is the Fourier transform of h(t). With K
subcarriers, the total PSD satisfies (take positive f only)
S(f ) ∝
K−1
k=0
S
k
(f)
=
K−1
k=0
|H(f −kf
Δ
−f
0
)|
2
,f
0
Kf
Δ
. (3)
The normalized time–bandwidth product (NTB) of this
2
transmission is
NTB
WT
Δ
2RK
Hz-s/bit, (4)
where W is a measure of the positive frequency bandwidth
of the entire Eq. (3), T
Δ
≤ T is the actual symbol time,
and R is the bits carried by each subcarrier symbol. Each
subcarrier carries 2R/T
Δ
bit/s, counting I and Q. For the
single-subcarrier Eq. (1), W is the positive baseband band-
width of just H(f). The NTB measures time–bandwidth on
a per data bit basis.
1
Simple time scaling of s(t) does not
affect the NTB, since the spectrum is then scaled by the
inverse factor.
As T
Δ
drops for the same h, the FTN signal be-
comes “faster”, and its NTB drops. In the baseband single-
subcarrier case, the total PSD is just |H(f)|
2
and W
measures its width, and as T
Δ
drops, the symbol-normalized
spectrum and the NTB are scaled down. In this view the
pulses come faster. It is equivalent to fix T and scale the
pulse h wider by the factor T/T
Δ
; this reduces the PSD
and the NTB narrows in an identical way. In either case,
the time–bandwidth per data bit is less. With K subcarriers
the calculation is more complicated but the outcome is the
same.
Mazo’s paper[1] envisioned one subcarrier and binary
sinc( ) pulses and claimed the surprising result that d
2
min
is in fact d
2
MF
=2for T
Δ
>.802T . No asymptotic error
rate is lost by increasing the symbol rate 24.7% above the
Nyquist limit, yet the NTB has dropped from 1 Hz-s/bit
to 0.802. A full MLSE detection is required in principle,
which compares all N-symbol signals to the full noisy
received signal. The reason for this behavior can be seen
by analyzing the error events that can occur (as in Section
II). As T
Δ
declines and the pulse rate grows, another error
event eventually has a distance less than the d
2
=2
antipodal event that leads to d
MF
. But this does not occur
immediately.
Later research has shown that a similar phenomenon
occurs with other orthogonal h(t) than the sinc pulse [2],
[3]. For the 30% root RC pulse for example, T
Δ
can be
as small as 0.703T . There is a least T
Δ
as well for pulses
such as the Gaussian, that are not orthogonal for any T .
Moreover, such a limit appears with nonbinary transmission,
with precoding, and with linear coded modulation based on
heavy filtering [3], [4]. All these cases can be summed up
as follows: The error performance of the linear transmission
of form (1) remains unchanged under downward scaling of
the normalized spectrum shape until a surprisingly narrow
bandwidth, after which it suddenly drops. This occurs
despite escalating ISI. We call this threshold bandwidth the
Mazo limit. Its significance is that it is pointless to transmit
in a wider bandwidth in a linear channel with AWGN,
if sufficient receiver processing is available. We will see
in this paper that a Mazo phenomenon applies as well to
concatenations of traditional coding with FTN modulators
and to multicarrier FTN.
We introduced the idea of multistream FTN (MFTN)
signaling in [5]. It is useful to think of it as two-dimensional
FTN signaling because the symbols can be associated with
1
Others have called the NTB the normalized bandwidth or the
bit normalized bandwidth; it could as well be called the bandwidth
normalized bit time. We prefer the neutral term time–bandwidth
product.
points in a lattice spaced every f
Δ
and T
Δ
. This is illustrated
in Fig. 1. Pulses are “hung” on each point. The signaling
is not two-dimensional in the sense of coding over the
magnetic domains on a multitrack tape, although receivers
have some similarity. Reference [5] gave examples that
simultaneous frequency and time squeezing can indeed
increase the symbols transmitted in a given time–bandwidth
at the same P
e
. Neither compression alone can achieve the
same increase.
T
Δ
f
Δ
f
t
Fig. 1. Two dimensional Mazo signaling, in time and frequency.
Dots represent symbols separated by f
Δ
and T
Δ
.
What total time–bandwidth product is occupied by the
transmission? The question may be approached in several
ways. Spectral and temporal sidelobes interfere with ad-
jacent users of frequency and time, and contribute to the
signal’s occupancy. For a moderate NK product, a packet of
say 100–10000 bits, the sidelobes make a significant contri-
bution. In a companion paper [6] we treat this contribution,
seek to minimize it, and find rather different outcomes than
given here. In this paper we let N and K grow large, so
that the sidelobes are insignificant. The lattice area is about
NKf
Δ
T
Δ
Hz-s and the NTB tends to f
Δ
T
Δ
Hz-s/bit. The
ratio of N and K can be changed at will so long as they
have the same product. The orthogonal binary sinc(t/T )
pulse case has f
Δ
=1/T and T
Δ
= T , and the NTB is
f
Δ
T
Δ
=1Hz-s/bit. This provides a useful benchmark for
other pulses and systems. Since the value of T does not
change the NTB, we henceforth take T =1.
When f
Δ
is less than the subcarrier bandwidth, the signal
interrelations that produce d
min
work in new ways and the
distance structure is time varying. Analytical results are
available only in special cases [15], [16]. Finding d
min
in
this new situation is challenging but possible. The subject
of minimum distance is taken up in Section II. Distance
studies with various pulses h(t) show that d
2
min
=2can
occur at 0.5 Hz-s/bit, i.e., half the sinc benchmark. The
section also gives a number of properties. Euclidean distance
computations appear in Appendix 1. An advanced algorithm
to find minimum distance appears in the Appendix 2.
Section III proposes several MFTN receiver designs
and introduces concatenation of convolutional codes with
MFTN. An iterated receiver works particularly well here,
and these concatenations form a successful practical appli-
cation of MFTN.
3
II. SIGNALS,PROPERTIES AND MINIMUM DISTANCE
The real part of Eq. (2) is
2E
s
/T [I(t)cos2πf
0
t − Q(t)sin2πf
0
t]
where the in-phase and quadrature signals I(t) and Q(t)
are
I(t)=
K−1
k=0
N−1
n=0
a
I
k,n
h(t − nT )cos2πf
k
t
−a
Q
k,n
h(t − nT )sin2πf
k
t
Q(t)=
K−1
k=0
N−1
n=0
a
Q
k,n
h(t − nT )cos2πf
k
t
+a
I
k,n
h(t − nT )sin2πf
k
t
. (5)
The signals cos 2πf
k
t and sin 2πf
k
t are the subcarriers.
They exist only mathematically; the physical modulation
carriers are cos 2π(f
0
+ f
k
)t and sin 2π(f
0
+ f
k
)t, for k =
0,...,K−1. Figure 2 shows example 2-carrier signals with
T
Δ
= .8 and f
Δ
= .625, in which a
I
0,0
= a
Q
0,5
=1for
carrier 0 and a
I
1,3
= a
Q
1,8
=1for carrier 1, with all other
symbols set to zero. The carriers lie at f
0
and f
0
+ .625
Hz. Arrows show the location of the 1-symbols, and I
0
,I
1
and Q
0
,Q
1
are the respective carriers’ I and Q signals. The
subcarrier cos 2πf
1
t is shown dashed for reference. Note
that I
1
symbols have an effect on Q
1
and vice versa.
−2 0 2 4 6 8 10
−
1
0
1
−
1
0
1
−
1
0
1
Q
0
Q
1
Q
tot
t
−1
0
1
−1
0
1
−1
0
1
I
0
I
1
I
tot
Fig. 2. Upper three: Component and total in-phase signals
for 2-carrier MFTN example with f
Δ
= .8 and T
Δ
= .625.
Bottom three: Component and total quadrature signals. Time scale
in multiples of T
Δ
.
The normalized Euclidean distance between two signals
s
(1)
(t) and s
(2)
(t) of form (2) is
d
2
=
|s
(1)
(t) − s
(2)
(t)|
2
dt
2E
b
. (6)
Because of the summations in (2), only the difference
between the symbol streams matters and d
2
becomes
1
2E
b
K−1
k=0
N−1
n=0
Δa
k,n
h(t − nT ) e
j2π(f
0
+f
k
)t
2
dt.
(7)
Here Δa
k,n
are the elements of the complex symbol dif-
ference matrix
ΔA =
Δa
0,0
Δa
0,1
... Δa
0,N−1
... ...
Δa
K−1,0
Δa
K−1,1
... Δa
K−1,N−1
.
Equation (7) can as well be written
|ΔI(t) − ΔQ(t)|
2
dt (8)
as f
0
→∞, in which ΔI(t) and ΔQ(t) are as in (5) with
ΔA instead of A. For binary signaling the elements in ΔA
take values in {±2 ± 2j, ±2, ±2j, 0}.
An error event is a region of nonzero difference compo-
nents that begins at some position (n, k), which we may
as well take as (0, 0). Without loss of generality, we can
restrict Δa
0,0
to {2, 2+2j}. The minimum distance d
min
of a signal set is the minimum of (7) over all such events,
and an event leading to d
min
is called a critical event.
Since there are very many error events, finding d
min
is
difficult, but it is possible to find reliable estimates (i.e., tight
overbounds) by searching over limited sets of events. In this
paper the critical error events are typically of size 3 × 3 or
smaller. Distance may be computed by direct integration of
the difference signal, but a much more efficient method is
based on autocorrelations of h(t); this is given in Appendix
1.
An important property of MFTN distance is that it
depends on the start time of the event. That is, for a given
error event ΔA whose first column corresponds to pulses
centered at t =0, the distance will change if the event starts
with pulses centered at t = nT
Δ
. The fundamental reason
is that the pulse rate is not in general synchronized with
the subcarrier frequencies; it is mathematically seen in the
derivations in Appendix 1. An example of the phenomena
appears in Figure 3 for a 30% root RC pulse, T
Δ
= .7,
f
Δ
= .8, and the event
ΔA =
2 −2+2j −2j
−22+2j −2j
. (9)
The figure plots square distance against the event start time
t
0
. Since the plot repeats every 1/f
Δ
=1.25 s, the time
axis can be taken as t
0
mod 1/f
Δ
. Dots show this event’s
distance when it starts at times t
0
=0,.8, 1.6,..., which
lead mod 1.25 to all the multiples of .05.If1/f
Δ
T
Δ
is not a
rational number, then in principle starts at all t
0
mod 1/f
Δ
in [0, 1/f
Δ
) are possible, and the worst case distance is the
minimum, which is 1.13 at start t
0
mod 1/f
Δ
≈ .24.
Synchronous MFTN. When finitely many modulo start
times can occur, the multistream FTN is synchronous.It
is easy to see that when f
Δ
T
Δ
= i/, i and positive
integers without a common factor, then the MFTN signals
are synchronous and the distances occur at multiples of 1/.
4
The worst-case distance is the minimum of the allowed time
points. Synchronism can thus lead to a better distance, but
in reality there is little gain unless i and are very small
integers. In the figure, with 25 points, there is virtually
no distance gain. But the MFTN case shown in Figure 2
is a synchronous one with f
Δ
T
Δ
=1/2; the subcarrier
structure repeats precisely every 2T
Δ
. With error event (9),
the distance from start time t
0
= T
Δ
or 2T
Δ
are both 8.72.
These avoid the worst-case start time, which is 0.25, with
the much poorer distance 0.77. (However, there exist other
error events with distance less than 2, so this MFTN is
beyond the Mazo limit). Synchronism can thus be of benefit
when the MFTN parameters allow it, but only a few cases
of synchronous FTN are worth reporting. We will ordinarily
take distance to be the minimum of the continuous distance
versus t
0
curve.
0 0.25 0.5 0.75 1.00 1.25
0
4
8
1
2
1
6
2
0
d
2
t
0
mod 1/ f
Δ
min =
1.13
Fig. 3. Distance versus event start time t
0
modulo 1/f
Δ
, for
f
Δ
= .8 and T
Δ
= .7 with error event (9).
When not more than two carriers overlap in spectrum,
the curve in Figure 3 can be shown to be a sinusoid, and
its minimum can be computed easily from any three points.
Another simplifying property is that very few error events
lead to a distance near d
min
. As well, groups of symmetric
events, typically 4–8 in number, have identical distance for
the entire range of T
Δ
and f
Δ
. These we call event families.
The results presented in this paper in fact stem from only
20 families.
Delayed Pulses. Subjecting subcarriers to different delays
has the potential to improve minimum distance because
pulses can come into more favorable alignments with each
other and with the sines and cosines in the I and Q signals
in (5). Many ways to execute the delays can be imagined.
A simple but effective way is to delay the I and Q pulses
h(t−nT
Δ
) in carriers 0, 1,...,K−1 by δ
0
,δ
1
,...,δ
K−1
,
where each δ satisfies 0 ≤ δ<T
Δ
. Delaying all K
carrier pulses by the same δ is the same as a time shift
to the subcarrier system; this can prevent a subcarrier in
(2) from passing through zero at the moment when a
pulse is largest, which can severely reduce distance. Pulse
trains k =0, 1,... can be delayed by linearly growing
amounts, e.g., δ
0
,δ
1
,...=(0,.2,.4,...)T
Δ
. A particularly
successful scheme is delays of the form .5, 0,.5, 0,...times
T
Δ
. Instead of pulse delays, schemes can be based on
delaying the subcarriers by fractions of their own period
1/f
Δ
.
The Mazo Limit for Root RC and Gaussian Pulses.
Figures 4 and 5 plot the outcome of a search for non-
synchronous combinations of f
Δ
and T
Δ
that have least
product. Dotted lines show contours of constant f
Δ
T
Δ
product. Figure 4 plots the case for the Gaussian pulse
h(t)=
1/2πσ
2
exp(−t
2
/2σ
2
), normalized, with σ
2
=
.399.
2
The minimum distance searches here are over all
start times in the error events. Overlapping curves show
the trajectory for each critical event family; the “northeast”-
most of all curves is the Mazo limit. Consider the trajectory
for one critical error sequence, with distance d
2
.AsT
Δ
drops, the f
Δ
needed to maintain d
2
=2rises, creating,
typically, a convex-up f
Δ
,T
Δ
relationship. Eventually, time
compression alone prevents d
2
=2;nof
Δ
allows it, and
the result is a horizontal section at the lower right end of the
convex section (this is most visible in Fig. 5). At the upper
left of a convex section, T
Δ
is large and it can happen that
no f
Δ
leads to d
2
< 2. The section simply stops at some
f
2
Δ
,T
2
Δ
(square blocks mark two such points in Fig. 4).
If this section is part of the ultimate Mazo limit, then at
T
Δ
= T
2
Δ
the Mazo limit must move horizontally left to
another event’s convex section.
Note that h(t) here is not orthogonal for any T . The
Gauss pulse has important properties when simultaneous
time and frequency side lobes are important [6].
0.55 0.65 0.75 0.85
0
.65
0
.75
0
.85
0
.95
T
Δ
f
Δ
Gauss
.5
.6
.52
.54
Fig. 4. Estimated position of the two-dimensional Mazo limit
for non-synchronous binary Gaussian pulse signaling. Dashed line
shows limit with alternate pulse trains delayed 0.5 symbol. Each of
the five component curves here represents an event family. Dotted
curves are contours of constant f
Δ
T
Δ
.
Figure 5 plots the non-synchronous Mazo limit for 10
and 30% root RC pulses, plus the 10% case when alternate
pulse trains are delayed by T
Δ
/2 (dashed). The searches
are again over all start times in the error events. It can be
seen that the least product for 30% is about 0.60, at (f
Δ
≈
.67,T
Δ
≈ .88); for 10% pulses this improves to product
0.556 at (f
Δ
= .660,T
Δ
= .843). The delays improve
the 10% case to 0.534 at (f
Δ
≈ .66,T
Δ
≈ .80); the 30%
pulse is similarly improved by delays. These products are
excellent but we have found a few synchronous 10% cases,
with T
Δ
in the range 0.78–0.9, for which f
Δ
T
Δ
=1/2.
This is a doubling of the spectral efficiency of the sinc
benchmark and OFDM.
The estimated minimum distance of all combinations is
2. Searches are generally performed with the method in
Appendix 2 and are over error events out to size 4 × 7
2
This is the Gaussian pulse that has itself as transform. A new
pulse begins each T
Δ
as always.
