4206 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO. 7, JULY 2017
We assume that M
out of the M inputs in Fig. 1 are fed by
data symbols and the rest of the inputs are zero. Also, there
are N PAM symbols across time. Thus, following Fig. 1, one
finds that [17], [18], [34]
s(k) =
m∈
N−1
n=0
a
m,n
g
k −n
M
2
e
j2πmk/M
e
j (m+n)π/2
g
m,n
(k)
,
(1)
where is the set of M
active subcarriers, g
m,n
(k) is the
modulated pulse carrying the (m, n)-th PAM symbol. When
an FBMC signal is transmitted through a channel that varies
slowly with time and its delay spread is significantly shorter
than the symbol interval, the channel transfer function over
each subchannel may be approximated by a flat gain. Let H
m,n
denote this gain for the m-th subchannel at the n-th time index.
With the slow varying assumption, we omit the subscript n
from H
m,n
for simplicity of presentation. Then, the m
0
th
output of the receiver analysis filter bank at the n
0
th time
index is obtained as, [18], [34],
r
m
0
,n
0
=
∞
k=−∞
g
∗
m
0
,n
0
(k)
M−1
m=0
n∈Z
H
m
a
m,n
g
m,n
(k)
≈ H
m
0
a
m
0
,n
0
+ H
m
0
(m,n)=(m
0
,n
0
)
a
m,n
ζ
m
0
,n
0
m,n
(2)
where
ζ
m
0
,n
0
m,n
=
∞
k=−∞
g
∗
m
0
,n
0
(k)g
m,n
(k). (3)
It is noteworthy that for a well-designed prototype filter
g(k), ζ
m
0
,n
0
m,n
= 1when(m, n) = (m
0
, n
0
), and is zero or a
pure imaginary value when (m, n) = (m
0
, n
0
), e.g., see [17].
To be more specific, ζ
m
0
,n
0
m,n
,for(m, n) = (m
0
, n
0
),represents
the imaginary interference (also called intrinsic interference)
to a
m,n
, which could be removed by taking the real part after
channel equalization.
There are two common structures that may be adopted
for the implementation of FBMC: (i) the polyphase network
FBMC (PPN-FBMC) [35], [36] and (ii) the frequency spread-
ing FBMC (FS-FBMC) [37]–[39]. The results developed in
this paper can be integrated into either of the two structures.
We assume burst transmission in this paper, and the burst
length L
burst
is equal to (N − 1)M/2 + ηM − 1. From this,
the first ηM/2 samples correspond to the ramp-up period and
the last ηM/2 samples correspond to the ramp-down period
of the packet. The contribution of this paper is to minimize
these ramp-up and ramp-down periods. Our study reveals that
starting with the typical ramp-up and ramp-down periods 2M
(corresponding to the choice of η = 4), in the new design one
can reduce these to M/2, without any significant impact to the
OOB emissions of the resulting waveform.
III. R
EVIEW OF TAIL-SHORTENING METHODS
Since FBMC systems adopt filters with rather long length,
the resulting ramp-up at the beginning and ramp-down at the
end of each data burst cover multiple symbol intervals. This
considerably reduces the spectral efficiency of FBMC when
data bursts are short and are isolated from one another in
time. In what follows, we use the tail at the end of the burst
to illustrate the tail-shortening methods. The processing at the
start of the burst is similar.
The most trivial method of shortening the burst length is
hard truncation of the tails. For the end part of the burst, this
is expressed as
s
h.trunc
(k) =
s(k), k ≤ K
e.burst
0, k > K
e.burst
,
(4)
where K
e.burst
is the end of the data burst after truncation,
and subscript “h.trunc” on s
h.trunc
(k) is to emphasize it is hard
truncated. The cost of this method is increased OOB emissions
and ISI/ICI due to the truncation.
Another method, called truncation with windowing, uses
a generalized weighting window to shorten the tail. The
windowed signal is given as
s
w.trunc
(k) = s(k)w(k), (5)
where w(k) is a window function. Specifically, we consider
the raised-cosine window function
w(k) =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
1, k < K
b.ro
1
2
+
1
2
cos
(k − K
b.ro
)π
L
ro
, K
b.ro
≤ k ≤ K
e.burst
,
0, k > K
e.burst
(6)
where L
ro
is the roll-off period of w(k) and K
b.ro
= K
e.burst
−
L
ro
is the beginning of the roll-off. Although this method
smoothens the edge transitions introduced by truncation,
hence, reduces the OOB emissions, it results in a greater
ISI/ICI than the hard truncation method. This is because the
roll-off introduces an additional distortion to the signal. Note
that in both cases of hard truncation and truncation with
windowing, the packet ends at the same position K
e.burst
.
IV. T
AIL-SHORTENING BY VIRTUAL SYMBOLS
In this paper, we propose adding a few dummy symbols at
the two sides of each data burst, as demonstrated in Fig. 2,
to suppress the tails of s(k). After suppressing the tails, the
residual tails may be truncated with a minimum effect on
OOB emissions and/or ISI/ICI interference. Alternatively, the
suppressed tails may be allowed to overlap among successive
data packets.
A. Tail Cancelation by Virtual Symbols
We refer to the dummy symbols as virtual symbols.One
may think of virtual symbols as a means of generating a
cancelation signal at the tails of each FBMC burst, hence,
removing the tails of the bursts. Moreover, since this signal is
synthesized following the FBMC construction, it suppresses
the tails without introducing any OOB emissions. In addition,
in the absence of channel, the proposed method does not
introduce any ISI/ICI.
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