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International Journal of Scientific & Engineering Research, Volume 4, Issue 8, August-2013 1781
ISSN 2229-5518
Tone-Reservation based on fractional Fourier
Transform for Chirp-based OFDM
Shi Pengfei, M. R. Anjum, Zhao Yue, Riaz Ahmed Soomro, Farhan Manzoor
Abstract—As an adaptive method to combat doubly selective channel, chirp-based orthogonal frequency-division multiplexing (Chirped- OFDM) also suffers from a high peak-to average power ratio (PAPR). In this paper, an efficient optimization for tone-reservation technique based on the structure of fractional Fourier Transform is developed to reduce the number of variables in the quadratically constrained quadratic program (QCQP) for Chirped-OFDM subcarrier modulation, which reduces complexity considerably. Simulation results yield its performance close to conventional tone reservation with less computational complexity.
Index Terms—Chirped-OFDM, fractional FFT, PAPR, tone reservation
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HIRPED-OFDM system obtains good performance in doubly dispersive (i.e., time and frequency selective)
X (0)
S/P IDFrFT
X − 1)
CP
adding
P/S
channel[1]. However, as the multicarrier system, the relatively high PAPR is also a major drawback of Chirped– OFDM system. The problem directly influences operation cost and efficiency of the system. In order to relieve the problem of high PAPR in Chirped-OFDM system, Conventional PAPR reduction techniques in OFDM system in OFDM system are
introduced, such as Clipping, Selecting Mapping(SLM) ，
( N
�X (0)
P/S DFrFT
�X ( N − 1)
CP
removing
S/P
Wireless channel
Partial Transmit Sequence(PTS), etc[2]. Besides, an efficient tone reservation technique is proposed by Tellado in [3] to overcome those drawbacks, especially, in a large number of subcarriers. A small number of subcarriers are reserved in transmitting to reduce PAPR. However, this algorithm also suffers large of FFT/IFFT operations due to large number of iterations.
In this paper, simplification for tone-reservation technique based on the structure of fractional Fourier Transform is pro- posed to reduce the number of variables in the QCQP for
Fig. 1. Block diagram of Chirped-OFDM system
At the transmitter side, an N-point IFRFT is applied to data block symbols after transforming the high-speed data stream with digital modulated into low-speed parallel data streams. In the Chirped-OFDM system, the exponential fundamental basis waveforms are replaced by the chirp fundamental basis waveforms. The signal modulated onto subcarriers is ex- pressed as [4]
N −1
Chirped-OFDM subcarrier modulation. The reminder of this paper is organized as follows. In section 2, we introduce the Chirped-OFDM system model and the traditional PAPR re-
x(t ) = ∑ X (k )�ck , −α (t )
k = 0
(1)
t ∈[0,Ts )
duction schemes. Section 3 describes conventional tone reser-
where
X = [ X (0), X (1),, X ( N − 1)]
is the transmitted data
th
vation technique. The proposed technique based on Radix-2
FRFT algorithm and its simulations are presented in section 4.
with the number of subcarriers N . ck , −α (t ) is the k chirped
subcarrier function given by,
ck ,α (t ) =
(2)
1 − j cot α �exp (
jp [cot α �t
2 − 2kt / T + cot α (k sin α / T )2 ])
with the Chirped-OFDM symbol period Ts
related to the FRFT transform order p .
and α = pp / 2
The structure of Chirped- OFDM system is shown in Fig.1.
Correspondingly, the normalized discrete transmitted sig- nal modulated onto subcarriers can be expressed as:
x(n) =
(1 − j ⋅ cot α ) / N ⋅ exp (− j / 2�cot α �n2 �∆t 2 )
N −1
(3)
×∑ exp (− j / 2�cot α �k 2 �∆u 2 + j�2p �n�k / N )�X (k ) ,
k = 0
n = 0,1,, N − 1
————————————————
where ∆t
is the sample interval in time domain, and
• Shi Pengfei is currently pursuing Ph. D degree program in Information & Communication Engineering in Beijing Institute of Technology, China,
100081. E-mail: shipengfeibit@gmail.com
∆µ = 2p �
sin α 
/ ( N �∆t )
domain.
denotes sample interval in fractional
• M. R. Anjum is currently pursuing Ph. D degree program in School of
Information & Electronics in Beijing Institute of Technology,Beijing,
For
x(n)
in (3) is the summation of independently multi-
China, 100081. E-mail: engr.muhammadrizwan@gmail.com
carriers in the transmitter, the Chirped-OFDM symbol may
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International Journal of Scientific & Engineering Research, Volume 4, Issue 8, August-2013 1782
ISSN 2229-5518
have large peak, which introduces large peak to average pow-
2v−1 η
er ratio. Based on the traditional system, the definition of
PAPR for the Chirped OFDM system is the ratio of peak pow-
x(n) = ∑ x (n) =
η =1
(1 − j ⋅ cot α ) / N ×
er and average power expressed as:
2v−1
N / 2
v−1
−1 η
�
( 2 2
2 )
n k
∑ ∑ −
� α � �∆ + �∆
W
( 2 2 )
η =1
k = 0
X (n) exp
j / 2 cot
n t k u
N / 2v−1
PAPR(dB) = 10 log10
(4)
max{ s(n) } / E{ s(n) }
(10)
where W donates to DFT, which can be expressed in matrix
PAPR performance of a multicarrier system is evaluated in
formation:
terms of the complementary cumulative distribution function
1
Q1
�X 1
(CCDF) of PAPR in general. The CCDF of PAPR is defined as
the probability that the PAPR of the transmitted signal ex-
ceeds a given threshold PAPR0 ,
η = Qη
�X η
x
CCDF = P{PAPR > PAPR0 }
(5)
v−1
x
r v−1
Qr �X
r v−1
(11)
with each blocks of FRFT on stage v satisfying:
In the Chirped-OFDM system, tone reservation signal
Q1 = = Qη =
v −1
Q =
C = [C (0),C (1),...,C ( N − 1)]T
is added to the original signal
1 1 1
(12)
X=[X (0), X (1),, X ( N − 1)]T , then the new transform-domain
− 2 2 2
2 2 2
v −1 2 2
signal can be expressed as[5]:
j / 2�cot α n �∆t + k �∆u
1 e W
− j / 2�cot α � n �∆t + ( N / 2 −1) �∆u v −1
e W
N / 2v −1
( −
N / 2v −1
x (n) = IDFRFT{X + C} = x(n) + c(n)
(6)
− �
v −1 − 2 � 2
�∆ 2
v −1 2 2
v −1 2 2
where IDFRFT donates to the inverse discrete fractional Fouri-
1
e j / 2 cot α �( N / 2 1) ∆t + k
u ( N / 2v −1 −1) k
N / 2v −1
− j / 2�cot α ( N / 2 −1) �∆t + ( N / 2 −1) �∆u ( N / 2v −1 −1)2
N / 2v −1
er transform with X + C
satisfying the condition:
By using the intermediate stage v of FRFT algorithm, the
C (k )
X + C =
X (k )
k ∈{i1 ,i2,...,iL }
k ∉{i1 ,i2,...,iL }
(7)
subset Λe of peak reduction tones (PRT) for generating C are
divided into equal subsets with length α = L / 2v −1 as follow-
ing, where L donates to the number of tones reservation and
where L and i1 ,i2 ,,iL donate to the number of tone reserva-
Λe is defined:
tion, and the subset of tone reservation, respectively. In order
1 1 η η
r v −1
r v −1
to reduce the computation burden, C (k )(k = i1 ,i2 ,...,iL ) always
Λe = λ1 ,, λα ,, λ1 ,, λα ,, λ1
,, λα
are chosen from the collection {1, −1}
or {±1, ± j} .
(13)
Then, the optimization problem for PAPR of Chirped-
Then, the component 1 1 1 1 T
[C (λ1 ),, C (λα )]
are randomly cho-
sen from {1, −1, j, − j} with subsets λ1 ,, λ1 randomly from the
OFDM system is defined as
1 1 α η
min ξ (8)
subset of �X (k ) . The other Cη are reserved in �X
1
(k ) with the
C
subject to: ξ = max x(n) + c(n) 2
same PRT locations of �X (k ) . Thus, the matrices C�
can be expressed as:
and Q�
1 η
2v −1
1 1 1 T
which can be formulated as a quadratically constrained quad-
ratic program (QCQP).
According to (3), Pei proposed the FRFT algorithm by com-
bining FFT with chirp-rate modulation in time and frequency
domain, respectively. So we could optimize the problem (8)
C� = C ,, C ,, C
(14)
and
Q1
= C ,, C ,, C
Q1
through radix-2 FRFT algorithm, either. The principle of the
proposed frame is depicted in Fig. 2.
Q� = Qη
= Q1
Usefull data
2v−1
Q1
X S/P
v stage
IDFRFT
Select the Chirped- OFDM signal
(15)
Q 2v−1 × 2v−1
2v−1 × 2v−1
Peak-cancelling signal
C
S/P
⊕
v stage
IDFRFT
change C
with the minimum PAPR
Then, the optimization problem in (8) could be simplified
as following:
min ξ (16)
Cη
Fig. 2. The proposed tone reservation technique based on radix-2
subject to:
η η η 2
x + Q C ≤ τ
FRFT algorithm.
Define the symbol x and index k for an intermediate stage v within the FRFT process are represented for the input X and frequency index k of each block, respectively. The new FRFT outputs on stage v :
Compared with (8), the proposed technique keeps the same number of constraints while reducing the number of variables from L to L / 2v −1 .
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International Journal of Scientific & Engineering Research, Volume 4, Issue 8, August-2013 1783
ISSN 2229-5518
According to [6], the PAPR was extended to the case of Chirped-OFDM system, whose simulation results showed that the PAPR of Chirped-OFDM system was slightly relative to TABLE 1
SIMULATION PARAMETERS
Parameters Value
[1]. M. Martone, “A Multicarrier System Based on the Fractional Fourier Transform for Time-Frequency-Selective Channel”, IEEE trans. on Commun., vol. 49, no. 6, pp.1011-1020, Jun. 2001
[2]. T. Jiang, and Y. Wu, “An Overview Peak-to-Average Power Ratio Reduction Techniques for OFDM Signals”, IEEE trans. on Broadcasting, vol. 54, no. 2, pp.257-268, Jun. 2008
[3]. J. Tellado and J. Cioffi. “Controlling clipping probability in DMT
Number of FRFT-OFDM
symbols
100000
transmission”, ANSI Document, T1E1.4 Technique Subcommittee, no.97-
397, pp.1-7, Dec. 8,1997
Number of subcarriers 256
Modulation mode QPSK Channel Type AWGN The value of C {1, −1, j, − j}
the fractional order p . Without loss of generality, in this sec- tion, the fractional order for the proposed technique based on
[4]. Y. Ju and et al. “Analysis of Peak-to-average power ratio of a multi- carrier system based on the fractional Fourier transform”. IEEE Singa- pore International Conference on Communication System. 2004, 165-168.
[5]. A. Ghassemi, and T. A. Gulliver, “Efficient Optimization for Tone
Reservation OFDM”, ISITA, 2008
[6]. Y. Ju and B. Barkat, “Analysis of Peak-to-average power ratio of a multicarrier system based on the fractional Fourier transform”, ICCS,
2004
radix-2 FRFT algorithm could be
p = 0.1 . The other simulation
parameters are in TABLE 1 and the result is presented in Fig.
3.
0
10
Orignal PAPR
Tone Reservation
The Proposed
-1
10
-2
10
-3
10
5 6 7 8 9 10 11 12 13
PAPR0 [dB]
Fig. 3. PAPR comparison among traditional tone reservation and the proposed technique in Chirped-OFDM system
As is shown in Fig. 3, we can see that the same Cη only makes the tone reservation performance worse negligibly while it could reduce the calculation meaningfully through the structure of radix-2 FRFT algorithm.
A new PAPR reduction technique based on FRFT structure has been extended to the Chirped-OFDM system via tone reserva- tion. For the symmetry and decimation in frequency of FRFT algorithm, the transform matrices on an intermediate stage are used to simplify the peak reduction signal. The derivation and simulation results show that its performance is close to con- ventional tone reservation with less computational complexi- ty.
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