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CHAPTER 4
QUADRILATERAL COMPANDING TRANSFORM FOR
PAPR REDUCTION IN OFDM SYSTEMS
As discussed in chapter 3, the time domain OFDM signal has large envelope fluctuation;
therefore, PAPR of OFDM signal is very high. Thus HPAs with large linear range and high
resolution ADCs/DACs are required to amplify the OFDM signal, and for its conversion
between analog and digital domains respectively. Many companding schemes [27]-[33] have
been proposed in the literature to reduce the PAPR of OFDM signal. The conventional μ -
law and A-law companding schemes can be used for PAPR reduction, by choosing the
suitable value of the parameters μ or A , controlling the nonlinearity of the μ -law [27] or
A -law companding function respectively. But the error performance of both the schemes
degrades as both of them introduce high companding distortion in OFDM signal at higher
values of μ or A . A nonlinear companding transform [28] has been proposed by Jiang et al.
to effectively reduce the PAPR of the OFDM signal. In this scheme [28], the Gaussian
distributed in-phase (I) and quadrature-phase (Q) components of discrete time complex
OFDM signal are transformed into a quasi-uniform distribution. In this scheme, the
companding function is separately applied to I and Q components of the OFDM signal. The
large values of I or Q components of the OFDM signal are compressed, whereas those with
small I and Q components are enlarged. The PAPR reduction capability and BER
performance of this scheme [28], can be optimized by properly choosing the parameters of
the companding function. Jiang et al. proposed “Exponential Companding (EC)” scheme [29]
to transform Rayleigh distributed OFDM signal magnitude into uniform distribution. EC can
effectively reduce the PAPR of the OFDM signal but its companding function has no design
flexibility and therefore, a good trade-off between BER and PAPR performances cannot be
achieved. Hunag et al. proposed four companding transformation functions [30] to reduce the
PAPR of the OFDM signal. These include: linear symmetrical transform (LST), linear non
symmetrical transform (LNST), non-linear symmetrical transform (NLST) and non-linear
non-symmetrical transform (NLNST). It has been shown in [30] that LNST performs the best
among four companding functions [30]. In LNST an inflexion point has been introduced to
60
treat large and small signals on different scales to achieve better BER and PAPR
performances. Linear companding transform (LCT) [31], proposed by Aburakhia et al., also
treats large and small signals on different scales, but has two inflexion points to achieve more
flexibility in designing the companding function. The transformed signal changes abruptly at
the inflexion points and hence degrades the power spectral density (PSD) of OFDM signal.
Trapezoidal companding (TC) [32] proposed by Hou et al., is an efficient PAPR reduction
method with low BER. This scheme [32] transforms the Rayleigh distributed OFDM signal
magnitude into trapezoidal distribution and is called “Trapezoidal Companding”. Trapezoidal
companding uses a piecewise function defined in three intervals of OFDM signal magnitude.
Jeng et al. proposed [33] trapezium distribution based companding (TDBC), to transform the
Rayleigh distributed OFDM signal magnitude into a biased linear distribution called
“Trapezium Distribution”.
In this chapter, we propose a novel non-linear generalized companding scheme called
“Quadrilateral Companding Transform (QCT)” to reduce the PAPR of OFDM signal. The
proposed method provides additional degrees of freedom in comparison to existing
trapezoidal companding, exponential companding and trapezium distribution based
companding schemes. This allows more flexibility in designing the companding function,
which is useful for the overall OFDM system to achieve low BER with good PAPR reduction
capability.
The remainder of this chapter is organized as follows: In section 4.1 we discuss the
motivation to quadrilateral companding transform. Section 4.2 deals with OFDM system
model with quadrilateral companding. The proposed quadrilateral companding and
decompanding functions are derived in section 4.3. Mathematical analysis of the PAPR
performance of proposed scheme is presented in section 4.4. Some special cases of the
proposed scheme are discussed in section 4.5. The simulation results for PAPR and BER
performances of the proposed scheme are presented and discussed in section 4.6. Finally this
chapter is concluded in section 4.7.
4.1 MOTIVATION TO QUADRILATERAL COMPANDING
TRANSFORM
As discussed earlier, the magnitude of OFDM signal has Rayleigh distribution. Rayleigh
distribution has a long tail, therefore, the peak value of the OFDM signal is very high as
61
compared to its average value; hence it has very high PAPR . But in Rayleigh distribution,
the probability of having large magnitude is very less than the probability of signal having
small magnitude. In TC, the magnitude of the transformed OFDM signal follows trapezoidal
distribution, in which the probability of having large magnitude is also very less than the
probability of the signal having small magnitude. Whereas, in EC, the magnitude of
companded signal follows uniform distribution, in which the probability of all OFDM signal
magnitude in the defined interval is equal, which makes it more lossy. In EC, the companding
function used for transformation has no flexibility to design the companding function i.e. the
parameters of the companding function controlling the nonlinearity are fixed due to the
constraints on the average power and PDF of transformed OFDM signal. Therefore, EC [29]
has no degree of freedom. On the other hand, in TC we can control the nonlinearity of the
companding function by properly choosing the values of normalized projection of two
hypotenuses (a, b) on down hemline. Therefore, its [32] degree of freedom is two. Also, with
appropriate choice of these parameters, the scheme can be made less lossy to provide better
BER performance for a given PAPR.
Motivated by this observation, we propose to transform the Rayleigh distributed OFDM
signal magnitude into Quadrilateral distribution function shown in Fig. 4.1, to achieve an
additional degree of freedom over TC. The parameters of quadrilateral distribution are chosen
in such a way that it produces least possible companding distortion to achieve low BER for a
given PAPR.
Figure 4.1: Quadrilateral distribution for proposed QCT
62
4.2 SYSTEM MODEL
The block diagram of an OFDM system using companding scheme for PAPR reduction is
shown in Fig. 4.2. Here, we have considered an OFDM system with N subcarriers, in which
each of the subcarrier is modulated by M-PSK or M-QAM. As shown in Fig. 4.2 the input
binary data sequence is first converted into N parallel data substreams and then these are
mapped to the constellation points of M-PSK or M-QAM to achieve desired modulation on
each of the subcarriers. After this, subcarrier modulation is performed using IFFT block to
obtain the discrete time domain OFDM signal. Let { } 10−=
NkkX be the N complex modulated
data symbols to be transmitted over N subcarriers. The discrete time domain OFDM signal
generated after taking IFFT of a block of N modulated data symbols, is given by (2.9).
Figure 4.2: Block diagram of OFDM system with companding transform
Discrete time domain OFDM signal is passed through the parallel to serial (S/P) converter
and then applied to the compander for reducing the dynamic range or PAPR of the OFDM
signal. The companded OFDM signal is applied to digital to analog (D/A) converter to get
analog signal and then finally amplified using HPA. At the receiver, the received signal is
first converted into digital signal using A/D converter. The digital signal is then expanded by
inverse companding function known as decomapnding function. After that subcarrier
demodulation is performed by taking the FFT of OFDM signal obtained from expander.
Finally, M-PSK or M-QAM decoder is used to decode the received data signal.
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The symbols notation used throughout this chapter are listed in Table 4.1 for convenience.
Table 4.1: List of symbols used in QCT
thk modulated data symbol kX
thn sample of discrete time domain OFDM signal ][nx
PDF of original OFDM signal (without companding) (.)|][|
onx
f
CDF of original OFDM signal (without companding) )(.|][| nxoF
PDF of OFDM signal after companding (.)|][|
cnx
f
CDF of OFDM signal after companding (.)cF
Upper-bound of the peak value of OFDM signal l
Quadrilateral Companding function (.)h
Quadrilateral Decompanding function (.)1−h
thn sample of discrete time OFDM signal after companding ])[( nxh
4.3 QUADRILATERAL COMPANDING AND DECOMPANDING
FUNCTIONS
The quadrilateral companding function )(xh is a nonlinear companding function. It
transforms the original probability distribution function of OFDM signal magnitude into a
quadrilateral distribution as shown in Fig. 4.1, and hence the name “Quadrilateral
Companding Transform”.
The pdf )(|][|
xf cnx
of quadrilateral distribution can be read from Fig. 4.1 as
( )
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
≤<−−
−
−≤<−−−
−+
≤
=
otherwise0
)1(,)(
(4.1))1(,)(
,
)(
2
121
1
|][|
lxlbbl
lxh
lbxalalxblall
hhh
alxal
xh
xf cnx
64
where balhh and,,, 21 are the parameters of quadrilateral distribution shown in Fig. 4.1.
These parameters ( balhh and,,, 21 ) control the nonlinearity of the companding functions.
The cumulative distribution function (CDF) of quadrilateral distribution function given by
(4.1) can be calculated using the following relationship
∫=l
cnx
c dxxfxF0
|][|)()( (4.2)
Using (4.1) and (4.2) we have
( )
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
>
≤<−−
−−−+
++
−≤<−−−
−+⎟
⎠⎞
⎜⎝⎛ −
≤
=
lx
lxlbbl
xlhblallhhlbhah
lbxalalxblall
hhalxh
alxalxh
xF c
1
)1(,2
)(2
)()(2
)(
(4.3))1(,)(22
,2
)(2
22121
2121
21
Quadrilateral distribution function is bounded in the interval ],0[ l . Like EC, TC and TDBC,
in this scheme also average power of the OFDM signal before and after companding is kept
same, therefore, we have
dzznxfz cl
)(|][|0
22 ∫=σ
(4.4)
( ) dzbl
lxhzdzalzblall
hhhzdzal
xhz
l
bll
bll
al
al
⎭⎬⎫
⎩⎨⎧ −−+
⎭⎬⎫
⎩⎨⎧
−−−
−++
⎭⎬⎫
⎩⎨⎧
= ∫∫∫−
− )()(
22121
2
0
12
12
)368(
12)}1()1({)(
4
)1(
12
)1(
12
3232
2212
3332
331
331 bbblhbabahhalblhblhlah −−
−−+−+−
−−
+−
+−=
which can be further simplified to
[ ] )5.4()]1()}1(2){1(3[}])1()1){(1[(121 223
2223
12 ++−+−−−+++−+−−= aaaabbalhababblhσ
65
As shown in Fig. 4.1, the PDF of quadrilateral companded OFDM signal lies in the interval
],0[ l , therefore, we have,
1)(|][|0
=∫ dzznxf cl
(4.6)
1)()(
0
2)1(
121
0
1 =−
−+⎭⎬⎫
⎩⎨⎧
−−−−
++ ∫∫∫−
dzbl
lzhdzblall
alzhhhdzal
zh llb
al
al
122
)()(2
2211 =+−−+
+lahlblalhhlah
2)()( 21 =−+− lalhlblh (4.7)
For given values of bal and, , the parameters ( 21 , hh ) of the companding function )(xh can
be easily calculated using (4.5) and (4.7). Therefore, three parameters bal and,( ) can be
chosen independently to control the nonlinearity of companding function )(xh . Hence the
proposed QCT has three degree of freedoms. The values of bal and, should be chosen
independently to provide low PAPR and BER.
As we can see from Fig. 4.3, TC [32] for the companding function parameters (a=0.4, b=0.1),
yields h1=h2=h=0.8165 and l=1.633, i.e. a single value of l, whereas in QCT for the same
values of a and b, a complete range ([1.29, 2.12]) of l can be utilized to design the
companding function. It is therefore confirmed from Fig. 4.3 that QCT provides more design
flexibility in comparison to TC. Hence, QCT can achieve a better trade-off between PAPR
and BER performances. The same design flexibility can also be achieved for other possible
combinations of a and b. Another example depicting the same is shown in Fig. 4.4.
It can also be seen from Fig. 4.4, EC [29] for the comapnding function parameters 0=a and
0=b , yields 5773.021 === hhh and l=1.732, i.e. a single value of l, whereas in QCT for
the same values of a and b, a complete range ([1.414, 2.42]) of l can be utilized to design the
companding function. Hence, a better trade-off between PAPR and BER performances can be
achieved by QCT.
66
Figure 4.3: 1h and 2h vs. l graphs for 4.0=a and 1.0=b
Figure 4.4: 1h and 2h vs. l graphs for 0=a and 0=b
0 0.5 1 1.5 2 2.5 3-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
l
a=0.4 and b=0.1
h1
h2
0 0.5 1 1.5 2 2.5 3-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
l
h1
h2
67
Let us derive the quadrilateral companding function )(xh , which converts the Rayleigh
distributed OFDM signal magnitude into the required quadrilateral distribution. Assuming
)(xh to be monotonically increasing function, the expression of QCT function )(xh can be
derived after equating the CDF of original and companded OFDM signal. Therefore, we have
))((}[n]Prob{}|][Prob{|)(|][| xhFh(x)|h(|xxnxxnxF co =≤=≤= (4.8)
where )(|][| xnxF o is the CDF of original OFDM signal given by following
⎟⎟⎠
⎞⎜⎜⎝
⎛−−= 2
2exp1)(|][| σ
xxnxF o , 0≥x (4.9)
Using (4.8), we have
)sgn()).(|][|()(1
xxnxFFxh oc −= (4.10)
where sgn(.) is the sign function. Using (4.3) and (4.9), the quadrilateral companding
function )(xh of (4.10) can be expressed as
)11.4(
))((22ln||
2||
exp))(()(2)sgn(
))((22ln||
22ln
||exp22
)()()()sgn(
22ln||
||exp12)sgn(
)(
211
2
2
21212
2111
2
2
1122
112
1
2
2
1
,
,
,
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪
⎨
⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛−−+−−
>
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−⎟⎟⎠
⎞⎜⎜⎝
⎛−+−−+++⎟⎟
⎠
⎞⎜⎜⎝
⎛−
⎟⎟⎠
⎞⎜⎜⎝
⎛−−+−−
≤<⎟⎟⎠
⎞⎜⎜⎝
⎛−
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛−+−
−−−
−−−−
+
⎟⎟⎠
⎞⎜⎜⎝
⎛−
≤
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−−
=
blallhhalhx
xblallhhlbhah
hbllx
blallhhalhx
alh
xalh
blallhh
hhh
blallalx
alhx
xhalx
xh
σ
σ
σσ
σ
σ
σ
68
The QCT function given by (4.11) with suitable values of companding function parameters
bal and,( ) can be utilized to perform the companding operation.
In order to eliminate the effect of quadrilateral companding transform, at the receiving end an
inverse transformation of QCT has to be used, which is known as quadrilateral decompanding
function and is denoted by )(1 xh − . The quadrilateral decompanding function )(1 xh− can be
derived after calculating the inverse of companding function )(xh , which, from (4.11),
becomes
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
−>⎟⎟⎠
⎞⎜⎜⎝
⎛
−+−−+−+−
−≤<⎟⎟⎠
⎞⎜⎜⎝
⎛
−−−−−−+−−
≤⎟⎟⎠
⎞⎜⎜⎝
⎛
−
=−
lbxxlhblbahhbhahbl
blx
lbxalalxhhblallxhalh
blallx
alxxhal
alx
xh
)1(||,|)|()}1)(({2(
2ln)sgn(
)12.4()1(||,)|)(|()|)(|22(
)(2ln)sgn(
||,||2
2ln)sgn(
)(
22
22121
21211
21
1
σ
σ
σ
4.4 PAPR PERFORMANCE OF QUADRILATERAL COMPANDING
TRANSFORM
In order to find PAPR reduction capability of the proposed scheme over original OFDM
signal, a parameter known as transformation gain (G) is used, which is defined as the ratio of
PAPR of the original OFDM signal to that of the companded OFDM signal.
]))[((])[(nxhPAPR
nxPAPRG = (4.13)
Let 1A and 2A are the peak values of companded and original OFDM signal respectively and
we denote the peak and the average power of companded OFDM signal as 21A and 2
1M
respectively. Similarly, we denote the peak and the average power of original OFDM signal
by 22A and 2
2M . As discussed earlier, the average power of the OFDM signal before and after
applying QCT remains same i.e. 22212 σ== MM . Therefore, the transformation gain (G) will
simply be the ratio of peak powers of the original and companded OFDM signals. Hence,
69
21
21
22
/
/ 22
MA
MAG =
2
1
2⎟⎟⎠
⎞⎜⎜⎝
⎛=
AA
)|][|max(
)|][max(|2
2
nxhnxG = (4.14)
The CCDF of PAPR of original OFDM signal is defined in (3.20), similarly we can write the
mathematical expression for CCDF of PAPR of companded OFDM signal as
)))((Pr()( 00 γγ >= xhPAPRCCDF c
(4.15)
⎟⎟⎠
⎞⎜⎜⎝
⎛>= 02
1
21Pr γ
MA
After simplifying (4.15), we get
⎟⎟⎠
⎞⎜⎜⎝
⎛>= 02
1
22
22
22
0 Pr)( γγAA
MA
CCDF c
(4.16)
The peak value of the companded OFDM signal ( 1A ) can be found by substituting 2|| Ax =
in (4.11) and the peak power of companded OFDM signal )( 21A can be calculated as
70
)17.4(
))((22ln
,2||
exp))(()(2
))((22ln
22ln
,||
exp22)(
)()(
22ln
,||
exp12
211
22
2
2
2
21212
211
22
1
2
2
2
2
1122
112
1
22
2
22
1
21
2
2
2
2
2
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪
⎨
⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛−−+−−
>
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−+−−+++⎟⎟
⎠
⎞⎜⎜⎝
⎛−
⎟⎟⎠
⎞⎜⎜⎝
⎛−−+−−
≤<⎟⎟⎠
⎞⎜⎜⎝
⎛−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−+−
−−−
−−−−
+
⎟⎟⎠
⎞⎜⎜⎝
⎛−
≤
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛−−
=
blallhhalhA
Ablallhhlbhah
hbll
blallhhalhA
alh
Aalh
blallhh
hhh
blallal
alhA
Ahal
A
σ
σ
σσ
σ
σ
σ
Substituting 21A into (4.18) yields
+⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−
≤
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−−
>=alh
AA
al
hAMA
CCDFc
1
220
2
21
22
22
22
0 22lnPr
||exp12
Pr)( 2
2
σγ
σ
γ
+⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−−+−−
≤<⎟⎟⎠
⎞⎜⎜⎝
⎛−
×
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−+−
−−−
−−−+−
−>
))((22ln
22lnPr
||exp22
)()(
)()(
)(Pr
211
22
1
2
02
2
2
1122
112
212
22
22
2
2
2
2
blallhhalhA
alh
Aalh
blallhh
hblallhhal
hhAM
A
σσ
γ
σ
71
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−−+−−
>×
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−⎟⎟⎠
⎞⎜⎜⎝
⎛−+−−+++⎟⎟
⎠
⎞⎜⎜⎝
⎛−
>
))((22lnPr
)18.4(
2||
exp))(()(2
Pr
211
22
02
2
22
21212
22
22
22
2 blallhhalhA
Ablallhhlbhah
hbll
AMA
σ
γ
σ
The expression for CCDF of companded OFDM signal can be further simplified to the
following form
+⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−
−
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−−
=alh
CCDFA
al
hACCDFCCDFc
10
2
22
122
0 22ln1
||exp12
)( γ
σ
γ
+⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−−+−−
−⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−
×
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛−+−
−−−
−−−+−
−
))((22ln
22ln
||exp22
)()(
)()(
)(
2111
02
2
22
1122
112
212
22
blallhhalhCCDF
alhCCDF
Aalh
blallhh
hblallhhal
hhACCDF γ
σ
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−−+−−
×
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−⎟⎟⎠
⎞⎜⎜⎝
⎛−+−−+++⎟⎟
⎠
⎞⎜⎜⎝
⎛−
))((22ln
)19.4(
2||
exp))(()(2
211
02
2
22
21212
22
blallhhalhCCDF
Ablallhhlbhah
hbll
ACCDF γ
σ
where CCDF(.) is the complementary cumulative distribution function defined in (3.20)
72
Upper Bound on transformation gain (G) for proposed scheme:
It can be observed from Fig. 4.1 that the peak value of the OFDM signal after companding is
l . If an OFDM signal with unity average power is applied as an input to the compander, the
PAPR of companded OFDM signal cannot exceed 2l . Hence the transformation gain (G) of
(4.14), is bounded by
2
2
max)|][max(|
lnxG ≤
(4.20)
Therefore, PAPR reduction capability of the proposed companding scheme is governed by
the choice of l , i.e., lower the value of l , better is the PAPR reduction.
Minimizing PAPR of companded OFDM signal:
Average power of OFDM signal before and after companding remains same, therefore PAPR
of the companded OFDM signal can only be controlled by its peak value l. Hence, the
minimum value of PAPR can be achieved by minimizing l. The minimum value of l can be
calculated using (4.5) and (4.7) subject to the constraints 0 ≤ a ≤ 1, 0 ≤ b ≤ 1, a+b≤1, h1 ≥0
and h2 ≥0. By solving (4.5) and (4.7) under given constraints, we have found out lmin→1,
when a→1 and b→0.
It can be seen from Figs. 4.5 and 4.6 that for 9999.0=a and 0001.0=b the minimum
possible value of l for which 01 ≥h and 02 ≥h is 1.01. Therefore, its PAPR performance is
very close to 0dB. We have evaluated the PAPR performance for this case to verify the same
(see Fig. 4.7).
Figure 4.5: 1h vs. l plot for 9999.0=a and 0001.0=b
0 0.5 1 1.5 2 2.5 3-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
l
h1 h1
73
Figure 4.6: 2h vs. l plot for 9999.0=a and 0001.0=b
Figure 4.7: PAPR of QCT for 9999.0=a and 0001.0=b
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5x 10
4
l
h2
0 0.5 1 1.5 2 2.5 3 3.5 410-2
10-1
100
PAPR0 [dB]
CC
DF
(Pr[P
AP
R>P
AP
R0]
)
74
4.5 SPECIAL CASES OF QCT
Many distribution functions like uniform distribution, trapezium distribution, trapezoidal
distribution etc. can be easily obtained from quadrilateral distribution by properly setting its
parameters ( 21,,, hhba and l ).
Trapezoidal distribution of TC can be obtained by taking hhh == 21 in quadrilateral
distribution. The obtained PDF of trapezoidal companding [32] is shown in Fig. 4.8 (a).
Similarly, if we set the parameters of quadrilateral distribution as
hhhba ==== 21and0,0 , then companded OFDM signal has uniform distribution,
which is the basis function of exponential companding [29]. The obtained uniform
distribution function is depicted in Fig. 4.8(b).
Finally, if we set 21and0,0 hhba ≠== then QCT becomes the equivalent of trapezium
companding [33]. The obtained trapezium distribution is depicted in Fig 4.8(c).
Hence exponential companding [29], trapezoidal companding [32], and trapezium
distribution based companding [33] are special cases of the proposed QCT and can be
obtained by properly setting the parameters of QCT. Therefore, we can say that quadrilateral
companding transform (QCT) is the generalized non-linear companding transform.
In the proposed quadrilateral companding transform for defining the companding function,
we require three parameters ( lba and, ), the remaining two parameters 21 and hh are
function of ( lba and, ) and can be calculated using (4.5) and (4.7) for specified value of 2σ .
But in case of TC the value of l is a function of ba and and cannot be chosen independently
for given values of ba and . Hence, the degree of freedom in TC is two. Therefore, QCT
provides extra design flexibility over TC.
.
Figur
4.6 RE
The ov
transfor
Here, w
scheme
symbol
subcarr
white G
results,
of prop
channel
(
re 4.8: Distrib
ESULTS A
verall perfor
rm is evalu
we use the
e wherein, th
s ( )(±12/1
iers are allo
Gaussian no
transmissio
posed comp
l.
(b)
bution function
AND DIS
rmance of
uated by pe
OFDM sy
he randomly
)j±1 on eac
ocated for C
oise is assu
on of 10,00
panding tra
h
l
n for special c
Unif
SCUSSIO
the OFDM
rforming M
stem mode
y generated
ch subcarrie
CP. In order
umed in th
0 OFDM sy
ansform is
75
(a)
cases of propo
form (c) Trape
ON
M system w
Monte Carlo
el presented
d data bits ar
r. We have
r to evaluate
he simulatio
ymbols is ta
evaluated o
osed compand
ezium
with and w
o simulation
d in section
re mapped o
taken N=25
e the BER p
on. To ensu
aken into ac
over AWGN
h1
ding transform
without usin
ns in MAT
n 4.2, with
onto equi-pr
56 subcarrie
performance
ure the vali
ccount. The
N and 3-ta
1
(c)
m (a) Trapezoid
ng the com
TLAB envir
QPSK mo
robable unit
ers and 8 ad
e, complex
idity of sim
e BER perfo
ap Rayleigh
l
dal (b)
mpanding
ronment.
dulation
t-energy
dditional
additive
mulation
ormance
h fading
h2
76
All non-linear companding schemes [28]-[32] result in spectral spreading but its effect can be
eliminated by iterative oversampling and filtering operations as suggested by Armstrong et al.
for clipping in [25] and also proposed by Jiang for companding schemes in [28]. Therefore,
its effect is not taken into consideration.
In [32], the PAPR and BER performance of TC has been evaluated for
)633.1and1.0,4.0( === lba , )164.2and7.0,2.0( === lba , )732.1and0,0( === lba ,
,9.0( =a )488.1and1.0 == lb and )449.2and1,0( === lba , here we refer to them as ‘TC-
1’, ‘TC-2’, ‘EC’, ‘TC-3’ and ‘TC-4’ respectively. In [32], it has been shown that TC-3
provides the best PAPR reduction capability among all the cases under consideration, but its
BER performance is very poor, on the other extreme TC-4 provides very less PAPR
reduction. Therefore, we ignore these two cases (TC-3 and TC-4) and the remaining three
cases i.e. (TC-1, TC-2 and EC), which offer reasonable PAPR and BER results are
considered in our simulations for comparison with the proposed scheme.
To show the outperformance of the proposed scheme (QCT), we have used the same values
of ba and as used in TC-1 and TC-2 but increased the value of l by 0.01 to keep PAPR
performance almost same and to see its effect in BER performance. Therefore, we have
evaluated the performance of QCT for two sets of companding function parameters i.e.
)8275.0and8596.0,174.2,7.0,2.0( 21 ===== hhlba and ==== 1,643.1,1.0,4.0( hlba )7874.0and8276.0 2 =h . Here, we call them as ‘QCT-1’ and ‘QCT-2’. The reason for
selection of these two sets of parameters is given in Appendix A.
To show the outperformance of the proposed scheme (QCT), the PAPR and BER
performances are evaluated for two sets of companding function parameters i.e.
)8275.0and8596.0,174.2,7.0,2.0( 21 ===== hhlba and ==== 1,643.1,1.0,4.0( hlba )7874.0and8276.0 2 =h . Here, we call them as ‘QCT-1’ and ‘QCT-2’. The reason for
selection of these two sets of parameters is given in Appendix A.
The values of 1h and 2h of quadrilateral companding function with parameters
)174.2and7.0,2.0( === lba and )643.1and1.0,4.0( === lba can be calculated using
(4.5) and (4.7) and these are given in Table 4.2.
77
Table 4.2: Parameters of Quadrilateral Companding Function
Proposed
Scheme
Parameters Controlling Nonlinearity
of )(xh 1h 2h
a band l
QCT-1 7.0,2.0 174.2 8596.0 8275.0
QCT-2 1.0,4.0 643.1 8276.0 7874.0
The QCT functions for the parameters depicted in Table 4.2 are plotted in Fig. 4.9. It is found
that the nonlinearity of companding function for parameters )643.1and1.0,4.0( === lba
is more and the OFDM signals with a magnitude greater than 1.2 are more compressed in
comparison to QCT-1. Therefore, the peak value of the companded OFDM signal will be less
in case of QCT-2 and hence QCT-2 has more PAPR reduction capability in comparison to
QCT-1.
Figure 4.9: Plots of proposed companding functions QCT-1 and QCT-2 vs. input signal magnitude
0 0.5 1 1.5 2 2.5 3 3.50
0.5
1
1.5
2
2.5
Input |x|
Out
put |
h(x)
|
QCT-2QCT-1
78
The histograms of the magnitude of original OFDM signal and the signal magnitude after
applying QCT-2 and QCT-1 are shown in Figs. 4.10 (a), (b) and (c) respectively. Histograms
shown in Figs. 4.10 (b) and (c) resemble the shape of the PDF of QCT shown in Fig. 4.1.
Figure 4.10: Histogram of signal magnitude (a) Original OFDM signal (b) OFDM signal with QCT 2 (c)
OFDM signal with QCT 1
-1 0 1 2 3 4 50
2
4
6
8
10
12x 104 Histogram of Normal OFDM Signal
Num
ber o
f Sam
ples
Magnitude of OFDM Signal0 0.5 1 1.5 2
0
2
4
6
8
10
12x 10
4
Magnitude of Compnaded OFDM Signal
Num
ber o
f Sam
ples
Histogram of Companded OFDM Signal
0 0.5 1 1.5 2 2.50
2
4
6
8
10
12x 104
Magnitude of Companded OFDM Signal
Num
ber o
f Sam
ples
Histogram of Companded OFDM Signal
79
It can be seen from Fig. 4.10 (a) that the peak value of the OFDM signal before companding
is very high in comparison to the peak value of the signal after companding operation.
Initially, the peak value of the OFDM signal is more than 3.8, whereas its value for QCT-1
and QCT-2 are 2.174 and 1.643 respectively. In case of QCT-2, the peak value of the
companded OFDM signal is lesser in comparison to QCT-1 and hence its PAPR reduction
capability is superior than QCT-1.
CCDF of PAPR defined in section 4.4 and BER of the OFDM signal are used as metrics for
performance comparison of different methods. The BER performance of the proposed QCT is
evaluated over both AWGN and fading channels. The PAPR and BER performances of
TDBC [33] are also evaluated for companding function parameters ,897.1,0,0( === lba
)3518.0and7025.0 21 == hh . All these combinations of lba and, , which offers reasonable
PAPR and BER results have been considered in our simulations. The BER and PAPR
performances of TC-1, TC-2, EC, TDBC, QCT-1 and QCT-2 are shown in Table 4.3.
Table 4.3: BER and PAPR performance comparison of various companding schemes
Schemes
Parameters of companding function )dB(0/Required
NbE
PAPR
(dB) lba and, 2and1 hh BER
(10-4)
BER
(10-5)
Original
OFDM _ _ 8.40 9.55 10.75
QCT-1 0.2, 0.7 and 2.174 0.8596 and 0.8275 8.50 9.70 6.80
QCT-2 0.4, 0.1 and 1.643 0.82766 and 0.7874 10.40 11.70 4.40
TC-2 0.2, 0.7 and 2.164 0.8443 and 0.8443 8.90 10.05 6.75
TC-1 0.4, 0.1 and 1.633 0.8165 and 0.8165 11.50 13.00 4.30
EC 0, 0 and 1.732 0.5773 and 0.5773 12.95 14.20 5.05
TDBC 0, 0 and 1.897 0.7025 and 0.7025 11.20 12.70 5.50
Figs. 4
PAPR a
compan
compan
respecti
easily s
techniqu
(withou
From th
least BE
achieve
requires
here E
.11 and 4.1
and BER pe
nding (TC)
nding (TD
ively for va
seen from F
ues in term
ut compandi
Figure 4.1
he results pr
ER degrada
es a PAPR
s 0.10dB an
ob NE / repr
12 show th
erformance
[32], expo
BC) [33]
arious comb
Fig. 4.12 th
s of BER p
ing).
11: PAPR per
resented in
ation in com
reduction
nd 0.15dB m
resents per
e complem
respectively
onential co
and prop
binations of
hat QCT-1 o
erformance
rformance com
Table 4.3 a
mparison to a
capability o
more b NE /
bit signal
80
mentary cum
y of OFDM
ompanding
posed Qua
f design para
outperforms
e, which lies
mparison of or
and Fig. 4.1
all compand
of 3.85dB
oN to achiev
to noise ra
mulative dis
M signal wit
(EC) [28] ,
drilateral c
ameters pre
s in compa
s very close
riginal and com
12, it is obse
ding scheme
over origin
ve a BER o
atio. Hence
stribution fu
thout compa
, trapezium
companding
esented in T
rison to all
e to the orig
mpanded OFD
erved that Q
es under con
nal OFDM
of 410− and
e, QCT-1 a
unction (CC
anding, trap
m distributio
g transform
Table 4.3. I
l other com
ginal OFDM
DM signals
QCT-1 prov
nsideration
signal and
d 510− respe
achieves sig
CDF) of
pezoidal
on based
m(QCT)
It can be
mpanding
M system
vides the
. QCT-1
d merely
ectively,
gnificant
81
PAPR reduction over original OFDM signal with insignificant degradation in BER
performance.
Figure 4.12: BER performance comparison of various companding schemes
As seen from Fig. 4.11 that PAPR reduction capabilities of QCT-1 and TC-2 are almost
coinciding but QCT-1 has better BER performance in comparison to TC-2 because if we
compare the companding function of QCT-1 with TC-2 (as shown in Fig. 4.13) then we find
that the peak value of the companded OFDM signals are only differ by 0.01, which cause an
almost negligible change in PAPR performance. But it can be clearly observed from Fig.
4.133 that QCT-1 results in lesser amplitude compression of the OFDM signal as compared
to TC-2, thereby introducing less non-linear distortion in comparison to TC-2. Hence, QCT-1
results in better BER performance in comparison to TC-2.
4 5 6 7 8 9 10 11 12 13 14
10-5
10-4
10-3
10-2
10-1
Eb/No(dB)
BE
R
Performance BoundQCT-1TC-2QCT-2TDBCTC-1EC
82
Figure 4.13: Companding function plots for QCT-1 and TC-2
The PAPR reduction capability of the proposed QCT can be further improved if we choose
8276.0,643.1,1.0,4.0( 1 ==== hlba and )7874.02 =h as parameters of the proposed
companding function. This case is referred to as “QCT-2”.
The QCT-2 provides PAPR reduction capability of 6.2dB over original OFDM signal, the
price paid for this PAPR reduction capability is the small BER performance degradation. The
QCT-2 requires ob NE / = 10.4dB to achieve a BER of 410− , which is about 2dB more in
comparison to original OFDM.
It is worth mentioning that QCT-2 outperforms in terms of both PAPR reduction and BER
performance over the best case of trapezoidal companding (TC-1), EC and TDBC.
The PAPR reduction capability of the proposed scheme (QCT-2) is 0.65 dB and 1.1 dB better
in comparison to EC and TDBC respectively, for a CCDF =10-3, and at the same time its
BER performance is also superior in comparison to EC and TDBC. For QCT-2, the required
ob NE / to achieve a BER of 410− and 510− respectively is about 1dB lesser than EC and
TDBC.
1.5 2 2.5 3 3.5 4 4.51.5
1.6
1.7
1.8
1.9
2
2.1
2.2
input |x|
Out
put |
h(x)
|
QCT-1TC-2
83
The PAPR reduction capabilities of proposed scheme (QCT-2) and trapezoidal companding
(TC-1) are almost same, but required ob NE / to achieve a BER of 410− and 510−
respectively are 1.1dB and 1.3 dB less than TC-1. QCT-2 has almost same PAPR reduction
capability as QCT-1 but provides better BER performance in comparison to TC-1 because it
can be seen from the graphs of companding functions (as shown in Fig. 4.14) for QCT-2 and
TC-1 that the difference between the peak values of two companded signals is same(0.01) as
kept between QCT-1 and TC-2. But the difference between their non-linearity has now
increased. Therefore, QCT-2 provides higher BER gain over TC-1 in comparison to the gain
achieved by QCT-1 over TC-2.
Figure 4.14: Companding function plots for QCT-2 and TC-1
The maximum value of the transformation gain Gmax given by (4.22) for QCT-1 and QCT-2
are 4.85dB and 7.28dB. As seen from Fig. 4.11 the values of the transformation gain of 0.1%
of OFDM signal are 3.95dB and 6.35dB respectively.
Fig. 4.15 depicts the BER performance of original OFDM signal, proposed scheme i.e. QCT-
1 and QCT-2 over fading channel. In this simulation, we have considered Stanford
University Interim-5 (SUI-5) channel as a 3 tap multipath fading channel with average path
1.5 2 2.5 3 3.5 4 4.5
1.4
1.45
1.5
1.55
1.6
1.65
input |x|
Out
put |
h(x)
|
TC-1QCT-2
84
gains [0dB,-5dB, -10dB] , with path delays [0 µs 4 µs 10 µs]. Besides N=256 subcarriers, 8
subcarriers are utilized for CP to mitigate the effect of ISI.
Figure 4.15: BER performance comparison of QCT over fading channel
In Fig. 4.15, the BER performance of original OFDM signal without companding scheme,
QCT-1 and QCT-2 are denoted by ‘OFDM (without companding)’, ‘QCT-1’ and ‘QCT-2’.
OFDM signal without companding require a ob NE / =13.5dB to achieve a BER of 10-4. As
seen from Fig. 4.15, the BER performance of QCT-1 over multipath fading channel is very
close to the original OFDM signal without companding , whereas the QCT-2 requires merely
1.9 dB more ob NE / =13.5dB to achieve a BER of 10-4 in comparison to QCT-1.
4.7 CONCLUSION
In this chapter, we have proposed a novel non-linear companding transform (QCT) that can
effectively reduce PAPR and provide good BER performance. The proposed scheme is
applicable for all modulations schemes and can work for any number of subcarriers. The
existing companding schemes, like exponential companding, trapezoidal companding and
4 6 8 10 12 14 16
10-4
10-3
10-2
10-1
Eb/No(dB)
BE
R
QCT-1OFDM(without companding)QCT-2
85
trapezium distribution based companding techniques are special cases of the proposed
method. The QCT provides extra degrees of freedom to design the companding function and
hence by choosing the suitable values of design parameters of the proposed companding
function, a good trade-off between the PAPR reduction and the BER can be achieved. The
proposed QCT provides better PAPR reduction and BER performance in comparison to TC,
EC and TDBC. QCT can achieve a minimum PAPR of 0dB, whereas TC and EC can achieve
a minimum PAPR of 3dB and 4.771dB respectively. QCT-2 has superior PAPR performance
in comparison to QCT-1 but its BER performance is inferior in comparison to QCT-1.
