Comp080726.qxd


The Journal of Engineering Research  Vol. 6, No. 2 (2009)  47-51

1.  Introduction

The Multi-Carrier MC technology has received much
attention in modern communication systems. In MC sys-
tems a number of data symbols are transmitted at separat-
ed sub-carriers in parallel thus increasing the symbol
length, which reduces the sensitivity to interference.
Multi-Carrier Modulation MCM divides the available fre-
quency band into a large number of orthogonal tones
which can be implemented in all digital realization by
exploiting Discrete Fourier Transform DFT methods as in
DMT and OFDM multi-carrier modulation Hnzo et al.
2004; Reimers 2001 and Al-Dhahir and Minn, 2006).
Discrete Multi-tone DMT a currently standard for
Asynchronous Digital Subscriber Line ADSL units man-
ages bandwidth for data transmission. It is one of the tech-
nologies that provides high speed Internet access in resi-
dence and offices economically via wire technology
(Gagnaire, 2003 and Daly 2003). Orthogonal Frequency
Division Multiplexing OFDM gives high resistance to fre-
_____________________________________________
Corresponding author’s e-mail: sameh-eng@yahoo.com

quency selective fading. It is often used in mobile radio
systems such as digital video broadcasting DVB/ digital
audio broadcasting DAB  (Reimers, 2001 and Al-Dhahir
and Minn, 2006).   It is proposed for 4th generation wire-
less communication systems (Bria, 2001). However, due
to the large number of sub carriers, the MC signal owns a
large Peak to Average Power Ratio PAPR. Thus, the trans-
mitted signal is sensitive to non-linear distortions that will
degrade the error performance and introduce high adja-
cent channel interference with spectrum regrowth. Several
techniques for reducing PAPR of multi-carriers signal
have appeared in the literature (Mustafa, 2007 and Friese,
1996).  However, most of these techniques introduce addi-
tional complexity. Moreover, there have been intense
research efforts aimed at designing different multi-carrier
transceiver scheme with different spectral containment.
Among those designs are the Discrete Cosine Transform
DCT-MCM (Al-Dhahir and Minn, 2006 and Ak-Dhahir
and Minn, 2005) and Discrete Wavelet Multi-Tone
DWMT (Daly, 2003 and Farhang-Boroujeny and Chin,
2000).

In this paper DCT-based multi-carrier system was con-

Simulation of a Multi-Carrier System in a Non-Linear Flat
Fading Channel

Samah A.B. Mustafa

Department of Electrical Engineering, University of Salahaddin, Erbil, Iraq

Received  26 July 2008; accepted 15 February 2009

Abstract: This paper is on multi-carrier wireless communication system, focusing on the physical layer in such
modulation system. Multi-Carrier MC system has received much attention in modern communication technol-
ogy and is finding attractive applications. The paper introduces and discusses the discrete cosine transform
DCT-based MC. Similarities and differences with respect to the Fourier transform-based system are pointed out.
It was found that a reduction up to 5dB in peak power can be gained over the conventional system. Also, the
simulation over flat fading channel depicted that QAM signaling in MC scheme based on DCT performed bet-
ter than that in FFT-based scheme, although it depends strongly on the channel parameters.

Keywords: Multi-carrier  transmission, MCM,  Discrete  multi-tone  DMT,  Orthogonal  frequency   division
multiplexing  OFDM,  Peak-to  average  power ratio  PAPR  in  MC  system, Flat fading channel
Flat fading channel

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48

The Journal of Engineering Research  Vol. 6, No. 2 (2009)  47-51

sidered.  Many simulations have been carried to outline
the differences over linear AWGN, non-linear AWGN and
a flat faded channel with respect to the DFT-based system.
Section II reviews in brief the properties and representa-
tions of the DCT, and section III depicts DCT-based
scheme. Simulation results and some further discussions
of the error performance over different channel parame-
ters are presented in section IV. Finally, section V con-
cludes the results.

2.  Discrete Cosine Transform

The DCT has been considered as one of the best tools
in digital signal processing and therefore, it has many
applications, e.g., in the area of multimedia and telecom-
munications. In this section, an introduction to the DCT is
given in order to provide background and motivation for
our work.

The orthogonal DCT is classified into many different
types with slightly different even/odd boundary condi-
tions at the two ends of the matrix. The first definitions of
the forward and inverse DCT were given by:

(1a)

(1b)

The DCT operators map an N-size real sequence into
another N-size real sequence. It is a linear Fourier-related
transformation similar to the DFT using only real numbers
of the DFT. By repeating the samples in a time reversed
order and performing a DFT on the length sample set a
DCT is obtained. Also, if we represent the N-size
sequence as X=[x1,…., xN] and Y=[y1,…., yN] and denote
the NXN DCT matrix by:

(2)

Thus  Eq. 1 can be rewritten as Y=XC and X=YC-1.
DCT is computationally simpler and the DCT matrices are
orthogonal, ie. CCT=I, the inverse transform matrix are
obtained with a matrix transpose (where T denotes trans-
position and I is the identity matrix) (Strang, 1999;
Oppenheim, et al. 1999 and Rao and Yip, 2003). 

3.  MC System Model

Figure 1 illustrates the MCM system used for multi-
carrier modulation. In DCT-based scheme, the IFFT &
FFT blocks are simply replaced by an IDCT & DCT or
vice versa, respectively, since IDCT matrix is the trans-
pose of DCT matrix.

In MC-based transceiver, the binary data is encoded
into a set of M-PSK or M-QAM symbols, called sub-sym-
bols, and converted into lower rate sequences via serial to
parallel conversion, which are then multiplexed by an
IDCT.  The outputs are serialized and transmitted. At the
receiver, inverse operations take place in reverse order. In
Fourier-based system, the Hermition symmetry is
enforced to ensure a real waveform at the output of the
IFFT.  This can be achieved by conjugate mirroring the
complex symbols and transmitting zero signals on the DC
and Nyquist tones of IFFT block sequence. However, as
complex signals are mapped onto orthogonal sub-chan-
nels without imposing the conjugate symmetry condition,
the multi-carrier system transmits twice the data in twice
the bandwidth required when the condition is imposed.
Also, it is not necessary in our proposed system, as even a
real signal can be mapped onto orthogonal sub-channels. 

DCT converts real signals to real signals (as depicted
earlier), and hence binary signaling or parsed in-phase and
quadrature components of complex data word must be
used in each sub-channel individually to keep the same
data rate in DCT systems.

4.  Simulation Results

Much of this paper compares the error performance of
DCT-based MC system, which transmits a real waveform
with that of the Fourier-based; therefore the conjugate
symmetry condition is imposed on FFT-based system
under the same transformation size N=256 to provide
fairest comparison possible. In the simulation we also
considered a system operating without guard space in the
resulted time domain symbols.

The scheme was simulated in Matlab ver. 7 with a con-
stant bit allocation of log2(M) bits per sub-channel is
assumed. The number of simulated bits was about 106 bits
and the results was computed and averaged over 3 itera-
tions for the demodulator.

4.1  Linear AWGN Channel
The error performance curves presented in Fig. 2 are

based on the signal-to-noise ratio SNR of the channel,
assuming the channel is a pure AWGN channel. As seen in
the figure, a differential phase shift keying (DPSK) in
both MC system will limit the number of bits per symbol
and results in an 11dB loss in SNR. By comparison the
error curves, a gain in SNR can be observed at the low

1

2 1 1
2

Nk
k n

n

w ( n ) ( k )
y x c o s

NN
π

∑
=

− −
= ⋅      

                                                 1k , . . . . . , N=

1

1 2 1 1
2

N
n k k

k

( n )( k )
x w y c o s

NN
π

∑
=

− −
= ⋅ ⋅  

                                                1n , . . . . , N=
1 1

2 2k
kwhere w

k N
=⎧= ⎨ ≤ ≤⎩

 

2 1 1
2

kw ( n )( k )C cos
NN

π − −
= ⋅

Mapper

Mapper

IDCT

P/S

I

Q

I

Q

Figure 1.  Block   diagram  of   DCT-based   MC 
modulator



49

The Journal of Engineering Research  Vol. 6, No. 2 (2009)  47-51

error rates using DCT-based scheme with QAM modulat-
ed tones. The resulted real and imaginary parts from the
constellation are used separately to modulate different
tones to guarantee real time domain samples without
transmitting zero signals on the DC and Nyquist tones as
have been applied in IFFT block sequence. Thus the trans-
mission data rate is improved only by 2*log2M than that
of FFT-based system.

However, a DCT scheme can be used with all the sub-
carriers allocated to transmit only the estimated phase of
the different constellation points (QPSK, DPSK), where
the amplitude is constant, thereby improving the data rate
to the double. 

4.2  Non-Linear AWGN Channel
We also analyzed the distribution of time domain trans-
formed amplitude as depicted in Fig. 3 & Fig. 4. The prob-
ability density function (PDF) depends on the probability
of occurrence of each discrete sample level. 

The distribution of MC signal with 256 sub-carriers and
16-QAM is founded where the amplitude Sn has Rayleigh
distribution with PDF given by:

(3)

where Sn the transformed (using IFFT or IDCT) time
domain samples have real parts only as stated before.

For Rayleigh distribution, as seen the signal levels
around the mean value have higher probability than other
levels, while the occurrence of the large signals has the
smallest probability and moreover different level in DCT
and FFT-based schemes, as shown in Fig. 3 & Fig. 4. It is
reasonable cast a way to change the statistic of the ampli-
tude for the benefit of PAPR reduction. Notice that PAPR
is a random variable for each transmitted block. It has
been found from a large number of independent runs for
the DCT and FFT-based MC scheme that within each
block, the peak power is reduced using DCT-based system
of about 5dB and a reduction of PAPR of about 1.3dB are
achieved.

For the simulated error in Fig. 5 a non-linear power
amplifier is applied with AM/AM response of clipping
scheme at the saturation point at three different Output
Back Off OBO; 4, 5 & 6dB with no compensation of the
non-linearity in the conventional MC system. OBO is
defined as the ratio of the maximum possible amplifier
output power to the average output power. This indicates
the power efficiency of the amplifier (Al-Dhahir and
Minn, 2006). OBO=6dB guarantees linear transmission
while OBO=4 & 5dB in FFT-MCM scheme generates an
almost flattened error curves indicating saturation of the
power amplifier. These results can be directly compared
with the results for MC scheme using DCT.  Error curves
with OBO=4, 5 & 6dB are also measured. It is found that
when OBO=5dB the simulated approaches the linear

Figure 2.  Pe in DCT-based and FFT-based MC sche-
me versus SNR of linear AWGN channel

Figure 3.  Distribution of  DCT-based  scheme  signal
amplitude

Figure 4.  Distribution  of  FFT-based  scheme  signal
amplitude

2 22
2

0

0

S /( )n n s
n

sn

S
e S

P( s )

otherwise

σ

σ
−⎧ ≥⎪⎪= ⎨

⎪
⎪⎩



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The Journal of Engineering Research  Vol. 6, No. 2 (2009)  47-51

transmission due to the overall net improvement in the
PAPR using DCT, while when OBO decreases to 4dB the
performance diverges from the linear case also, indicating
non-linear distortion occurs.

4.3  Flat Fading Channel
Flat fading has an impulse response given by:

(4)

where g (t, τ) is the impulse response at observation time
t to an impulse applied at time t-τ. This channel is consid-
ered  to be slowly  time-varying  such that the amplitude
β(t) and the phase shift  θ(t) can be considered constant
during one symbol interval. 

To compare and contrast the performance of the
schemes over fading channel, the error curves are exam-
ined at similar parameters. Fig. 6 shows that, for QAM
multi-carrier signal the performance are incompatible
exactly for 16-QAM signal where the DCT-based system
approaches the linear transmission. The distortion arose
from fading channel can be compensated by increasing
SNR, while multi-amplitude signaling based on FFT with-
out channel estimation induces irreducible error curves.
As depicted the DPSK signals in both MC systems have
the same performance depending on the phase tracking of
the decoder. The simulated error rates as shown in Fig. 7
are depicted for various flat fading parameters, which
affects on the overall DCT-based MC system performance
and the error performance limit. It is clearly observed that
the amplitude and phase shift of the reflected signals have
a much stronger influence on the systems using QAM
than on DPSK modulation systems.

The results could be greatly improved to meet the spec-
ification requirement that could be implemented in the
future to further enhance the simulator, where the results
can be extended to any system which uses the MCM tech-
nique. 

5.  Conclusions

Extensive computer simulations have been carried out
to demonstrate and compare the performance of DCT
multi-carrier scheme with that of FFT scheme. The results
indicate quite similar performance over linear AWGN
channel. Concerning the overall system performance,
QAM is often more robust and more spectrally efficient
than DPSK modulation. Performance enhancement of the
proposed system has been recorded over non-linear
AWGN channel due to the reduction in PAPR power of
the DCT multi-carrier signal. It is more flexible to present
linear transmission with no compensation of the non-lin-
earity in the system.  Moreover, the simulated results con-
cludes that the new scheme over flat faded channel with-
out channel estimation performs better corresponding to

Figure 5.  Pe in FFT-based MC scheme versus SNR of
non-linear AWGN channel

j ( t )g( t , ) ( t ).e . ( t )θτ β δ=

Figure 6.  Pe in DCT-based and FFT-based MC sche-
me versus SNR over flat fading channel

Figure 7.  Pe  in  DCT-based MC scheme  versus SNR
with  16-QAM  (discrete line) and 64-QAM
(solid line)  modulated  sub-carriers    over
three different  flat  fading channel charac-
ters (ch.1, ch.2, and ch.3)



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The Journal of Engineering Research  Vol. 6, No. 2 (2009)  47-51

that of the traditional scheme, especially with QAM mod-
ulated tones depending  too much on the reflected path
amplitude and phase. Such modulated tones in FFT-based
scheme results in an irreducible error even at high SNR. 

However, the results could be greatly improved to meet
the specification requirement that could be implemented
in the future to further enhance the simulator, where the
results can be extended to any system which uses the
MCM technique. 

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