Analysis of Power Spectral Density of
Digitally-Modulated Combined Pulse Trains
Wachira Chongburee
Kasetsart Advanced RF and Electromagnetic Laboratory (KAREN)
Department of Electrical Engineering, Kasetsart University, Thailand
Email: fengwrc@ku.ac.th
ABSTRACT
OFDM, a mutlicarrier modulation technique, is
widely adopted because of its many advantages. Most of
the time, in-band spectrum of the multicarrier signal is
assumed to be flat. A recent literature analyzes the PSD
and concludes that the assumption of flat spectrum is not
completely correct. This paper analyzes the PSD with
different approach. The modulated carriers are cropped
in an interval of symbol length and viewed as a
modulated pulse. An expression for the PSD with the
necessary and sufficient conditions is revealed. The
validity of the flat PSD assumption of multicarrier signals
is also suggested.
Keywords: Orthogonal Frequency Division
Multiplexing, Orthogonal Pulse, Power Spectral Density,
Spectrum Analysis
2. PSD OF COMBINED WAVEFORMS
A set of functions, {f0(t), f1(t), f2(t), … } is said to be
orthogonal over a symbol interval [-Ts/2, Ts/2] if
Ts / 2
f
Ts / 2
p
 0,
(t ) f q (t )dt 
 pp ,
Orthogonal
frequency
division
multiplexing
(OFDM), a form of multicarrier modulation techniques, is
one of the most widely-adopted modulation techniques
because of its many advantages, for example, the ease of
implementation using discrete Fourier transform and the
immunity to multipath fading gained from the extended
symbol length [1].
Figuratively speaking, OFDM
partially overlays the spectra of a number of narrow band
modulated signals and makes use of the orthogonality of
the carriers to extract the data from the combined
waveform. In some literatures, e.g., [2], OFDM power
spectral density (PSD) is treated as a flat spectrum. An
intensive analysis in a very recent work shows that the
assumption of flat PSD is not completely true, especially
for the case of the OFDM adopted by the 802.11a
standard [3].
In this paper, a different approach is used to analyze
the PSD of OFDM. The orthogonal carriers are cropped
at a size of the symbol length and they are viewed as an
orthogonal symbol. The results are identical to the result
found in [3] but in this work, the remark is that
orthogonality is a not required condition. The result is
then taken to analyze the PSD of OFDM signals. A
conclusion on the validity of the flat PSD is drawn,
finally.
(1)
where pp is the energy of the function fp(t).
Let s(t) be a transmit waveform of which digital
information is kept on the coefficients of the orthogonal
basis pulse fp(t). Such a modulation technique is named a
P-dimensional orthogonal scheme.
The transmit
waveform s(t) can be written as
s(t ) 

P 1
 a
n   p  0
1. INTRODUCTION
pq
pq
n, p
f p (t  nTs )
(2)
where an,p denotes the digital data of the nth symbol
carried on the pth pulse (dimension). It is shown in [4]
that Ps(f) the power spectral density of one-dimensional
s(t), which only f0(t) is available, can be written as
Ps ( f ) 
1
 

| F0 ( f ) |2   R0 (k )e jk 2fTs 
Ts
k 

(3)
where F0(f) is the spectrum of the basis pulse f0(t) and
R0 (k )  an ank . More specifically, the signal is said to be
antipodal if the binary data is used to sign the basis pulse
f0(t), i.e., an,p  {-1, 1}. Assuming uncorrelated binary
data, R0(k) = 1 for only k = 0 and zero elsewhere. The
summation term in (3) is removed. As a result, the PSD
of the antipodal signal is just the magnitude squared of
the basis pulse Fourier transform divided by the bit
period.
Now, let us generalize the results for the PSD of s(t).
PSD can be determined either directly by its definition or
indirectly using Wiener-Khintchine theorem. The choice
of the following analysis is based on the direct method.
By definition, the power spectral density (PSD) of a
random process s(t) is given by
 | S ( f ) |2 

Ps ( f )  lim  T

T  
T


(4)
where ST(f) is Fourier transform of the truncated version
of infinite waveform s(t).
Let us observe the original signal s(t) for 2N+1 symbol
periods, from –Nth to Nth symbol. The truncated signal
sT(t) can be written as
sT (t ) 
N
P 1
 a
n   N p 0
n, p
f p (t  nTs )
(5)
Fp ( f )e  jnTs
(6)
Its Fourier transform is then
ST ( f ) 
N
P 1
 a
n   N p 0
n, p
where Fp(f) is Fourier transform of the basis pulse fp(t).
Thus,
| S T ( f ) | 2  S T ( f ) S T* ( f )
(7)



*
    a n, p Fp ( f )e  jnTs    a m,q Fq ( f )e  jmTs 
 n   N p 0
 m N q 0

N

N
P 1
N
N
P 1 P 1
   a
Fp ( f )e jnTs am, q H q ( f )e jmTs
*
n, p
n  N m N p 0 q 0
P 1
(8)
N
N
1
G (k , f )


N  T (2 N  1)
n N m N
s
Ps ( f )  lim
(14)
For a given N, the outer and the inner summations
together produce a total of (2N+1)2 adding terms. The
values of k range from -2N to 2N. For a given k, (14)
evaluates G(k,f) in total of (2N+1) - |k| times. By letting
m = k-n and summing over the outer summation results in
Ps ( f ) 
1
(2 N  1) N  n
lim
 G(k , f )
Ts N  (2 N  1) k   N  n
(15)
Note that as pointed out on p.400 in [4] for the case of
one dimensional signal, replacing the outer summation
with 2N+1 is not strictly correct since the inner
summation is also a function of n. However, as N
approaches , the below result is valid. As a result, the
final form of the PSD is
Ps ( f ) 
1
Ts

 G (k , f )
(16)
k  
It is possible to further simplify G(-k,f) + G(k,f) by using
the cross correlation property Rpq(k) = Rqp(-k). However,
it is beyond the interest of this investigation.
If the data carried on one dimension (pulse) are

1  N N P 1 P 1
*
 jnTs
 jmTs uncorrelated to the ones carried on other pulses, Rpq(k),

Ps ( f )  lim     an , p Fp ( f )e
am , q Fq ( f )e
 where p  q, is equal to zero for all k. Consequently, the
T  T
 n N m N p 0 p 0

cross-spectral component Gcross(k,f) is zero. The PSD is
(9) then simplified to
In (8) n and m represent the time index while p and q
identify the basis pulse numbers. Thus, the PSD for sT(t)
becomes
where T = Ts(2N+1). Since only an,p and am,q are random,
the ensemble averaging operator affects only on them.
Equation (9) reduces to
1
Ps ( f )  lim

N  T ( 2 N  1)
s
   a
N
N
P 1 P 1
n   N m   N p 0 q 0
n, p
(10)

am ,q e  j ( n m )Ts Fp ( f ) Fq ( f )
*

Let us define a cross spectral function G(k, f) by
P 1 P 1
G(k , f )   Rpq (k )e j ( k )Ts Fp ( f ) Fq ( f )
*
(11)
p 0 q 0
G(k,f) can be decomposed into co-spectral component
Gco(k,f) and cross-spectral component Gcross(k,f) where
P 1
Gco (k , f )   R pp (k )e  j ( k )Ts | Fp ( f ) |2
Moreover, assuming the digital data on the same pulse,
an,p and am,p, are independent, Rpp(k) = 0 for k  0.
Therefore the power spectral density of such a signal is
just the sum of the magnitudes of the Fourier transform
squared of the basis pulses.
Gcross (k , f )   R pq (k )e
p 0 q 0
q p
 j ( k )Ts
Ps ( f )   Rpp (0) | Fp ( f ) |2
(18)
p 0
This result is a generalized version of the PSD
formula for multi-dimensional antipodal waveforms
developed in [5]. An essential remark is that the
derivation does not require the orthogonality of the basis
pulses at all. Therefore orthogonality is not a necessary
condition. The sufficient condition for the PSD is that the
data must be uncorrelated. Note that a sequence with
channel coding does not satisfy this condition. Another
note is that a pulse train modulated by random data
appears orthogonal to another modulated pulse train when
observed for a sufficiently long time.
(12)
p 0
P 1 P 1
(17)
p  0 k  
P 1
The term an, p am, q is in fact a cross correlation function of
the nth and mth data bits carried on the pth and qth pulses,
respectively. It is referred by Rpq(m, n). Assuming the
random processes associated with pth and qth dimensions
are jointly wide sense stationary, Rpq(m, n) can be written
as Rpq(m-n) = Rpq(k) where k = m-n.
The PSD becomes
P 1
Ps ( f )    Rpp (k )e j ( k )Ts | Fp ( f ) |2
Fp ( f ) Fq ( f ) (13)
*
3. COMPARISON OF SIMULATION AND
THEORETICAL RESULTS
Simulation of PSD is implemented based on a
commonly known rectangular pulse, (t/T) = 1 for |t| <
T/2 and zero elsewhere. The Fourier transform pair of the
pulse is expressed as
sin fT 
t
    Ts
T
fT
 
Let p1 (t ) 
T
T1


t
T1
and p2 (t ) 
T
T2

(19)

t
T2
be the basis
pulses. Both pulses are unity-energy and obviously, not
orthogonal to each other. A truncated transmit signal can
be written as
sT (t ) 
N
a
n N
n ,1
p1 (t  nTs )  an , 2 p2 (t  nTs )
(20)
s (t ) 

Nc / 2
 a
c ,k
n   k   Nc / 2
 t  nTs 

cos[ 2 ( f c kf )(t  nTs )] 
 Ts  (22)
 t  nTs 

 as ,k sin[ 2 ( f c  kf )(t  nTs )] 
 Ts 
where fc is frequency of the middle carrier and (t/Ts) is a
rectangular window as defined earlier.
where the coefficients an,I  {-1,1} contains the transmit
independent digital data and Ts is the symbol period. Let
T1 and T2 are equal to 0.2Ts and Ts, respectively.
Q
3
1
I
10
-3
Simulation
Theoretical
-1
1
3
-1
0
-3
dB-PSD
-10
-20
Fig.2: The 16-QAM signal constellation used in the
analysis.
-30
-40
-50
-20
-15
-10
-5
0
fTs
5
10
15
20
Fig.1: Theoretical and simulation PSD of the
signal consisting of two non-orthogonal
rectangular basis pulses vs. normalized
frequency.
According to the derivation, the theoretical PSD of
sT(t) is described by
sin fTs 
Ps ( f )  Ts

fTs
2
Ts
sin f 0.2Ts 
0.2Ts
0.2Ts
f 0.2Ts
2
(21)
In the simulation, 200 symbols are randomly generated
at a time. The 200-symbol waveform is then Fourier
transformed to determine the magnitude squared. The
simulation repeats for 20 times and the final simulated
PSD is the average of the results from the 20 trials. The
consistency of the simulation and theoretical results is
shown in Fig. 1. The comparison confirms that the
orthogonality of the basis pulses is not a necessary
condition for the evaluation of the PSD.
The
independency of the random data is a sufficient condition.
4. FLATNESS OF PSD OF A SIMPLE 16-QAM
OFDM
The flatness of PSD of an OFDM signal is investigated
in this section. Let s(t) be a simple OFDM signal
adopting a 16-QAM modulation technique. The QAM
signal constellation is shown in Fig. 2. For a Ncdimensional or Nc-multicarrier signal, s(t) can be
expressed as
Let us treat the windowed carriers as a basis pulse.
The orthogonality of the basis pulses is guaranteed over
the pulse duration of [-Ts/2, Ts/2] if the carrier frequency
spacing f is equal to Rs = 1/Ts. Note that the variable k
is reused in this section as the carrier index. In this
particular signal constellation, the values of the digital
data ac,k and as,k  {-3, -1, 1, 3}, independently and
equiprobably,. The results from the previous derivation
suggest that the multicarrier 16-QAM PSD can be written
as
Ps ( f ) 
Nc
R
k   Nc / 2
ac , k
(0) | Fc , k ( f ) |2  Ras, k (0) | Fs , k ( f ) |2
where
Rac, k (0)  Ras, k (0)  an  an 

(23)

1
(3) 2  (1) 2  12  32  5
4
and F ( f ) 2  F ( f ) 2  T sin  ( f  kf )Ts 
c ,k
s ,k
s
 ( f  kf )Ts
2
The final form of the PSD becomes
Ps ( f )  10
Nc / 2

k   Nc / 2
Ts
sin  ( f  kf )Ts 
 ( f  kf )Ts
2
(24)
Analytical PSD of the OFDM using 5 and 41 carriers
with 16-QAM modulation scheme are shown in Fig. 3. It
is found that, for the case of 41 carriers, the PSD in the
middle of the passband approaches a flat spectrum as the
number of carriers increases. The ripples in the middle of
the overall PSD are symmetrically filled by the side lobes
of the PSD associated with carriers on both sides. In
contrast, the flatness of spectrum close to the cut off
frequencies is not improved by a large number of carriers
because of asymmetric fill up. Strictly speaking, the PSD
is not flat. Nevertheless, the ripple on the PSD is well
kept within -0.5 dB, even with a small number of carriers,
says 5, as illustrated in Fig 4. For the rectangular
windowing, the PSD is reasonably considered flat.
Therefore the assumption of flat spectrum is practically
valid.
20
15
Nc = 41
10
5
Nc = 5
5. CONCLUSION AND FUTURE WORKS
The analysis shows that the PSD of such a digitally
modulated signal can be evaluated from the sum of the
Fourier magnitude squared. Independent random data is
a sufficient condition whereas the basis pulse
orthogonality is not necessary. It is shown that PSD of an
OFDM with rectangular time window is not perfectly flat
but reasonably considered flat. The future works will
focus on the analysis of the PSD of the 802.11 a/g
standard, in which a raised-cosine based time window is
used.
dB-PSD
0
6. REFERENCES
-5
-10
-15
-20
-25
-30
-30
-20
-10
0
fTs relative to fc
10
20
30
Fig. 3: PSD of 5-carrier and 41-carrier
OFDM with a 16-QAM modulation scheme.
11
Nc = 41
10.5
Nc = 41
dB-PSD
10
9.5
9
Nc = 5
8.5
8
-4
-3
-2
-1
0
1
fTs relative to fc
2
3
4
Fig. 4: PSD in the middle of the band of the 5carrier and 41-carrier OFDM with a 16-QAM
modulation scheme.
Nc = 5
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mobile channel using orthogonal frequency division
multiplexing,” IEEE Trans. Commun., vol COM-33,
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techniques for digital terrestrial TV broadcasting,”
IEEE Commun. Mag., vol. 33, pp. 100-109, Feb
1995.
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signals for WLAN”, IEE Electronics Letters, Vol.
40, No. 22, October 2004.
[4] L. W. Couch, Digital and Analog Modulation
Systems, Prentice Hall 5th ed., New Jersey, 1997.
[5] W. Chongburee, “Digital transmission by Hermite Ndimensional antipodal scheme,” Ph.D. dissertation,
Dept. Elect. Eng., Virginia Tech, VA 2004.