ICAO ACP/Working Group F
21st meeting Bangkok , 10-18 Dec 2009
Agenda Item 4 : Development of material for ITU-R meetings
WGF 21-IP12
Information paper
Derivation of an approximation for the ICAO-standardized MLS DPSK
unwanted emissions model
1. Introduction
This information paper (IP) provides an analytical derivation of a suitable approximation for the MLS
DPSK transmit spectrum model for large frequency offsets with respect to the considered MLS transmit
frequency and for reference bandwidth much greater than the DPSK modulation frequency.
A short description of the MLS system can be found in the ICAO contribution to ITU 4C/210 with
WGF20/TMP 6 as the source for this contribution.
This IP provides the justification for the MLS DPSK unwanted emissions model quoted in WGF20-WP 13
ATT. C, giving the attenuation or Roll-Off of the MLS DPSK transmit spectrum at frequency offset f from
the considered MLS frequency on-tune transmit level, i.e. Roll  Off ( f , Bw) 
-
1 f d  Bw
where
2 ² f 2 
f is the frequency offset from the considered MLS on-tune frequency,
Bw the reference bandwidth and
fd the MLS DPSK modulation rate
and referred to in the NSP/SSG Mar’ 09 Bretigny meeting ( WGF20-WP 19 refers) as contained in “Flimsy
3 rev 2” .
The purpose of this IP is to bring this justification to the attention of WGF21 as the contents of this
Flimsy 3 are not available any more, neither from the ACP nor NSP ICAO web sites.
2. MLS DPSK emissions power spectral density (PSD) theoretical model.
Below is an extract of the ICAO NSP/WGW March /WP11 paper:
‘’ 4. Theoretical Power Spectral Density (PSD)
The Power Spectral Density (PSD) of a DPSK-modulated signal can be expressed as follows:

f
P  sin   f d
PSD f  

f d    ffd

With:

2
Eq. 1

P = Signal Power in W =Watts
f = offset from carrier frequency in Hz
fd = 15625 Hz =15.625 kHz = data rate of DPSK-signal
The unit of the PSD is W/Hz = Watts per Hertz bandwidth
For the sake of simplicity the PSD can be normalized to the Power, P:

f
PSD f  1  sin   f d
S f  


P
f d    ffd

2


Eq. 2
The unit of this normalized PSD is 1/Hz.The envelope of the normalized PSD can be obtained by setting:

f 
sin      1
fd 

Hence:
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ACP/WGF 21-IP ---


2
f
1  sin   f d 
1  1 
S f  






f d    ffd 
f d    ffd 
2
Eq. 3
This can be further simplified as follows
1
Envelope f  
fd
 1

f
   f d
2

fd
 
  f 2

Eq. 4
The unit oft the envelope of the normalized PSD is 1/Hz.’’
4. Close approximations of the integration of the MLS DPSK PSD over the reference bandwidth Bw
4.1 First approximation by taking the 1/f² enveloppe
A first MLS DPSK unwanted emission upper-bounding model can be derived on the basis of an 1/f² PSD

f 
envelop, obtained by setting sin     equal to 1 in Eq. 1.
 f 
d 

This yields the PSD of Eq. 4 above .
4. 2 Second approximation derived by integrating the sinusoidal arches of the sinc funtion sin² x/x
A more precise 2nd approximation can be obtained by piecemeal integration of the sinusoidal “arches” of

 sin   ff
d
the sinc function itself, (see figure 1 below) 
f
   f d

2
 , assuming that a lower bound is taken for1/(fk)²

i.e. 1/(fk+1)² . Only a few of the “ arches” are shown on this figure in order to avoid cluttering it.
1/f²
1/(fk)²
1/(fk+fd/2)²
1/(fk+i+fd/2)²
1/(fk+1)²
1/(fk+i)²
1/(fk+n)²
1/(fk+i+1)²
fd
fd
fd
f1
fk- 1
fk
fk+1
fk+i
fd
fk+i+1
fk+n-1 d
fk+n
f2
fk+n+1
Figure 1 : MLS DPSK signal PSD partial representation
Detailed mathematical calculations given in section 6 hereafter show
that for Bw >> fd the transmit out-of-band power falling in [ f1, f2], which is also the MLS transmit Roll-off
at frequency f’= (f1+ f2)/2 over Bw bandwidth (since the transmit power is normalized to 1 watts) can be
formulated as:
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ACP/WGF 21-IP ---

f '  Bw / 2
f '  Bw / 2
S ( f )df = Roll  Off ( f , Bw)  1
2 ²
f d  Bw
  Bw
2
f '  1   '
 2f




2




where f’ denotes the centre frequency (f1+f2)/2. Further assuming that
Roll  Off ( f , Bw)

1 f d  Bw
2 ² f ' 2 
Compared to Eq. 4 from which one can derive PSD(f). Bw =
Bw,
Bw <<f one can write:
Eq. 5
fd
Bw assuming PSD(f) constant over
  f 2
this second approximation using Eq. 5 introduces an ½ factor .
5 . Experimental verification : comparison with actual MLS TX spectrum measurement:
The following figure is an MLS spectrum measurement snapshot provided by the Dutch C.A.A. (LNVL) to
an ICAO/EUR FMG ad-hoc MLS meeting , in 2006, in London :
By taking f = 300 KHz, one can then calculate that this 2nd approximation gives an upper-bound of -30,6
dB for the MLS unwanted emission falling in the Bw =150 KHz , at an offset of 300 Khz, This upperbound is 2,2 dB above the spectrum -measured value of -32.83 dB shown in above figure.
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ACP/WGF 21-IP ---
5. Derivation of a close upper- and lower-bound approximations of the MLS DPSK
transmit spectrum integrated over a bandwidth, f2 –f1=Bw , with f1, f2 and Bw much greater
than fd, the DPSK data modulation frequency
This is practically the case with fd = 15, 625 KHz, Bw = 150 Khz , when looking at the centre frequency of
the adjacent channel , centred at multiple of 300 KHz (equal to the MLS Channel spacing )
The figure 1 above illustrates part of the MLS DPSK power spectrum density (PSD) curve expressed as :
P
S(f)= PSD f  
fd

 sin   ff
d

f
   f d

2


Eq. 1
where: P = Signal Power in W =Watts
f = offset from carrier frequency in Hz
fd = 15625 Hz =15.625 kHz = data rate of DPSK-signal
as quoted in paper ICAO/ NSP WGW , Bretigny, 2009 /WP11, we assumed that S(f) is the normalized PSD,
i.e. P = 1 W
The derivation at hand aims at finding an approximation of the transmit power of the MLS out of band
(OOB) emissions PSD , S(f), due to the DPSK modulation, integrated over the bandwidth [f1, f2] of Bw
KHz , typically . f2-f1 = 150 KHz , as both an upper bound (UB) and lower bound (LB) , i.e. LB ≤

f2
f1
S ( f )df ≤UB and showing that under the conditions :
fd << Bw << f they converge to the formulation of Eq. 5
Expressing f1 as k fd +Δ f1 and f2= (k+n) fd + Δ f2 , with k = INT (f1/fd), and k+n = INT (f2/fd), the integer
parts respectively of the ratios f1/fd , and f2/fd and with both Δ f1 and Δ f2 smaller than fd , the MLS out of

band (OOB) transmit power
f2
f1
S ( f ).df falling into[ f1, f2]can be calculated as the sum of 3 integrals:
f k 1
i n
f k i 1
f2
f1
i 1
f k i
f k n
 S ( f )df    S ( f )df   S ( f )df
As it can be seen from the above figure 1, UB and LB can be expressed as
i  n 1 f k i 1
LB =
  S ( f )df
i 1
≤
f k i

f2
f1
S ( f )df ≤ UB =
in
f k i 1
i 0
f k i
  S ( f )df
Both bounds are the cumulative sums of the elementary integrals ai , itself a function of the sinc function
sin (f/fd) / (f/fd) squared:
n 1
LB =
a
i
1
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n
and UB =
a
i
0
with
1
ai 
fd
f k i 1

f k i
 sin f / f d 

  df
 f / f d 
2
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ACP/WGF 21-IP --Accordingly by noting that 1 / f k i 1 is smaller than any 1/f in the interval
] f k  i , f k i 1 [, and that 1/fk+i
is greater than any 1/f in this interval , ai can be then lower- and upper-bounded by :
f d2 sin 2  (k  i  x)
f d2 sin 2  (k  i  x)
≤
≤
a
dx
dx .
i
 2 0
f k2 i 1
 2 0
f k2 i
1
1
By expressing f k+i as (k+i)fd , f, in the interval [f k+i , f k+i+1] as (k+i) fd + x fd , with x varying from 0 to 1, ai
f d2
can be re-written as
 ² f k2 i

the integral
f2
f1
 sin x  dx
1
2
 sin xdx = ½
1
. This allows to calculate that
0
2
Therefore ,
0
S ( f )df is lower- and upper-bounded by the sums:
f2
f d2 i  n 1 1
f d2 i  n 1
S
(
f
)
df
≤LB
≤
≤UB≤


f1
2 2 i  0 f k21 i
2 2 i  0 f k2 i
Eq 6
By noting δf the increment in f, i.e δf = f k+i+1- f k+i (in fact δf equals fd), it is easily shown than the sum
f
i
1
2
j i
can itself be lower- and upper- bounded by an integral of the form
df
f
2
, i.e. as shown by the
following figure 2 :
Figure 2 ;
1
f
f j  n 1

i n
df
f

2
i 0
fj
1
f
2
j i
1

f
f j n

df
f
2
from which one can compute the following inequality:
f j 1
i n
1
1 f j  n  f j 1
1 f j  n 1  f j
 2 
f f j  n 1. f j
f f j  n . f j 1
i 0 f j i
By applying Eq. 7 to both sides of Eq. 6, and with f =fd one can deduct :
i) with j = k :
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
f2
f1
S ( f )df ≤UB ≤
f d2 in 1
f
f  f k 1
≤ d 2 k n
2 
2
2 i0 f k i 2
f k  n . f k 1
Eq. 7
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ACP/WGF 21-IP --and noting by inspection of figure 1 that the difference f k  n  f k 1 can be upper bounded by f2 –(f1-2fd), i.e.
Bw+ 2fd and f k  n and f k 1 lower bounded respectively by f2- fd , equal to f’ + Bw/2- fd and by f1-2 fd, equal
to f’ - Bw/2-2 fd one can achieve the ultimate upper bound :

f2
f1
fd
2 2
S ( f )df ≤

ii) with j= k-1:
f2
f1
Bw(1  2 f d / Bw)
 Bw / 2  f d  Bw / 2  2 f d
1 
f '2 1 
f'
f'





Eq.8
f d2 i  n 1 1
f
f  f k 1
S ( f )df ≥ LB ≥
≥ d 2 k n
2 
2
2 i  0 f k 1 i 2
f k  n . f k 1
using the same approach as above i) on can achieve the ultimate lower bound :

f2
f1
S ( f )df ≥
fd
2 2
Bw (1  2 f d / Bw )
Bw / 2  Bw / 2  f d 

1 

f '2 1 
f ' 
f'


Eq. 9
Assuming further that Bw is much greater than f d and f ' in its turn much greater than Bw , both ultimate
upper and lower bounds converge to the same limit
Accordingly

f2
f1
S ( f )df 
f d Bw
2 2 f '2
f d Bw
with f’ >> Bw >> fd
2 2 f '2
______________
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