Matched-filter detection of binary signals is discussed in terms of the probability of bit error. The equations for the probability of error are derived for coherent phase shift keying, and coherent frequency shift keying (FSK). Suboptimum detection of FSK signals is also discussed for discriminators

Batson, B. H.

English

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https://ntrs.nasa.gov/search.jsp?R=19770016345 2020-03-22T08:46:40+00:00Z
JW 97
W-73-11
WS-73-257
N77-23289
Unclas
G3/32 24553
PERFORMANCE OF BINARY  FSK
DATA TRANSMISSION SYSTEMS
7
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t	 AVIONICS SYSTEMS ENGINEERING DIVISION
CO[ SOTMICAT TONS POWER Ah'D DATA SYSTEMS BRA=
a^n^r^r • • _ .,lam
National Aeronautics and Space Administration
LYNDON B. JOHNSON SPACE CENTER
Hou anon, TPZas
July 24, 1973
JSC-08097
EJ-73-11
F.J5-73-257
PERFORMAXE OF BINARY FSK
	
•
DATA TRANSMISSION SYSTEMS
Prepared by
B	 Batson
AVIONICS SYSTEMS ENGINEERING DIVISION
COMMUNICATIONS, POWER, AND DATA SYSTEMS BRANCH
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
LYNDON B. JOHNSON SPACE CENTER
Houston, Texas
July 24, 1973
u 4. f ^
l
PERFORMANCE OF BINARY FSK DATA TRANSMISSION SYSTEMS
MATCHED-FILTER DETECTION OF BINARY SIGNALS 	
W_
It is well known that matched-filter (or correlation) detection is
the best means of detecting any class of binary signals, in the sense
that the probability of bit error at the detector output is minimized.
Although matched-filter detection is somewhat difficult to instrument
because it is a coherent detection scheme and requires a knowledge of
the RF phase of the signal, there are several good reasons for
considering such schemes.
a. The --tched-filte' system is particularly easy to analyss.
b. Since . patched-filter detection is the best detection
technique, the perfon-ance of a matched filter sytee represents a :our__
which can only be approached by systems utilizing other detection
schemes.
c. Using the bounds este.biished by watched-filter detection
and the results of typical non-matched-filter detectors, we can "guess"
at the performance of systems which have not been analyzed in det4;.I.
For matched-filter detection of binary signals, the probability of
error is given by
OF POOR U` LM
iwhere Eb is the average energy o•r bit and p is the correlation
coefficient between the two signal waveforms; S1(t) and S 2 (t), or
T
p	 F r S1(t)S2(t)dt
b 
o
where T is the bit duration.
Note that, for matched-filter detection of binary signals, 'Fe i4i
a function of only two parameters -- EbINa and p. The only parameter
r
	
	 that is a function of the particular signal set being transmitted is
the correlation coefficient, o, which can assume values between. -1
OF POOR QUA.I, TY
iD	 ^
3
and +1. For any particular signal set, we need onil to deters:ine p
in order to plot P e as a function of EbINo . '-his will row be
accomplished for a few familiar cases.
Coherent_PSK:
S1(t) - A sin Wct 7 F^	 n A`T
U 1	 2
52(t) - -A sin w ct t Eb, 
s A2T
1
E + E
	
bl	 b2	 A2T
-b 
s	 2.4	 a	 2
T
p s 2	 - ! ^ [1 - cos (2w t )^ dt . -1
	
A2T	 L	 JJ
0
2
T
P e	 ezfc 2N
	
2 erfZ-b
	
C)
	 o
Coherent FSK (Orthogonal)
A2T
	
C 
	 ..b
1
S2(t) - A sin w t; E	 x A2T
	
c2	
b2	 2
(2)
ORIGINAL PAGE IS
OF POOR QUALITY
gg
-b l 	 2	 A2T
2	 2
p = 0 if sin w
cl t and sin w c2 t are 
orthogonal
1	 A?T
erfc	 m l erfc	 (3)
Ve = 
2	 4NJ	 2	 2N0L
Coherent ASK (on-off Keying):
A2TS 1 (t) = A sin ' wC1 t t	 =
.	 bl	 2
S2 (t) = 0 i Eb = 0
2
F + E
b}	 b2	 D.`T
Eb
	
2	 4
P
	 0
	
2	 E
F	 1 erfc A T a l erfc	
b
	
827	 2	 2N (4)e	 2 0	 0
The results given by (2), (3), and (4) are plotted in figure 1.
Note that the performance of coherent orthscrona.l FSK -s always 3 db
4
'k
ORIG1242T, PAGE, 1S
OF, FOOF4 Qiir%,I,ITY	 .
s
10 1
Coherent
;' Coherent orthogonal
FSK (p - 0) or
	 A
t`= coherent ASK	 ..=
. ^san^ av_g 
_ R9wCJ^. - --
{^.'T! Coherent ASKo
—,10
yam
.. ;:,^:^^^.^^"Y:-1^ _^.	 ,' -- r----^- -r--- ^ -•-^-- ^
•-^-_.
i..	 7
10_ 5,--
-
-
10	 - 
0	 2	 4	 6	 8	 10	 12	 14
E b /No, db
Figure 1.- Matched-filter detection of binary signals
ti
.	 i	 --i-.	 I ,_..........
ORIGRIAI; PAGE, IS
1DF POOR QUALITY
10-1
Coherent
10-2 t=' PSK
t;- Coherent orthogonal
F -: FSK (p - 0) or
I __. ^ __ I
10
S
10 j
10-4
ILn
r---------,
10- S'
7
	
--} .--^- - — 
-^ -	 -
i
0	 2	 4	 6	 8	 10	 12	 14
E b/No, db
Figure 1.- Matched-filter detection of binary oigrals
6worse than that of coherent PSK, while that of coherent ASK (on-off
keying) is either 3 db worse than coherent PSK for equal allSMS power
(or equal Eb) or E db worse than coherent PSK for equal peak powor
(or equal signal amplitude, A).	 do-
WITCHED FILTER DETECTION Of FSK SIOWILB
Figure 2 is a run:tional illustration of the generation of an FSK
waveform. Here FSK is visualized as being the sum of two MK (on-off
keyed) waveforms, or as the switched outputs of two sir_usoidal tone
generators. An alternate means of obtaining FSK is to use the binary
data sequence to control the frequency of a single oscillator. This
could be done by using the binary sequence as the modulation input to
an PH transmitter.
Regardless of the technique used to generate the FSK signal, the
optimum detection scheme is the matched-filter which, for ISK, can
consist of two coherent multipliers followed by low-pass filters (to
reject unwanted products appearing at the multiplier outputs) and data
matched-filters. The multiplier/LPF conbinations perform the coherent
dowdulation process and provide noisy baseband data which must
subsequently be detected using appropriate binary decision devices.
Figure 3 illustrates this process of coherent detection of PSK. Mote
that a phase-coherent reference for each of the two FSK tones is
required, but that since a discrete spectral component is present at
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each of the tone frequsneies, l these coherent references can be readily
obtained using phase-locked tracking filters.
The results summarized in th-3 prev=ous section for coherent 	 w,
detection of PSK assumod orthogonality (p - 0) between the two signaling
waveforms (FSK ;ones) Sl ti) and S2 (t) . in fac', however, it is not
necessary that there be zero correlation between S 1 (t) and 92(t).
In general, the correlation coefficient of two FSK tones is given by
T
fSl (t)S2 (t)dt
Eb 0
T.
A2T , &Sin(acl t) *in (wC2Odt
0
T
T er sin(2wf t)sin(2xf t)dt 	 (S)t	 J	 c	 c1	 20
But fc 
1 
can be related to a center frequenwy f  by, say,
f	 f v Af	 (6)
c l	
and fc 
2 
can be likewise expressed as
f	 s f + Af
	 (7)
C2	 c
where 4f is the instantaneous carrier frequency deviation caused by
the modulating signal. Substituting (6) and (7) into (5) yields
'This is because each of the tone frequencies is effectively
modulated by a random binary sequence having a d.c. value of 1/2.
Therefore, half of the total transmitted power is contained in the two
	
discrete spectral components located at f	 and fcl	 c2.
10
p	 Y*in12u (fC - Af ) t)sin(2s ( fC + Af)t]dt
0
T('^
' 
J
JCOS [2w(fc + ' Af)t - 2w ( fc - Af)t]dt
0
/T`
T J cos [2n ( fc + Af ) t + 2n (fc - Af)t,dt
0
T	 'T
= T1—,
1 
cos[27r ( 2Af)t]dt - T cos[2a ( 2fc )t+	(8)
Assuming that an integral number of cycles of the center frequency f 
occurs in a bit period T, (8) . be:omes
TTf
P = TJcos[2x(2Af)t]dt
0
t T
_
1 sin [271 (2Af )] t
T	 2n(2Af)	 it	 0
sin [2n (2Af)T1
2n (2&f )T
Note that p is a function only of (Af)(T) and can assume either
positive, negative, or zero values. Ile are i.nterosted in the maximum
negative value of p, which can be found as fellows:
Let	 2n (2Cf)T - x	 (10)
sin x
Then p ^	 .:.`_h maxima occuring when
x
6&.
(9)
Mw s
^	 f
i
1 
11
kxcosx - sinx	 0
^tx	 x2
or when
sin x	
c,os x	 (11)
x
Equation (11) is satisfied for x - 0 (which corresp exids to Ot - 0)
but this is obviously not the point of interest. However, (11) is also
satisfied for x . 4.493 and this corresponds to (2Sf)T 0.715, or
to
Af	 (0.2151 T
0`.358 R	 (12)
where R - 1/T is the bit rate of the binary data being transmitted.
For this value of Af, the correlation coefficient given by (9) is
p - -0.22	 (13)
which is the maximum negative value of p achievable for FSK
transmIL. ion. Substitution of (13) into (1) yields
1 CO6 Eb
Pe	 rfc(14I
Equation (14) indicates that the best possible performance ;p - -0.22)
achievable using coherent detention of rSK is only 2 . 2 db worse than
coherent PSK. This represents a 0.8--db improvement over the achievable
performance using coherent detection of orthogonal FSK and'constitutes
a bound on the achievable performance of FSK systems utilizing
suboptimum (non-hatched -filter) detection schemes.
I	 r
12
SUSOPTIM K (NONCOHERENT) DETECTION OF FSK SIGNALS
Since systems employing coherent detection of FSK have about the
same complexity as coherent YSK systems and, at best, perform about
2.2 db worse than coherent FSK systems, it is difficult to conceive of 	 -
an application in which coherent FSK would be preferred. Coherent
detection of FSK is, in fact, rarely ( if ever) used in practical systems.
;ste primary attractiveness of FSK arises from the relative simplicity
associated with the various noncoherent (and, therefore, suboptimum)
detection techniques which car. be employed. Figure 4 illustrat two
noncoherent demodulation approaches that can be utilized, one
approach being based on the functional structure of the rsx signal as the
sum of two amplitude-modulated (ASK) signals which are subject to
envelope detection, and the othe3r approach being based on use of a
frequency discriminator. The frequency discriminator approach is
probably of more general interest and will be discussed here because
the same modulation/demodulation equipment used for transmission of
binary FSK data can then be used for transmission of information in
analog form. Thus a system employing discriminator detection of FSK is
by nature a somewhat versatile system. In addition, discriminator
detection of FSK is of considerable inter* t because it has been shown
to perform almost as well as coherent detection of optimum FSK..
The analysis of systems employing discriminator detection of FSK
is complicated by (1) the fact that it i^ very difficult to accri'mt
for the effects of signal distortion due to bandpass filtering and by
6- .
it
TO BIT
ENVELOPE	 ' '''DETECTOR
DETECTOR
BANDPASS
FILTER AT
BANDPASS
FILTER AT fc2 ENVELOPEDETECTOR
TO BIT
DETECTOR
13
(a) Technique #1
(envelope detection)
BANDPASS
	
TO BIT
FILTER. AT	 DETECTOR
t1+ fez	
LIMITER/DISCRIMIKATOR
_ I—
(b) Technique Y2
1., ^lYril. YAU f':
OF Y(OR QUALITY
14
(2) the presence of non-Gaussian noise at the discriminator output and
the resulting difficulties associated with computation of error
probabilities.
Several recent studies of error prohabilities in noncoherent FSK
systems have been performed. Klapper (ref. 1), Mazo and Salz (ref. 2),
and Schilling, et. at., (ref. 3) evaluated FSK error probabilities based
on Rice's (ref. 4) click theory of noise in FM. However, these papers
assumed a sufficiently broad bandpass filter in the system for negligible
distortion of the FSK signal. In fact, it is possible to make a
favorable tradeoff between signal distortion and input noise reduction,
so these results do not indicate error rate performance of the "optimum"
FSK system employ i na discriminator detection.
Bennett and Salz (ref. 5) determined error rates for a binary FSK
system, taking into account the effects of distortion due to a bandpass
filter. However, their receiver model  did not include a data matched
filter after the discriminator.
Tjhung and Wittke (ref. 6) evaluated error probabilities for a
binary FSK system (utilizing discriminator detection) taking into
account the effects of both a bandpass filter and a data matched filter.
In order to account for tie FM signal distortion du, to bandpass
filtering, a periodic modulating signal (a 30-bit pseudo random sequence)
was used. The particular sequence used was (11000 00101 10111 00111
11010 010001 and it was determined tha- the FM spectrum for thi:_
signal was a good approximation to the spectrum for FM by a random
a Y
OHIGINAi:TAGE LS
OF POOR QUALPI'Yf 	 is
binary signal. The predetection Landpass filter was assumed to have a
symmetrical passband and a linear phase characteristic. Results were
obtained for two filter yodels: rectangular passband and Gaussian
pa.ssband. Using Rice's click theory of FM noise, Tjhung and Nittke
computed overall error probabilities by taking the average of the
error probabilities for the individual bits. A number of error-rate
curves were calculated as functions of F, /N 	 (for the unfiltered M
signal), with 2Af and BT (the prod-act of the filter bandwidth and
the bit period or, al`ztnately, the ratio of the filter bandwidth
to the bit rate) as parameters. These curves are shown in figure S
(for rectangular bandpass filter) and Ln figure 6 (for Gaussian bandpass
filter). Figure 7 contains the data shown in figure 6, plotted in a
way that allows an interesting com parison of the effects of the
various parameters. The various sets of curves indicate that, for a
given filter type and bit rate, there is a bandwidth B and a frequency
deviation Af that minimize the probability of error. Tables I and II
were provided by Tjhung and wittke to allow some degree of precision
in determining the optLmum values of these parameters for an error
probability of 10' 4 . It can be seen from these tables that for both
the Gaussian and the rectangular bandpass filters, a value cf
2Af - 0.7R is best in that it requires the smallest value of Eb/ho
to achieve a 10-4
 bit error probability. The optimum IF bandwidth fer
Pe = 10-4
 is seen to be 1.2 times the bit rate for the rectangular
bandpass filter and 1.0 dries the bit rate for the Gaussian filter.
Optimum parameter values for error pro^jaLilities other than 10- 4 can
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1.OR 12.38 12.23 12.53
19
be obtained (with less precision) from the curves shown in figures 5
and 6. In general, it appears that a value of about 0.7R for 2Gf and
a value of about 1 . 0 (or slightly greater) for BT will minimize the
error probability for binary FSK systems employing discriminator
detection.
TABLE I.- EJ /:10 I:' DB R::; iIR=D TO ACHIM-E .A 10 -4 BIT ERROR
PROBABILITY IN BI::AR1 FSK SYST ':S EMPLOYING DISCIRLAM&TOR DETECTION
(RECTANGULAR Be:MPASS FILTER)
2Gf
BT
1.0 1.2 1.6 2.0 3.0
0.5R 12.27 10.95 11.7 12.63
0.7R 1.1.28 10.65 11.7 12.23
i.OR I	 13.8 ! !	 13.25 12.8
Ti.BLE II. - L/yo IN DB RENUIRED TO ACHIEVE A 10 -4 KT ERROR
PROBABILITY IN BINARY FSK SYST..:S EMPLOYING DISCR.MIUATQR DETEC:ION
(GAUSSIAN SAND?ASS FILTER)
F	 ^
Ii
W.
1' 1
20
It is very significant that (from table I), using discriminator
detection of binary FSK, it is possible to achieve an error probability
of 10-4 for Eb/No - 10.65 db. This is only 2.25 db more than is
required for coherent PSK and is within 0.1 db of the best performance
achievable using coherent detection of FSK. Thus the results of rjhung
and Wittke indicate that the performance bound represented by cohsrent
FSK is almost achievable using discriminator detection, given that some
discretion is exercised in choice of frequency deviation and IF filter
bandwidth.
.V
i21
t
j REFERENCES
1. Klepper, J.:	 Demodulator Threshold Performances and Error Rates in
r
Axle Modulated Digital Signala. 	 RCA Review, June 1966, pp. 236-244.
2. Mazo, J. E.; and Salz, J.:	 T%eory of Error Rates for Digital FM.
BaZZ Sys. Tech. J., vol. 45, November 1966, pp. 1511-1535.
3. Schilling, D. L.; Hoffman, E.; and Nelson, F. A.:	 Error Rates for
Digital Signals Demodulated by an FM Discriminator.	 IEEE Trans.
on Co ►mrunfcation Technology, vol. COM-15, August 1967, pp. 507-517.
4. Rice,	 S. 0.:
	 Noise in FM Receivers.	 _':.r^e Series AnaZyeie, ch.	 25,
M. Rosenblatt, ed.	 John Wiley G Sons, 1963.
S. Bennett, W. R.: and Salz, J.s	 Binary Data Transmission by FM
Over a Real Channel.	 BeZI Sys. Tears. J., vol. 42, September 1963,
pp.	 2387-2426.
6. Tjhung, T. T.; and Wittke, P. H.:	 Carrier Transmission of Binary
Data in a Restricted Barvl. 	 IEEE Trans. on Commnication Techwtogy.
vol. COM-18, August 1970	 pp. 295-304.
--


Performance of binary FSK data transmission systems

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