Performance of ideal quantum receiver of coherent signal of random phas

Helstrom, C. W.

English

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Per fo rmance  of an Ideal  Quantum Receiver  
of a Coherent  Signal of Random Phase 
.I, 
C a r l  W. Helstrom"' 
Department  of Applied Ele ct  r ophy s i c s 
University of California,  San Diego 
La Jol la ,  California 92037 
ABSTRACT 
, An idea l  quantum rece ive r  is to  de tec t  a 
coherent  narrowband optical  signal in the presence  
of thermal background radiation. 
both of the average  probabili ty of e r r o r  in  a binary 
communication s y s t e m  t ransmi t t ing  0's (blanks) 
and 1's (pulses)  with equal  probabi l i t ies ,  and of the 
probabili ty of detection for various fixed values  of 
the false-alarm probability. 
Curves  are  given 
i 
https://ntrs.nasa.gov/search.jsp?R=19690006419 2020-03-12T08:20:29+00:00Z
In analyzing a proposed optical communication or r a d a r  
r ece ive r  it is useful t o  compare  its per formance  with that of an idea l  
r ece ive r  l imited only by background radiation and by the quantum na ture  
of the s ignals  t o  be detected.  
formance  cu rves  fo r  a r ece ive r  of coherent  optical  signals of random 
phase in  the presence  of thermal background radiation of absolute t empera -  
t u r e  T .  
are a s sumed  known. 
In this correspondence,  we p resen t  per- 
The time of a r r i v a l  and the f o r m  of the envelope of each  s ignal  
The s ignal  is postulated to  be a coherent narrowband pulse such 
The idea l  r ece ive r  cons is t s  of a as would be emit ted by an idea l  laser. 
cavity with an ape r tu re  d i rec ted  toward the source  of the signal. The 
cavity is initially empty. 
t o  a r r ive ,  the a p e r t u r e  is opened to  admi t  it and then closed. 
makes the best possible measu remen t s  on the field in the cavity f o r  the 
purpose of deciding whether it contains a component due to  a s ignal  o r  not. 
It has been shown that the opt imum procedure f o r  detecting a signal of this 
During the period when the s ignal  is expected 
An observer  
kind is equivalent to counting the number  m of photons - -  or measur ing  the 
energy  - -  in  a composite mode that is  in effect matched to  the signal. 
If m exceeds a cer ta in  decision level p , the obse rve r  decides that a 
s ignal  is present .  
s igna l  occupied a single mode of the r ece ive r .  
1 
The result ing per formance  is the  same as though the 
1 
Let S be the average  number  of photons in the signal;  S = E / h v ,  
S 
where E 
carrier frequency of the signal.  
photons per, mode of the r ece ive r .  
t empera tu re  T ,  
is the energy  in the s ignal ,  h is Planck 's  constant,  and v is the 
S 
Let  N be the average  number of noise 
For t h e r m a l  radiation of absolute 
1 N = [exp ( h v / k T )  - 1 1 -  , 
where  k is Boltzmann's constant. 
In the absence of a s ignal  (hypothesis H the probabili ty that m 
0 
photons are counted is 
m P = ( 1  - v ) v  , v = N / ( N t l ) .  
Om 
When the s ignal  is  p re sen t  (hypothesis H 1, the probabili ty that m photons 
are counted is given by the Laguer re  distribution 
1 
2 
2 -(l-v)S m v L c- (1 - V I  w v l  
m 
p = ( l - v ) e  
lm (3) 
where  L (x) is the m-th Laguer re  polynomial. m 
We analyze first a binary communication s y s t e m  sending messages  
coded in too ' s  and 1's. 
0 by dispatching nothing. 
The opt imum receiver compares  the likelihood ra t io  P 
A 1 is t ransmi t ted  by dispatching a laser pulse,  a 
W e  suppose 0's and 1's t o  be equally probable ,  
/P Im om with the 
2 
decision level 1, deciding that  a 1 was s e n t  (choosing H ) whenever 
1 
> 1. The decision leve l  IJ. is therefore  the smallest integer  for  
PLm/POm 
which 
( 1 - v ) S  L [-(1-v12 S / ~ I  2 e 
P 
The probabili ty of an e r r o r  of the first kind (choosing hypothesis 
H when H is t r u e )  is 
1 0 
.. Qo=I  P O m = v  P + 1  
(4) 
The probability of an e r r o r  of the second kind (choosing H when H 0 1 is 
t r u e )  is 
Q1= PIm. 
m =  0 
The average  probability of error is 
1 
Pe = 5 (Qo t Q1). 
It is plotted in Fig.  1 v e r s u s  the average  number S of signal photons, and 
in Fig.  2 v e r s u s  the signal-to-noise ra t io  S / N ,  for various values of N. 
In Table 1 we give the values  of the r a t io  hv/kT corresponding to  the 
average numbers  N of noise photons used in the graphs.  
3 
TABLE 1 
N 0. 01 0. 03 0. 05 0 . 1  0 . 2  0. 3 0 . 4  
hv/kT 4 .61  3.54 3.04 2.40 1.792 1 .466  1.253 
N 0 . 5  1 2 3 5 7 9 
hv/kT 1.099 0.693 0.406 0.288 0.1823 0.  1334 0.1054 
When N is l a rge ,  the e r r o r  probabilities are near ly  the same as 
for  detecting a c l a s s i ca l  narrowband signal of r andom phase in white 
Gaussian noise,  
(8 1 
2 2  
xexp[-*(x Sa )]I (ax)dx,  
B 
0 
Q , =  1 - Q ( c Y , ~ ) =  
0 
where 
2 2 
f3 /2 = p /N,  CY / 2  = S / N  , (9) 
and Q (a ,  B )  is Marcum' s  Q-function. 
s y s t e m  the value of B is the solution of the equation 
F o r  this binary communication 
3 
3 
and the average probability of error is 
4 
It has been plotted as the curve marked  IIN = in Fig.  2. 
In a laser r a d a r  the rece ive r  wil l  be based on the Neyman-Pearson 
cr i te r ion  and will  maximize the  detection probability Q 
fixed false-alarm probability Q 
assigned value of Q 
0' 
F o r  all numbers  m of photons less than a cer ta in  number 1.1, hypothesis 
H is chosen; for  all m > p,  H is chosen. Whenever exactly p photons 
are counted, hypothesis M is chosen with probability f ,  H with probability 
= 1 - Q fo r  a 
d 1 
In o rde r  t o  attain exactly a p re -  
0' 
the decisions mus t  be randomized. 
0 1 
1 0 
(1 -f )*  
The f a l s e - a l a r m  probability is now 
I.L P + 1  = f ( l - v )  v t v , 
QO 
v = N / ( N t l ) ,  0 < f 5 1, 
f r o m  which P and f can be calculated for  given values  of Q and N; p is the 
0 
grea te s t  integer  in  the quotient (An Q /&nv) .  The probability of detection 
0 
is now P 
Q d = l -  1 P l m t f P  
1P 
m=O 
It has been plotted in Figs .  3-5 versus the average  number S of s ignal  
photons and in Figs .  6-8 versus the signal-to-noise r a t io  S/N,  for  var ious  
values of N and f o r  Qo = 10 -' , 10-9, and 10 . The curves  marked "N = d l  -6 
5 
in F igs .  6-8 r ep resen t  the detection probability for  a c l a s s i ca l  rece iver  
of a narrowband s ignal  of unknown phase,  
4 
1/ 2 
Qd = Q(a ,  P )  , a = ( 2 S / N )  , 
where P is determined by the f a l s e - a l a r m  probability, 
2 
Q, = exp (-P /2) , 
5 
In a previous paper  the probability of detection was erroneously 
s ta ted to  be given as in  (14) f o r  all values of N ,  but with 01 = [ 2S/(N +$) 1'. 
The detection s ta t i s t ic  was regarded  the re  as the modulus of a complex 
quantity, the mode amplitude,  whose r e a l  and imaginary pa r t s  a r e  
independent Gaussian random var iab les .  
1 
Those p a r t s  r ep resen t  non- 
commuting observables ,  however,  and their  joint probability density 
function, f r o m  which the distribution of the modulus was der ived,  cannot 
legit imately be specified.  
t o  the quantized var iab le  m ,  whose probability distributions a r e  given by 
(2) and (3) under the two hypotheses.  
The squared modulus is i n  fact  proportional 
6 
FOOTNOTES 
% This r e s e a r c h  was c a r r i e d  out under Grant No, NGR-05-009-079 
f r o m  the National Aeronaut ics  and  Space Administration. 
1. 
2,  
3 .  
4.  
5.  
C .  W .  Hels t rom,  "Fundamental  l imitations on the detectabil i ty of 
e lectromagnet ic  s ignals ,  I '  Int. J.  Theor .  Phys.  vol. 1, pp. 37-50; 
May, 1968. 
G. Lachs ,  "Theoret ical  aspec ts  of mix tures  of t he rma l  a n d  coherent  
rad ia t ion ,"  Phys .  Rev. vol.  138, pp. B 1012-B 1016; May 24, 1965. 
C.  W .  Hels t rom,  Stat is t ical  Theory of Signal Detection, Pergamon 
Press, Oxford; 2nd e d . ,  1968, pp. 192-193. 
Ib id . ,  pp. 166-171. 
C .  W.  Hels t rom,  l lQuantum Limitations on the Detection of Coherent 
and Incoherent Signals ,  ' I  Trans .  IEEE,  vol.  IT-11, pp. 482-490; 
October ,  1965. 
-
7 
FIGURE CAPTIONS 
FIG. 1: Average probability P of e r r o r  ve r sus  average  number S of e 
signal  photons. 
FIG. 2: Average probability P of e r r o r  ve r sus  signal-to-noise ra t io  S/N.  e 
FIG, 3 :  Probabi l i ty  Q of detection v e r s u s  average  number S of signal 
-2 
photons; Q =10 . 
d 
0 
FIG. 4: Probabi l i ty  Q of detection ve r sus  average  number S of s ignal  
-4 
d 
photons; Q =10 . 
0 
FIG. 5: Probabili ty Q of detection ve r sus  average  number S of s ignal  d 
photons; Q = 10 -6 . 0 
FIG. 6: Probabi l i ty  Q of detection versus  signal-to-noise ra t io  S / N ;  d 
-2 
Qo =10 * 
FIG. 7: Probabili ty Q of detection ve r sus  signal-to-noise r a t io  S / N ;  d 
-4 a =10 . 
0 
FIG. 8: Probabili ty Q of detection ve r sus  signal-to-noise ra t io  S /N;  d 
-6 
Q0=1O . 
8 
I 
I I i i i i i 
2 4 6 8 10 12 14 
S 
1S51 
1.0 - 1 I I I I I I I I I I I I I I I I 
- 
\ 
Figure 2 
I I I I I I I I f 1 I I I I I I 1 
IO 15 
I 
99.9 
9. 
IC 
F 
c 
I 
0. - 
Figure 3 
2 4 6 8 I 12 I4 I6 18 
99.99 
99.9 
50 
I 
0.1 
J 
Figure 4 
5 IO 15 20 
S 
99.99 
99.9 
18 
G 
U 
I 
0. I 
15 20 
S 
25 30 35 
99.99 
99. 
Q 
I 
5 
1 
0.1 
0 
Figure 6 
I f I I I I I I 
4 6 IO 
S I N  
12 14 16 
99. 
I 
I 
.I 
Figure '7 
Q~ = ls4 
2 4 
/ 
I 12 14 I I 
99.99 
99.9 
9 
9 
IO 
G 
u 
1 
0. I 
Qo = 
5 IO 15 
S / N  
20 


Performance of an ideal quantum receiver of a coherent signal of random phase

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