NASA/TM--2002-211703

Real-Time In Situ Signal-To-Noise Ratio

Estimation for the Assessment of

Operational Communications Links

Robert M. Manning

Glenn Research Center, Cleveland, Ohio

August 2002
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NASA/TM--2002-211703

Real-Time In Situ Signal-To-Noise Ratio

Estimation for the Assessment of

Operational Communications Links

Robert M. Manning

Glenn Research Center, Cleveland, Ohio

National Aeronautics and

Space Administration

Glenn Research Center

August 2002
NASA Center for Aerospace Information

7121 Standard Drive

Hanover, MD 21076

Available from

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22100

Available electronically at h__GLTRS
REAL-TIME in situ SIGNAL-TO-NOISE RATIO ESTIMATION

FOR THE ASSESSMENT OF OPERATIONAL

COMMUNICATIONS LINKS

Robert M. Manning

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

INTRODUCTION

The assessment of the propagation conditions that prevail on a communications

link, whether it is a point-to-point terrestrial link or an earth-space link, is of vital

importance for the optimal operation of the link. The significant indicator of the

communications quality of the link is, of course, the signal-to-noise ratio (SNR) 7.

If the value of this quantity goes below a certain given threshold due, for example, to

atmospheric conditions such as rain, the bit error rate on the link becomes unacceptable

for reliable communications integrity. Knowledge of the dynamic behavior of 7 is thus

essential for the optimal implementation of procedures to mitigate the further degradation

of 7. In many instances, a separate propagation receiver is employed to acquire an

associated signal transmitted by a beacon on the communications satellite. Measurement

of the fading conditions of this signal level is then extrapolated to that of the communications link. In an eflbrt to ease the operational and financial burden to terminal operation,

methods (to be briefly discussed below) have been advanced to estimate 7 of the

communications link by measuring various quantities of the operational communications

link itself.

Many methods have been advanced to estimate the SNR using an active

modulated communications channels. For example, the output of the receiver matched

filter can be sampled, i.e., the voltage level Vs for an output symbol, and the value is

compared to a pair of a priori determined voltage levels _+a, a > 0 a < _[. The

statistical frequency of occurrence nF of values which fail to fall within the interval

I-a, a] is calculated as well as the total number n r of samples considered. The ratio

nF/n r is related to the symbol error probability by

where E s is the symbol energy (i.e., energy per symbol), N Otwice the noise power at the

output of the matched filter and Q(---) is the well-known Q-function. This relationship

is then solved for the SNR 7 = Es/No by using the inverse function Q 1(...). Although

this method has several shortcomings, e.g., limited dynamic range and sensitivity to

automatic gain control variations and inter-symbol interference, it suffers from an

irreconcilable defect; the inversion of the function Q(---), necessary to obtain the estimate

of 7, is mathematically correct only if dQ/d7 _ O. In the event that dQ/d7 --_ O, the

problem of determining _ becomes ill-posed 1. That is, a small error in the estimate of

nF/n r leads to a large error in the estimate _. In fact, the ill-posedness of this problem is

the major source of the lack of dynamic range.

NASA/TM--2002-211703 1
An entirely different approach can be imagined which avoids the ill-posedness

of that above and thus tends to be more robust in the presence of measurement errors

inherent in the sampling of the matched filter output. In the BPSK case, the sampled

voltage V_,is related to the bi-phase signal amplitude _+A and the corresponding in-phase

noise component Ncby

V_,= K(_+A + N_)

where K is a constant proportionality coefficient incorporating receiver gain factors, etc.

Remembering that the goal here is to obtain an expression for the SNR y = Es/N o

2

= A2/2o'_ where _rx = (Nc2) for zero mean, white Gaussian noise, the method endeavors

to obtain this ratio solely from the measured values of V_,. Thus, to separate the noise

term, one would try to form the average (V_,) using the fact that (N_) = 0. However, the

random bipolar nature of the signal amplitude A also yields a zero average giving

(V_,) = 0. The technique that is then adopted in this approach is to form the absolute

value ]V_,]of each sample and then forming the ensemble average giving

(K(+AJrNo)>

=KA

so long as the condition A >> O" N prevails. Additionally, the sampled values V_,are used

to compute the variance

Hence, using the former expression to rid to the A 2 term to give

O- N =

therefore allowing one to write

A 2 (Iv ,l) =

Y- 2o-_ - 2((Vs2)_ (IV_,I)2)

solely in terms of the voltage samples V_,.

The major drawback of this method is the formation of the absolute value of

the random quantity V_,; such an operation can drastically change the statistical

characteristics of the random variable. Even though this was done to rid of the bipolar

nature of the communications signal amplitude, it is mathematically faulty and,

moreover, needless. One can, and in fact should, incorporate the bipolar nature of the

signal amplitude into a rigorous statistical analysis; this characteristic of the signal is

just as important as its other aspects. In addition, this approach requires one to use a

predetermined bit-stream format within the communications data composed of a series

of 1's thus necessitating a synchronization with, for example, a preamble within the

modulation format. This results in a further complication of its implementation and, as

mentioned above, is unnecessary.

It is the purpose of this work to formulate the rigorous statistical basis for the

correct estimation of BPSK signal SNR from what is known about its behavior at

the input and output of the receiver demodulator. Instead of employing tacit and

unwarranted assumptions concerning the nature of the communication signal for

analytical simplification, a complete consistent statistical description of a BPSK signal

will be provided to which the well-known techniques of maximum likelihood estimation

theory can be applied. By employing, rather than neglecting, all the subtitles of the

NASA/TM--2002-211703 2
statistics describing the BPSK signal, an unbiased estimation procedure will be derived

that makes simple use of its inherent phase characteristics at the demodulator. In what is

to follow, a preliminary review of BPSK signal representation will be given which will

lay the foundation for the statistical connection between Gaussian noise and SNR. Once

an appropriate probabilistic description is obtained that establishes a rigorous contact

between SNR and the measured phase error of the BPSK signal entering the receiver

demodulator, the methods of maximum likelihood estimation theory will be used to

obtain analytical expressions for biased and unbiased estimates of SNR from easily

measured phase errors. Finally, the straightforward modifications needed at the

demodulator to implement the required phase measurements will be given. It should

be noted that the resulting SNR estimation technique is also applicable for a QPSK

demodulator simply by applying it to one of the BPSK arms with appropriate

modifications for the SNR expression.

PRELIMINARIES OF SIGNAL REPRESENTATION

Consider the simple BPSK demodulator shown in Figure 1. The phase-modulated

signal sg(t) at the input is defined by

si(t)=A(t)cos(coot+Oi(t)), Oi(t) -27c(i-1) i=1,2;M=2 (1)

M '

where coo is the angular frequency of the carrier wave, A(t) is its time varying amplitude

and Oi(t ) is its bi-phase state.

given by

In terms of quadratures, the noise-free BPSK signal is

s,(t) = A(t)[cos(coot)cos(O,(t))-sin(coot)sin(O,(t)) ]

= A(t)cos(coot)Cos(O,(t))

= Ac(t)cos(coot ), Ac(t)- A(t)cos(O,(t)) (2)

ao(t ) - a(t)cos(Oi(t)) = +_a(t) (3)

showing the obvious fact that the noise free signal only has an in-phase component AC(t)

since the bi-phase states are 01(t ) = 0 °, 02(t ) = 180 ° thus relegating the sine factors to zero.

where

e(O= si(O+ n(O

2 cos(co0 0

I

;+_.

si (t) = A(t) cos(co0t + 0 i(t))

n( t) = N(t) cos( COOt + ON(t))

[.n4-Phase Output

c (t) + N c (t)

a c (t) = a(t) cos(Pi (t)) = +_a(t)

N c (t) = N(t) cos(0 N (t))

FIGURE 1.

BPSK demodulator showing relationship between signal parameters

NASA/TM--2002-211703 3
Similarly, the noise n(t) associated with the input is given by

n(t) = N(t)cos((Oot + ON(t)) (4)

where N(t) is the time varying amplitude of the noise and ON(t ) is the associated phase.

In terms of quadratures, the noise is given by

n(t) = N(t)[COS(OOot)CoS(ON(t))--sin(OOot)sin(ON(t)) ]

(5)

where

= Nc(t)cos(Wot ) - Ns(t)sin(Wot )

Nc(t ) - N(t)cos(Ox(t))

Ns(t ) - N(t)sin(ON(t))

and

(6)

(7)

showing that, unlike the noise-free BPSK signal component, the noise is characterized by

both in-phase and quadrature terms. The composite signal e(t) at the input of the

demodulator is given by

e(t)=si(t)+n(t)

= [Ac(t) + Nc(t)lcos(wot)+ Ns(t)sin(wot ) (S)

where use has been made of Eqs.(2) and (5). Writing the composite signal e(t) in terms

of its in-phase and quadrature components and employing Eq.(8) yields

e(t)- Ec(t)cos(COot ) - Es sin(COot) (9)

where

Ec(t)-ac(t)+ Nc(t ) (lO)

and

<(0--- Ns(O (11)

Writing Eq.(9) in the standard form, in which the signal and noise components are written

e(0= ecos( Oot+o (0)

E(t)=?E_2.(t)+E_2(t), Ou(t)=tanl[Esl

L .J

in Eqs.(1) and (4), finally gives

where

Thus,

and

<(0 =e(Ocos(O (O)

Es(t)=E(t)sin(Ou(t))

(12)

(13)

(14)

(15)

The geometrical depiction of the various quadrature components dealt with here in the

case of the bi-phase signal Sl(t ) = Ac(t ) = A(t)cos(Ol(t)) = +A(t) is shown in Figure 2. In

this figure, it is easily seen how the addition of the noise n(t) to the signal sl(t) yields a

total composite signal e(t)with a phase of Ou(t ) with respect to the in-phase axis. The

phase Ou(t ) becomes, in this bi-phase signaling case, the phase error of the received

signal sl(t), which represents the input to the demodulator.

Given the preceding development of the relationships of the various signal

components that enter into the demodulation of a BPSK signal, the problem to be

addressed here is the determination of the associated signal-to-noise ratio

NASA/TM--2002-211703 4
I

- A(t)

Es(t ) = Ns(t )

In-Phase Axis

_ , Es

I _dt'_ _ r I _ Ec

°lt_J + A(t)

I I

"_ Ac(t) _1_--_ Nc(t)

I

-4-_ E c (t): A c (t) + Nc (t) _1

FIGURE 2.

Relationship of the various signal and noise quadrature components of the input of the

BPSK demodulator

A2(t)

)

2

where the noise variance a N is given by

a_(t)-(N_) ÷ (N_2)

2 2

As shown in the next two sections, this problem can be placed on a

mathematically tractable basis and its correct solution rests on the statistical connection

between the easily measurable parameters, not of the demodulator output, but of the

demodulator input, i.e., E(t) and/or ON(t ), and the signal-to-noise ratio y. The fact that

the demodulator input is significant is that it contains both the in-phase and quadrature

components of the noise. Additionally, the bi-phase nature of the signal modulation, i.e.,

+_A(t), is a major impediment in the application of the methods mentioned above. This

aspect of the signal must be properly accounted for in a legitimate treatment of the

problem. Such a program begins with the use of what is known about the statistics of

the noise process.

NASA/TM--2002-211703 5
NOISE STATISTICS AND THE SIGNAL-TO-NOISE RATIO

The noise components Nc(t ) and Ns(t ) are random quantities whose statistics are

usually taken to be governed by a Gaussian random process. Additionally, it is known

2

that such quadrature components are statistically independent. Letting 0-X(t) be the

variance of each of these noise components, one has for their j oint probability density

P(Nc(t),Ns(t)) =

27co2 (t)-exp 20-2(0

(16)

Using Eqs.(10),(11), (14) and (15), one has in terms of the composite signal parameters

e(0 and of(0,

Nc(t ) = E(t)cos(OE(t))- Ac(t ) (17)

and

Us(t) : E(t)sin(Of(t)) (lS)

Transforming the probability density function of N Cand N s(dropping the time

dependence for notational clarity) given by Eq.(16) into one that is a function of E and

Of(t) involves the transformation

N "lO(Nc'Ns)

p(E, Os) = p(N C, s) _ (19)

which requires the Jacobian

O(Uc,U_.). = aNt aN t.

O(E, Of) - aS s : E (20)

OE O0f

where Eqs.(17) and (18) where used to obtain the indicated result. Thus, using Eqs.(16)-

(20), one has in terms of the composite signal parameters

P(E'Oe)=(_E--_-s lexp[_2;C0-x L (Ec°sOf-Ac)2+E2sin20f120-2 )

= (__--_E_2/ exp(- A--_22/ exp[ ES-2Afc°sOe (21)

_,2ZV0-s) _. 20-N ) L 20-2

The signal-to-noise ratio y characterizing the composite signal input to the demodulator

is given for the BPSK case as _/(t) = A(t)/0-2(t). Writing Eq. (21) in terms of y and

remembering Eq.(3) gives

p(E, OeI?',Oi,0-N):I., E_---E_---27-]exp(-LCOS20i)•

12 0-N) 2

I E2-2E0-N"fyc°sOic°sOeI (22) •exp 20.2

where the density function p(E, Of) is now written as a conditional probability density

p(E, Oel?',Oi,0-N) governing the values of E and 0E conditioned on the values of y, 0i,

and O-N" This is done with the hope of being able to obtain an expression, using Eq.(22),

connecting the measurable values of E and/or 0f at the output of the demodulator onls_

NASA/TM--2002-211703 6
to the associated value for 7 thus allowing one to statistically estimate 7 from such

easily measured values.

Having secured the above relationship governing the probability density of the

envelope and phase of the composite signal, i.e., E and 0 e, and the prevailing values of

2

the signal-to-noise ratio 7, the variance of the noise power O-N, and the phase state of the

transmitted signal 0i, one can obtain two relationships involving measurable statistics of

E, or 0e, and 7. For example, one can obtain the statistics, conditioned on the value

for 7, that describe the composite signal envelope over all possible signal phase states

simply by summing Eq.(22) over the two possible values for oi (t) and integrating over

all possible values of the phase 0 e, viz2.,

2

i=1

=(E)exp(_7]exp(_ E____22/. 2j k 2oN7

•[ IexpI+--cos0ul+exp- cos0 u dOe

L /

k:rc_N) \ 2./ k 20"N)"0 k O"N

= (2__E_E]exp(_ 7 ] exp(_ E___22/'o( _/TE ]

k aN) \ 2J k 2aN)_, aN )

=p(¢,,:,-N) (2s)

which is a form of the well-known generalized Rayleigh distribution for the bi-phase

signal envelope. Although this expression is useful for many purposes, it is not sufficient

for the estimation of the quantity 7 from measurable statistics involving E(t) since, as

2

can be seen from Eq.(23), it explicitly involves the parameter o-N. Thus, using this

2

approach, one must know, a priori, the prevailing values of o-N. Further consideration

shows that this must be the case since the noise n(t) has two degrees of freedom which

contributes to the composite signal envelope, i.e., N(t) and ON(t), as seen from Figure 2.

Integrating over all possible values of the phase characterizing the demodulator output

leaves the phase of the noise to be determined by knowledge of o-N. Thus, if one desires

to obtain an estimate of 7 from measurements of E or some statistic related to E, one

must also need an a priori estimate of o-N thus rendering useless a straightforward

estimate of 7 from easily measurable signal parameters.

NASA/TM--2002-211703 7
However, if one considers statistics that describe the composite signal phase error

Ou(t ) by summing Eq.(22) over the two possible values for Oi(t ) and integrating over all

possible values of the signal envelope E(t), one obtains 3

2

i=l

_ 2Jr_rx ) \ 2)

E 2 ._E

"[.IoEexp(--g-7--2]IexpI+--c°sOul+exp[---_N ZCyx) L [_ cyx j C°SOul} dE_.

= _(-1-7-_-] ]'/'O'N ) exp(---_)\2idol= Eexp/-E-ff-_-22/c°sh/_ 2CYN) _._ECOS0UcyN )dE

r 2

=(1)exp(-_)f_rexp(---_)cosh(_f_rcosOu) dr

=Vzzc _[1__ "_7cos (0 u ) exp(-_sin2Ou)erf(_cosOu)+ V z lexp(-_)

= p(Oulr) (24)

where the third integral results from the change of variables r - E/cr N and eft(---) in the

fourth line is the 'error function' or probability integral defined by

erf(x)-_I[exp(-t2)dt

This formulation does not involve any a priori information other than the signal-to-noise

ratio 7. Thus, Eq.(24) gives a relationship involving the conditional probability density

of the phase of the demodulator output given a value for the signal-to-noise ratio 7 of the

composite signal, with no other a priori information necessary. This expression is indeed

a candidate for the basis of estimating 7 from knowledge of values for 0 E . The proper

mathematical foundation for this procedure will now be given.

THE MAXIMUM LIKELIHOOD ESTIMATION OF SIGNAL-TO-NOISE RATIO

FROM PHASE MEASUREMENTS OF THE BPSK DEMODULATOR INPUT

Equation (24), giving the conditional probability density governing the values of

0 E given a prevailing value for 7, can formally be written in the opposite sense, i.e., a

conditional probability density governing the values of _' given a prevailing value for 0E,

viz.,

p(ylOu)= __cos(Ou)exp(-_sin2Ou)erf(_cosOu)+lexp(-_) (25)

Although not explicitly shown in Eq.(25) for notational simplicity, all signal components

that enter into this expression are still functions of time, 0u = Oh(t), etc. Assuming that a

time series of statistically independent phase measurements 0u(tl) , Oh(t2), ..., Ou(te)

can be obtained from the composite signal input to the demodulator that subtend a time

interval At = tk - t1 whose value is small enough such that 7(t) and o-(t) can be taken to

NASA/TM--2002-211703 8
be constant in At, one can use Eq.(25) to obtain a probability density function

conditioned on the series of phase measurements given by

k

p(_l OE(tl) , 0E(t2) , "'',OE(tk )) _- I-I p(_l OE(tj )) (26)

j-1

From this, one can form the likelihood functional defined by

L(_" OE(tI), OE(t2), "",OE(tk)) - ln{p(_" OE(tl) , OE(t2), "",0E(tk))}

k

=Eln{p(?'Ou(t,))} (27)

j-1

According to the method of maximum likelihood, the corresponding estimate y * of _" is

found from this functional as that value of _" for which L(---) is a maximum. Hence, the

estimate _, * of _" which prevails over the set of phase measurements 0E(tl) , Oe(t2),

•", OE(t_), of a BPSK modulated signal is given by

G_L(_IOE(tl)'OE(t2)'''"OE(tk)) :0 (28)

&

y=y*

It is important to remember that the bipolar nature of the modulated signal as well as the

usual assumptions of Gaussian noise are already convolved in the probabilistic

2

description of Eq.(25). Also, unlike the general case where y and o-x are functions of

time, they can now be taken as constant and independent of time during sufficiently small

measurement intervals At. A quantitative measure for 'sufficiently small' will be given

below.

It is now necessary to find the root y * of Eq.(28). Using Eqs.(25)-(27) in

Eq.(28), and, for analytical tractability, neglecting the last term of Eq.(25) (an

approximation which holds for large values of y), and performing the required

differentiation yields

c_L =0=-_ - sin2(Ou(ti))+

-'_ y=y* i=1

+_ a___lln(erf(_=_ cos(0E(ti))//l (29)

Completing the differentiation of the last term results in a rather unwieldy expression.

Using the fact that the eft(---) function tends to a constant for large values of the

argument, i.e., for large y, one can neglect this term (consistent with the neglect of the

second term of Eq.(25)) and find that in this case, the optimal estimate for y of a biphase signal with additive Gaussian noise based on a series of k phase measurements

Ou(ti) of the composite signal input is given by

I l _ I 1

y* = sin2(Ou(ti)) , y >>1 (30)

k

Because of the approximations involved with neglecting the second term of Eq.(25) and

the third term of Eq.(29), this maximum likelihood estimate is biased toward large values

of y. This bias must now be removed for small values of y. To this end, one uses the

NASA/TM--2002-211703 9
original expression of Eq.(25) retaining the second term since this will be appreciable for

small y. However, to maintain analytical flexibility, the eft(---) function of Eq.(25) will

continued to be neglected. Finally, the analysis becomes more amenable if the inverse

function 1/7, * given by Eq.(30) is used. Given these considerations, one has 4 for the

unbiased estimate _) of y * over all possible values of the phase Ou(t )

1 m (o )p(rlo )do 7:

= ( 2 )exp(-f_)I]Asin2 OudOu +

1 ^ 5/2.e Y.e

+2_'_-X/_ sln(Ou)cos(Ou)exp(--_sln(Ou))dOu

2 1 exp---Y (31)

Thus, the unbiased estimate _) of y is related to the biased estimate y *, given by

Eq.(30), through the non-linear relationship of Eq.(31). (It should be mentioned that the

ensemble average of sin2(0u) is calculated over the phase interval -_/2 < 0 u < _/2 since

p(_]O u) is periodic in 0 u with period/r. (Hence, phase values that differ by _+a:

essentially correspond to the same SNR values and the averaging interval chosen in

Eq.(31) implements the separate identification of these intervals.) Since there are no

other parameters of the problem that enter into Eq.(31), this relationship is universal for

BPSK modulation and is plotted in Figure 3. From Eq.(31) one has the following

limiting behavior:

lim/1____/ 1 lim/1____/ 1

'

Thus, for large _), the two estimates converge for the reasons explained above. In the

case of small _), the biased estimate approaches 2 since in this case, the phase errors

0 (ti) are randomly distributed within their range -7r/2 < 0 (ti)< u/2. Note that Eq.(31)

does not monotonically approach the value of 0.5 as y --+ 0; it overshoots the 0.55 level

before it turns toward the 0.5 limit. This defect is due to the approximation made above

in the neglect of the eft(---) function in Eq.(29). A more careful analysis may correct this

shortcoming.

The implementation of this process is straightforward. Considering now the

composite signal input to the demodulator in terms of Eqs.(9), and (14)-(15), instead of

the previous representation of Eq.(8), and keeping in mind the goal of securing the values

of the phase error Ou(t ), one arrives at the required modifications shown in Figure 4. As

k sampled values of Ou(t ) are obtained (just how to determine a value for k will be

discussed below), one then implements Eq.(30) to obtain the biased estimate y * of the

signal-to-noise ratio y. Once this has been obtained, Eq.(31) is employed and solved for

the corresponding unbiased estimate _ ; this latter procedure can be effected by use of a

'look-up' table that represents the universal graphical behavior of Figure 3.

NASA/TM--2002-211703 10
0.55

0.5

0.45

0.4

0.3

0.25

0.2

0.15

0.1

0.05

0

..... _:-:` ................................................................................

",,,i",

....... i\ i!, ...............................................................................

.....:: ...........................................................................

......... ................................ i . ....................

Eq.(31) : , , , _.,,: ! !

: : : --..:%. : : : : :

I I I I I I I I I

2 4 6 8 10 12 14 16 18 20

FIGURE 3.

Characteristic curve relating the biased estimate )' *, calculated from phase error

measurements Oe(t i) according to Eq.(31), to the unbiased estimate _).

DETERMINATION OF PHASE SAMPLING LENGTH TO ASSURE

2

CONSTANCY OF _ AND _rx DURING THE INTERVAL

2

It is imperative that the prevailing values for )" and O"N do not evolve during the

time period in which the k samples of composite signal phase 06 (ti) are formed to obtain

the biased estimate )' *. The fundamental constraint which must be addressed during the

measurement process is the fact that )" does not have spectral components that exceed 20

Hz for an atmospheric channel. (This is the upper limit for atmospheric scintillation at

communications satellite frequencies.) Thus, if the data rate of the communications link

is R D bps, and if a measurement of the phase OE(tz) occurs for k consecutive bit

intervals, one must satisfy the Nyquist sampling constraint

RD > 40Hz

k

This inequality serves to bound the number of phase samples.

NASA/TM--2002-211703 11
2 cos(cO 0t)

e(t)= si(t) +n(t)

= Ecos(eOot+ OE(t))

__ In[__.APhase Output

c (t) + Nc (t)

lEe(t) = E( t)cos( OE(t))= Ac(t)+ Nc(t)

Es(t) = H t)sin(OE( t))

= Ns(t)

Phase+90Shift [_ °

FIGURE 4

Modification to BPSK demodulator to obtain phase error in composite signal.

oE(t)

EXTENSION TO THE GENERAL M-ARY PHASE MODULATION CASE

The simplest application of the foregoing to QPSK and other M-ary phase

modulated cases involves using the method shown in Figure 4 in conjunction with

Eq.(30) with just one of the arms of the demodulator. In the interest of maintaining

mathematical rigor, one could return to the analysis given above and employ, e.g., M=4

in Eq.(1) for the QPSK case. Thus, Eq.(24) would then employ a sum over i = 4 possible

phase states. This, however, would complicate the SNR estimate analogous to that of

Eq.(30) for the BPSK case.

REFERENCES

. A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (John Wiley

& Sons, New York, 1977).

. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products

(Academic Press, New York, 1980). Eq.(3.534.2).

3. Ref. 2, Eq.(3.562.4).

. Ref. 2, Eq.(3.381.1) and use of the simplifying transformations embodied in

Eqs.(8.356.1) and (8.359.4).

NASA/TM--2002-211703 12
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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

August 2002 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Real-Time In Situ Signal-To-Noise Ratio Estimation for the Assessment of

Operational Communications Links

6. AUTHOR(S)

Robert M. Manning

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

John H. Glenn Research Center at Lewis Field

Cleveland, Ohio 44135-3191

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

WU-727-01-10-00

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REPORT NUMBER

E-13448

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA TM--2002-211703

11. SUPPLEMENTARY NOTES

Responsible person, Robert M. Manning, organization code 5640, 216-433-6750.

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Unclassified - Unlimited

Subject Categories: 04, 17, and 32 Distribution: Nonstandard

Available electronically at http://gltrs.grc.na_a.gov/GITRS

This publication is available from the NASA Center for AeroSpace Information, 301-621-0390.

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

The work presented here formulates the rigorous statistical basis for the correct estimation of communication link SNR of

a BPSK, QPSK, and for that matter, any M-ary phase-modulated digital signal from what is known about its statistical

behavior at the output of the receiver demodulator. Many methods to accomplish this have been proposed and implemented in the past but all of them are based on tacit and unwarranted assumptions and are thus defective. However, the

basic idea is well founded, i.e., the signal at the output of a communications demodulator has convolved within it the

prevailing SNR characteristic of the link. The acquisition of the SNR characteristic is of the utmost importance to a

communications system that must remain reliable in adverse propagation conditions. This work provides a correct and

consistent mathematical basis for the proper statistical 'deconvolution' of the output of a demodulator to yield a measure

of the SNR. The use of such techniques will alleviate the need and expense for a separate propagation link to assess the

propagation conditions prevailing on the communications link. Furthermore, they are applicable for every situation

involving the digital transmission of data over planetary and space communications links.

14. SUBJECT TERMS

Communication theory; Signal fading; Likelihood ratio

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