Signal-to-noise ratio estimation for autonomous ... - Semantic Scholar
Signal-to-Noise Ratio Estimation for Autonomous Receiver Operation* Marvin Simon, Sam Dolinar Jet Propulsion Laboratory, California Institute of Technology Abstract-A robust SNR estimator is proposed and its performance analyzed for a variety of cases dealing with its operation in receivers where little or no information is known regarding other system parameters such as carrier phase and frequency, order of the modulation, and data symbol knowledge. I. Introduction Of the many measures that characterize the performance of a communication receiver, signal-to-noise ratio (SNR) is perhaps the most fundamental in that many of the other measures directly depend on its knowledge for their evaluation. In the design of receivers for autonomous operation (often referred to as cognitive radios), it is desirable that the estimation of SNR take place with as little known information as possible regarding other system parameters, e.g., carrier phase and frequency, order of the modulation, data symbol stream, etc. While the maximumlikelihood (ML) approach to the problem results in the highest quality estimator, the resulting structure becomes quite complex unless the receiver is provided with some knowledge of the data symbols typically obtained from data estimates made at the receiver (which themselves depend on knowledge of the SNR). Such in-service SNR estimators and the evaluation of their performance have been considered in the literature [1]; however, our interest here is in SNR estimation performed without any such data symbol knowledge yet, despite the ad hoc nature of the estimators, maintaining a high level of quality and robustness with respect to other system parameter variations. One such ad hoc SNR estimator that has received considerable attention in the past is the so-called split-symbol moments estimator (SSME) [2-5] that forms its SNR estimation statistic from the sum and product of information extracted from the first and second halves of each received data symbol. Implicit in this estimation approach, as is also __________ *The research described in this paper was performed at the Jet Propulsion Laboratory, California Institute of Technology under a contract with the National Aeronautics and Space Administration.
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the case for the in-service estimators, is that the data rate and symbol timing are known or can be estimated. Previously, the performance of the SSME has been investigated only for binary phase-shift-keying (BPSK) modulations. The work in [5] extended the definition of the classic SSME from the real to the complex received domain and analyzed its performance with and without carrier frequency uncertainty. While it is stated in [1]. in reference to the SSME, that “none of these methods is easily extended to higher orders of modulations”, we shall demonstrate in this paper that such is not the case. In fact, the traditional SSME structure, as extended to the complex symbol domain in [5], is readily applicable to the class of M-PSK ( M ≥ 2) modulations, and furthermore its performance is independent of the value of M! In [6] the authors of this paper and of [5] reformulate an equivalent but more tractable definition of the complex symbol SSME using sums and differences from the two symbol halves rather than sums and products; this was the approach used in [4] for analyzing the SSME based on real symbols. The work in [6] proceeds to obtain accurate asymptotic expressions for the mean and variance of the SSME for a variety of different scenarios related to the degree of knowledge assumed for the carrier frequency uncertainty and to what extent it is compensated for in obtaining the SNR estimate. The performance results in [6] depend only mildly on the number of received samples per symbol. In this paper, we consider only one sample of information from each half symbol, e.g., the output of half-symbol matched filters, and thus we restrict ourselves here to the case of two samples per symbol. Furthermore, we consider the wideband case wherein the symbol pulse shape is assumed to be rectangular and thus the matched filters are in fact integrate-and-dump (I&D) filters. With this formulation, we are able to obtain exact, not just asymptotic, expressions for the mean and variance of the SSME under the various scenarios considered in [6]. II. Signal Model and Formation of Estimator
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Corresponding to the kth transmitted M-PSK symbol dk = e jφk i n t h e i n t e r v a l
It is straightforward to show that the squared norm of the signal components can be evaluated as
(k − 1)T ≤ t ≤ kT ,
the complex baseband received signal that is input to the first and second half I&Ds is given by
( ) = m h
1 ± W0 δ sy 2 sk± = m 2 W (δ ) 2
(1) y(t ) = mdk e j (ω t + φ ) + n(t ) where ω and φ are the carrier frequency and phase uncertainties and n(t ) is the zero mean
∆
2
±
(4)
where for simplicity of notation we define W δ = sinc 2 δ / 4 , W0 δ = cos δ / 2 , and
( )
(
(
( )
)
)
additive white Gaussian noise (AWGN). The outputs of these same I&Ds are given by
δ sy = ω − ω sy T . Note that the parameters h ± depend on whether or not phase
1 ( k −1 / 2 ) T j (ω t + φ ) 1 ( k −1 / 2 ) T y0 k = mdk ∫ e dt + ∫ n(t )dt T ( k −1)T T ( k −1)T
compensation is used and also the accuracy of the frequency uncertainty estimator but not on the random carrier phase φ nor the particular ±
symbol φ k . As such, h are independent of the order M of the M-PSK modulation. Next, we calculate N-symbol averages of the squared norms of these half-symbol sums to produce
= ( mdk / 2)e jφ e jω ( k − 3 / 4 )T sinc(δ / 4) + n0 k 1 kT 1 kT y1k = mdk ∫ e j (ω t + φ ) dt + ∫ n(t )dt 1 2 k T − / (2) ( ) T T ( k −1 / 2 ) T = ( mdk / 2)e jφ e jω ( k − 3 / 4 )T e jωT / 2 sinc(δ / 4) + n1k ∆
∆
where sinc x = sin x / x , δ = ωT , and m is a constant that reflects the signal amplitude. The complex noise variables n0 k and n1k are
U± =
zero mean Gaussian with variance σ for each (real and imaginary) part. If an estimate ωˆ of the carrier frequency uncertainty is available then we may choose to compensate for the phase due to this uncertainty by multiplying ˆ y1k by e − jωT / 2 prior to using ωˆ for the SNR estimate itself (as we shall see shortly). On the other hand, even though the estimate ωˆ is available, we might still choose not to use it to compensate for the phase due to the frequency uncertainty. Both of these options will be explored and their relative performance tradeoffs compared in terms of the degree of match between the estimate ωˆ and its true value ω . (setting
ω sy = ωˆ
y1k by e
E {U ± } = 2σ 2 + sk± = 2σ 2 (1 + h ± R) 2
4 4 var {U } = σ 2 ( h ± m 2 + σ 2 ) = σ 4 (1 + 2 h ± R) N N Finally, since from (6) R is expressible as
u = y0 k ± y1k e
= ( mdk / 2)e
[
+ n0 k + n1k e
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± k
(7)
h + E{U − } − h − E{U + }
ˆ is then the general form of the ad hoc SSME R ± obtained by substituting the sample values U
− jω syT / 2
for their expected values and the estimates for their true values, namely,
Rˆ =
jφ
U+ − U− hˆ +U − − hˆ −U +
hˆ ±
(8)
where hˆ are obtained from h defined in (4) by substituting ωˆ for ω (equivalently ±
]
j ω −ω T / 2 × e jω ( k − 3 / 4 )T sinc(δ / 4) 1 ± e ( sy ) (3) − jω syT / 2 ∆
E{U + } − E{U − }
R=
corresponds to half-symbol
− jω syT / 2
(6)
±
p h a s e c o m p e n s a t i o n w h i l e ω sy = 0 corresponds to no phase compensation), we form the sum and difference variables ± k
(5)
Making the key observation that the + − observables U and U are independent random variables (RVs) and defining the true SNR by R = m 2 / 2σ 2 , then it is straightforward to show that their means and variances are given by
2
After multiplying
1 N ±2 ∑ uk N k =1
δˆ = ωˆ T
± k
=s + n
for
δ
and
(
)
δˆsy = ωˆ − ω sy T
for
δ sy ).
For the case of real data symbols, i.e., BPSK, the estimator in (8) is exactly identical to the SSME considered in [2-5].
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RVs each with 2N degrees of freedom. However, with some degree of effort and
III. Mean and Variance of the SNR Estimator In this section we evaluate the mean and related to: 1) the absence or presence of carrier frequency uncertainty ω and likewise for its ˆ is estimation, 2) whether or not its estimate ω used for phase compensation, and 3) the ˆ matches ω . In all cases degree to which ω ˆ as involving frequency estimation, we treat ω a nonrandom parameter that is externally provided. Case 0: No frequency Uncertainty
(ω = ωˆ = ω
sy
{}
N 1 + h + R)1 F1 (1; N ; − Nh − R) − 1 ( N −1 + 2 N N − 1 (1 + 2h R) var Rˆ = N − 1 N − 2 N E Rˆ =
{}
)
= 0 ⇒ δ = δˆ = δ sy = δˆsy = 0
W (0) = W 0 (0) = 1, then, for this case, + + we have from (4) that h = hˆ = 1, h − = hˆ − = 0 and Rˆ = (U + − U − ) / U − . Since Rˆ + 1 = U + / U − is the ratio of a non-central to a central chi-square RV each with 2N degrees of freedom, then the mean and variance of Rˆ
Since
can be readily evaluated as
{}
E Rˆ =
{}
var Rˆ =
−
1 1
[
]}
− 1 F1 (1; N ; − Nh R)
2
(11)
where 1 F1 ( a; b; z ) is the confluent hyper± geometric function [7]. Since ω and thus h are now unknown, the bias of the estimator cannot be removed in this case. Furthermore, + since 1 F1 ( a; b; 0) = 1, then when h = 1 and − h = 0 , (11) immediately reduces to (9). Case 2a: Frequency Uncertainty, Perfect Frequency Estimation, No Phase Compensation sy
= 0 ⇒ δ = δˆ ≠ 0,
generic form of (8). Obtaining an exact compact closed-form expression in this case is much more difficult. since h and hˆ are now all non-zero. Nevertheless it is possible to obtain a closed-form expression in the form of an infinite series. In particular, letting ±
±
]
{ } 1 2 N − 1 var{Rˆ } = +R (1 + 2 R) N − 2 N
N
(
ξˆ =∆ hˆ − / hˆ + = tan 2 δˆsy / 4
)
(for
this
case,
ξˆ = tan 2 (δ / 4) ) and Λ = U + / U − , then after
E Rˆ 0 = R
considerable effort and manipulation, the
mean and variance of Rˆ can be evaluated in terms of the moments of Λ as
(10)
2
0
{}
( )∑ ξˆ
{} (
)
E Rˆ = −1 + 1 − ξˆ
Case 1: Frequency Uncertainty, No Frequency Estimation (and thus No Phase Compensation)
ω ≠ 0, ωˆ = ω sy = 0 ⇒ δ ≠ 0, δˆ = 0, δ sy = δ , δˆ = 0 . For this case, hˆ + = 1, hˆ − = 0, h ± =
∞
n −1
n =1
E{Λn }
2 ∞ var Rˆ = 1 − ξˆ ∑ (n − 1)ξˆ n − 2 E{Λn } n = 2
)
W (δ )[1 ± W0 (δ )] / 2, and Rˆ = (U + − U − ) / U − . − ˆ + 1 = U + / U − is Since h is non-zero, then R
(12)
2 ∞ ˆ n −1 n − ∑ ξ E{Λ } n =1
now the ratio of two non-central chi-square
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2
] F (2; N; − Nh R)
= W (δ )[1 ± W0 (δ )] / 2 and Rˆ is given by the
(9)
ˆ = ( N − 1) / N R − 1 / removed estimator R 0 whose mean and variance now become
sy
−(1 + h + R)
)
2
Since N is known, the bias of the estimator is easily removed in this case by defining a bias-
(
2
δ sy = δˆsy = δ . For this case, h ± = hˆ ±
2 N − 1 × (1 + 2 R) + R2 N
[
+(1 + h + R)
(ω ≠ 0,ωˆ = ω ,ω
N 1 R+ N −1 N −1 1 N N − 2 N − 1
Rˆ can
manipulation, the mean and variance of still be evaluated as
variance of Rˆ for a variety of special cases
where
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E{Λn } =
Γ ( N + n)Γ ( N − n) Γ 2(N)
obtained from (14)
it by
× 1F1 ( − n; N ; − Nh R)1 F1 (n; N ; − Nh R) For small frequency error, i.e., ξˆ small, (13) +
−
[
(ω ≠ 0,ωˆ ≠ ω ,ω = 0 ⇒ δ ,δˆ ≠ 0,δ = δ , δˆ = δˆ ) . Here, h = W (δ )[1 ± W (δ )] / 2 , hˆ = W (δˆ )[1 ± W (δˆ )] and Rˆ is given by the sy
]
E Rˆ = −1 + E{Λ} + ξˆ E{Λ2 } − E{Λ}
{} (
)
var Rˆ = 1 − 2ξˆ × var{Λ}
[
±
]
0
(15)
{}
E Rˆ = R + O(1 / N ) and thus, for this case,
{} (
δˆ ≠ 0, δ sy = δˆsy = 0 . Here h + = hˆ + = W (δ ), h − = hˆ − = 0 and Rˆ = (U + − U − ) / U − / hˆ + .
]
[
( )
2
)
sy
= ωˆ ⇒
)]
Here
h± =
()
[
]
Rˆ = (U + − U − ) / U − / hˆ + .
Hence, by analogy with Case 2b, the mean and variance of the SNR estimator are also given by (16) where now, however, one must make
{} {}
(
and once again
1 N + 1 E Rˆ = + h R+ ˆ ( N − 1) h N −1 2
1 / hˆ + and
W (δ ) 1 ± W0 δ − δˆ / 2, hˆ + = W δˆ , hˆ − = 0
hˆ + Rˆ + 1 = U + / U − , the
1 N N − 2 N − 1
by
δ , δˆ ≠ 0, δ sy = δ − δˆ, δˆsy = 0 .
hˆ + Rˆ can be directly obtained ˆ of Case 0 by replacing from the moments of R + R by h R . Thus,
1 hˆ +
2
(ω ≠ 0,ωˆ ≠ ω ,ω
Compensation
moments of
var Rˆ =
)
{}
E Rˆ
Case 3b: Frequency Uncertainty, Imperfect Frequency Estimation, Half-Symbol Phase
=ω ⇒ δ =
Recognizing then that
( )
ξˆ = tan 2 δˆ / 4 , the results are obtained from
var Rˆ by 1 / hˆ + .
Case 2b: Frequency Uncertainty, Perfect Frequency Estimation, Half-Symbol Phase Compensation
[
generic form of (8). The method used to obtain the moments of the SNR estimator is analogous to that used for Case 2a. In particular, noting that for this case
(13) by multiplying
the estimator is asymptotically (large N) unbiased.
)
0
sy
Although not obvious from (13), it can be shown [6] that the mean of the SNR estimator can be written in the form
sy
sy
±
+2ξˆ E{Λ3 } − E{Λ}E{Λ2 }
(ω ≠ 0,ωˆ = ω ,ω
[( N − 1) / N ] . 2
Case 3a: Frequency Uncertainty, Imperfect Frequency Estimation, No Phase Compensation
can be simply approximated by
{}
{}
var Rˆ of (16) by multiplying
the distinction between are not equal. (16)
h + and hˆ + since they
IV. Numerical Results and Comparisons To compare the performances of the estimator corresponding to the various cases just discussed, we first define a parameter
2 2 N − 1 × (1 + 2h + R) + ( h + R) N
{}
Nˆ = N var Rˆ / R2 (or in the cases where a
where for this case, as noted above, we can
h = hˆ . Once this is done in (16), + then since hˆ is known, we can once again
bias-removed
completely remove the bias from the estimator by defining the bias-removed estimator
number of symbols that are needed to achieve a fractional mean-square estimation error of 100% using that estimator. Then, if one wishes to achieve a smaller fractional mean-square
further set
+
+
( )
Rˆ 0 = [( N − 1) / N ]R − 1 / Nhˆ + whose mean
is given by
{ }
{ }
estimator
is
possible,
Nˆ 0 = N var Rˆ 0 / R2 ) that measures the
E Rˆ 0 = R and whose variance is
estimation error, say
{ }
{}
var Rˆ / R2 = ε 2 (or
var Rˆ 0 / R = ε ), then the required number IEEE Communications Society Globecom 2004
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2
2
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of symbols to achieve this level of performance would simply be
Nreq (ε 2 ) = Nˆ / ε 2 (or
Nreq (ε 2 ) = Nˆ 0 / ε 2 ). As an example, consider
the bias-removed SNR estimator for Case 2b for which
Nˆ 0 can be determined from (16) as
2 4 1 − 1 +1 + 2 N ( h + R) 2 h + R Nˆ 0 = 2 1− N
( )
(17)
Clearly, the above interpretation of the
ˆ is a bit circular in that Nˆ of meaning of N 0 0 (17) depends on N . However, this dependence is mild for reasonable values of N . Thus, to a good approximation one can
2
( h R)
2
+
4 +1 h+ R
(20) 2 4 h = + + 1 Nˆ * = lim 2 + ˆ+ 2 ˆ N →∞ + h hˆ R R h R Thus, by comparison with (18) we see that the asymptotic penalty for imperfect frequency estimation and thus the inability to remove the
{}
N var Rˆ
[( N − 1) / N ]2 ) by
R2 and equating the result 2 to ε results in a quadratic equation in N whose solution can be exactly expressed as
ˆ* Nˆ 0* − 1 2ε 2 Nˆ * < 2 + N0 Nreq (ε 2 ) = 1 + 02 1 + 1 − 2 ε2 2ε Nˆ 0* + 2ε 2 (19) Thus we see that the exact number of requisite symbols is not more than two extra symbols beyond the number that would be obtained from the approximation in (18).
)
ˆ versus R in dB Fig. 1 is a plot of N 0 with N as a parameter for the biased-removed estimator of Case 0 where the results are determined from (10). We observe that a value of N = 50 is virtually sufficient to approach 2
h + / hˆ + . ˆ versus R in Figs. 2 and 3 are plots of E R dB for fixed δ / 2π = fT and fractional frequency estimation error η = δ − δˆ / δ as
(
)
a parameter varying between 5% and 20% in steps of 5%. We observe that for a relative frequency uncertainty of half a cycle ( δ / 2π = .5 ), the amount of bias is quite small over the range of frequency estimation errors considered. When the relative frequency error increases to a full cycle ( δ / 2π = 1.0 ), then sensitivity of the bias to frequency estimation error becomes more pronounced. Fig. 4 is a
ˆ versus R in dB for a fixed fractional plot of N estimation error η = 5% and δ / 2π = fT as a parameter varying between 0.5 and 0.9. These curves are the analogous ones to Fig. 1 with the purpose of demonstrating the sensitivity of the number of symbols required for a given level of mean-square error performance to frequency uncertainty and estimation error.
We have extended the work of [6] by defining a suitable SSME when the split-symbol observables consist of the outputs of the two I&D filters per symbol, and we have computed exact expressions for the mean and variance of
+
ˆ versus h R in dB would be plot of N 0 identical to Fig. 1 in accordance with (16) and IEEE Communications Society Globecom 2004
( )
V. Conclusions
ˆ = 1 + 2(1 + 2 R) / R . the asymptotic value N For the biased-removed estimator of Case 2b, a * 0
2
{}
(18)
( )
)
+
bias is reflected entirely in the ratio
Alternatively, for this case one use the exact expression for the fractional mean-square 2 estimation error to solve directly for Nreq ε . In particular, dividing (16) (multiplied by
( (
)
{}
required number of symbols to achieve a 2 fractional mean-square estimation of ε would approximately be given by +
(
h+ E Rˆ = + R hˆ
ˆ by its limiting value Nˆ * replace N 0 0 corresponding to N = ∞ , in which case the
Nreq (ε 2 ) ≅ Nˆ 0* / ε 2 , Nˆ 0* =
the comments below this equation. Thus, the degradation in performance when frequency uncertainty is present but is perfectly estimated and fully compensated for is reflected in a horizontal shift of the curves in Fig. 1 to the right by an amount equal to h + = W δ = sinc 2 δ / 4 . Equivalently, a larger number of symbols is now required to achieve the same SNR estimation accuracy as for the case of no frequency uncertainty. For Case 3b where the frequency uncertainty estimate is imperfect but is still used for compensation, the asymptotic ( N large) behavior is obtained from (16). In particular, for N → ∞
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this SSME under various scenarios of different amounts of knowledge of the carrier frequency uncertainty and to what extent this knowledge is used in calculating the SSME. In the case of perfect knowledge and full compensation of the frequency uncertainty, the SSME is asymptotically unbiased and is easily converted to a bias-removed SSME with precisely zero bias for any number of symbols N. In the case of nonzero frequency estimation error but compensation using the best available frequency estimate, the resulting nonzero bias is a purely multiplicative factor determined by the frequency estimation error and independent of the true SNR.
10 7
{ }
Nˆ 0 = N var Rˆ0 / R2
10 50
3
( )
∞ Nˆ 0*
2
1.5 1 0
2
4 R, dB
6
8
10
Fig. 1 Case 0: No Frequency Offset, Perfect Frequency Estimate, No Phase Compensation
10 7
References
5
{}
E Rˆ
{}
E Rˆ = R
3
η = 5% − 20%
2 1.5 1 0
2
4
6
8
10
R, dB Fig. 2. Case 3b: Frequency Offset, Imperfect Frequency Estimate, Half-Symbol Phase Compensation; Relative Frequency Uncertainty δ/2π=fT=.5
10 7 5
{}
E Rˆ
3
η = 5% − 20%
{}
E Rˆ = R
2 1.5 1
0
2
4
R, dB
6
8
10
Fig. 3. Case 3b: Frequency Offset, Imperfect Frequency Estimate, Half-Symbol Phase Compensation; Relative Frequency Uncertainty δ/2π=fT=1.0
15 10
.9 .8 .7 .6 .5
7
{}
Nˆ = N var Rˆ / R 2
1. D. R. Pauluzzi and N. C. Beaulieu, “A comparison of SNR estimation techniques for the AWGN channel,” IEEE Trans. Commun., vol. 48, no. 10, October 2000, pp. 1681–1691. 2. M. K. Simon and A. Mileant, “SNR estimation for the baseband assembly,” Jet Propulsion Lab., Pasadena, CA, Telecommunications and Data Acquisition Prog. Rep. 42-85, May 15, 1986. 3. B. Shah and S. Hinedi, “The split symbol moments SNR estimator in narrow-band channels,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-26, September 1990, pp. 737–747. 4. S. Dolinar, “Exact closed-form expressions for the performance of the split-symbol moments estimator of signal-to-noise ratio,” Jet Propulsion Lab., Pasadena, CA, Telecommunications and Data Acquisition Prog. Rep. 42-100, Feb 15, 1990. 5. Y. Feria, “A complex symbol signal-to-noise ratio estiimator and its performance,” Jet Propulsion Lab., Pasadena, CA, Telecommunications and Data Acquisition Prog. Rep. 42-116, February 15, 1994. 6. S. Dolinar, Y. Feria and M. K. Simon, “A split-symbol signal-to-noise ratio estimator for autonomous receivers of M-PSK signals,” Jet Propulsion Lab., Pasadena, CA, to appear in an upcoming issue of Telecommunications and Data Acquisition Prog. Rep. 7. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. New York: Dover Press, 1972.
N =5
5
δ / 2π = 0
5 3 2
1.5 1 0
2
4
6
8
10
R, dB Fig. 4. Case 3b: Frequency Offset, Imperfect Frequency Estimate, Half-symbol Phase Compensation; Fractional Frequency Estimation Error η = 5%
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the case for the in-service estimators, is that the data rate and symbol timing are known or can be estimated. Previously, the performance of the SSME has been investigated only for binary phase-shift-keying (BPSK) modulations. The work in [5] extended the definition of the classic SSME from the real to the complex received domain and analyzed its performance with and without carrier frequency uncertainty. While it is stated in [1]. in reference to the SSME, that “none of these methods is easily extended to higher orders of modulations”, we shall demonstrate in this paper that such is not the case. In fact, the traditional SSME structure, as extended to the complex symbol domain in [5], is readily applicable to the class of M-PSK ( M ≥ 2) modulations, and furthermore its performance is independent of the value of M! In [6] the authors of this paper and of [5] reformulate an equivalent but more tractable definition of the complex symbol SSME using sums and differences from the two symbol halves rather than sums and products; this was the approach used in [4] for analyzing the SSME based on real symbols. The work in [6] proceeds to obtain accurate asymptotic expressions for the mean and variance of the SSME for a variety of different scenarios related to the degree of knowledge assumed for the carrier frequency uncertainty and to what extent it is compensated for in obtaining the SNR estimate. The performance results in [6] depend only mildly on the number of received samples per symbol. In this paper, we consider only one sample of information from each half symbol, e.g., the output of half-symbol matched filters, and thus we restrict ourselves here to the case of two samples per symbol. Furthermore, we consider the wideband case wherein the symbol pulse shape is assumed to be rectangular and thus the matched filters are in fact integrate-and-dump (I&D) filters. With this formulation, we are able to obtain exact, not just asymptotic, expressions for the mean and variance of the SSME under the various scenarios considered in [6]. II. Signal Model and Formation of Estimator
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Corresponding to the kth transmitted M-PSK symbol dk = e jφk i n t h e i n t e r v a l
It is straightforward to show that the squared norm of the signal components can be evaluated as
(k − 1)T ≤ t ≤ kT ,
the complex baseband received signal that is input to the first and second half I&Ds is given by
( ) = m h
1 ± W0 δ sy 2 sk± = m 2 W (δ ) 2
(1) y(t ) = mdk e j (ω t + φ ) + n(t ) where ω and φ are the carrier frequency and phase uncertainties and n(t ) is the zero mean
∆
2
±
(4)
where for simplicity of notation we define W δ = sinc 2 δ / 4 , W0 δ = cos δ / 2 , and
( )
(
(
( )
)
)
additive white Gaussian noise (AWGN). The outputs of these same I&Ds are given by
δ sy = ω − ω sy T . Note that the parameters h ± depend on whether or not phase
1 ( k −1 / 2 ) T j (ω t + φ ) 1 ( k −1 / 2 ) T y0 k = mdk ∫ e dt + ∫ n(t )dt T ( k −1)T T ( k −1)T
compensation is used and also the accuracy of the frequency uncertainty estimator but not on the random carrier phase φ nor the particular ±
symbol φ k . As such, h are independent of the order M of the M-PSK modulation. Next, we calculate N-symbol averages of the squared norms of these half-symbol sums to produce
= ( mdk / 2)e jφ e jω ( k − 3 / 4 )T sinc(δ / 4) + n0 k 1 kT 1 kT y1k = mdk ∫ e j (ω t + φ ) dt + ∫ n(t )dt 1 2 k T − / (2) ( ) T T ( k −1 / 2 ) T = ( mdk / 2)e jφ e jω ( k − 3 / 4 )T e jωT / 2 sinc(δ / 4) + n1k ∆
∆
where sinc x = sin x / x , δ = ωT , and m is a constant that reflects the signal amplitude. The complex noise variables n0 k and n1k are
U± =
zero mean Gaussian with variance σ for each (real and imaginary) part. If an estimate ωˆ of the carrier frequency uncertainty is available then we may choose to compensate for the phase due to this uncertainty by multiplying ˆ y1k by e − jωT / 2 prior to using ωˆ for the SNR estimate itself (as we shall see shortly). On the other hand, even though the estimate ωˆ is available, we might still choose not to use it to compensate for the phase due to the frequency uncertainty. Both of these options will be explored and their relative performance tradeoffs compared in terms of the degree of match between the estimate ωˆ and its true value ω . (setting
ω sy = ωˆ
y1k by e
E {U ± } = 2σ 2 + sk± = 2σ 2 (1 + h ± R) 2
4 4 var {U } = σ 2 ( h ± m 2 + σ 2 ) = σ 4 (1 + 2 h ± R) N N Finally, since from (6) R is expressible as
u = y0 k ± y1k e
= ( mdk / 2)e
[
+ n0 k + n1k e
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± k
(7)
h + E{U − } − h − E{U + }
ˆ is then the general form of the ad hoc SSME R ± obtained by substituting the sample values U
− jω syT / 2
for their expected values and the estimates for their true values, namely,
Rˆ =
jφ
U+ − U− hˆ +U − − hˆ −U +
hˆ ±
(8)
where hˆ are obtained from h defined in (4) by substituting ωˆ for ω (equivalently ±
]
j ω −ω T / 2 × e jω ( k − 3 / 4 )T sinc(δ / 4) 1 ± e ( sy ) (3) − jω syT / 2 ∆
E{U + } − E{U − }
R=
corresponds to half-symbol
− jω syT / 2
(6)
±
p h a s e c o m p e n s a t i o n w h i l e ω sy = 0 corresponds to no phase compensation), we form the sum and difference variables ± k
(5)
Making the key observation that the + − observables U and U are independent random variables (RVs) and defining the true SNR by R = m 2 / 2σ 2 , then it is straightforward to show that their means and variances are given by
2
After multiplying
1 N ±2 ∑ uk N k =1
δˆ = ωˆ T
± k
=s + n
for
δ
and
(
)
δˆsy = ωˆ − ω sy T
for
δ sy ).
For the case of real data symbols, i.e., BPSK, the estimator in (8) is exactly identical to the SSME considered in [2-5].
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RVs each with 2N degrees of freedom. However, with some degree of effort and
III. Mean and Variance of the SNR Estimator In this section we evaluate the mean and related to: 1) the absence or presence of carrier frequency uncertainty ω and likewise for its ˆ is estimation, 2) whether or not its estimate ω used for phase compensation, and 3) the ˆ matches ω . In all cases degree to which ω ˆ as involving frequency estimation, we treat ω a nonrandom parameter that is externally provided. Case 0: No frequency Uncertainty
(ω = ωˆ = ω
sy
{}
N 1 + h + R)1 F1 (1; N ; − Nh − R) − 1 ( N −1 + 2 N N − 1 (1 + 2h R) var Rˆ = N − 1 N − 2 N E Rˆ =
{}
)
= 0 ⇒ δ = δˆ = δ sy = δˆsy = 0
W (0) = W 0 (0) = 1, then, for this case, + + we have from (4) that h = hˆ = 1, h − = hˆ − = 0 and Rˆ = (U + − U − ) / U − . Since Rˆ + 1 = U + / U − is the ratio of a non-central to a central chi-square RV each with 2N degrees of freedom, then the mean and variance of Rˆ
Since
can be readily evaluated as
{}
E Rˆ =
{}
var Rˆ =
−
1 1
[
]}
− 1 F1 (1; N ; − Nh R)
2
(11)
where 1 F1 ( a; b; z ) is the confluent hyper± geometric function [7]. Since ω and thus h are now unknown, the bias of the estimator cannot be removed in this case. Furthermore, + since 1 F1 ( a; b; 0) = 1, then when h = 1 and − h = 0 , (11) immediately reduces to (9). Case 2a: Frequency Uncertainty, Perfect Frequency Estimation, No Phase Compensation sy
= 0 ⇒ δ = δˆ ≠ 0,
generic form of (8). Obtaining an exact compact closed-form expression in this case is much more difficult. since h and hˆ are now all non-zero. Nevertheless it is possible to obtain a closed-form expression in the form of an infinite series. In particular, letting ±
±
]
{ } 1 2 N − 1 var{Rˆ } = +R (1 + 2 R) N − 2 N
N
(
ξˆ =∆ hˆ − / hˆ + = tan 2 δˆsy / 4
)
(for
this
case,
ξˆ = tan 2 (δ / 4) ) and Λ = U + / U − , then after
E Rˆ 0 = R
considerable effort and manipulation, the
mean and variance of Rˆ can be evaluated in terms of the moments of Λ as
(10)
2
0
{}
( )∑ ξˆ
{} (
)
E Rˆ = −1 + 1 − ξˆ
Case 1: Frequency Uncertainty, No Frequency Estimation (and thus No Phase Compensation)
ω ≠ 0, ωˆ = ω sy = 0 ⇒ δ ≠ 0, δˆ = 0, δ sy = δ , δˆ = 0 . For this case, hˆ + = 1, hˆ − = 0, h ± =
∞
n −1
n =1
E{Λn }
2 ∞ var Rˆ = 1 − ξˆ ∑ (n − 1)ξˆ n − 2 E{Λn } n = 2
)
W (δ )[1 ± W0 (δ )] / 2, and Rˆ = (U + − U − ) / U − . − ˆ + 1 = U + / U − is Since h is non-zero, then R
(12)
2 ∞ ˆ n −1 n − ∑ ξ E{Λ } n =1
now the ratio of two non-central chi-square
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2
] F (2; N; − Nh R)
= W (δ )[1 ± W0 (δ )] / 2 and Rˆ is given by the
(9)
ˆ = ( N − 1) / N R − 1 / removed estimator R 0 whose mean and variance now become
sy
−(1 + h + R)
)
2
Since N is known, the bias of the estimator is easily removed in this case by defining a bias-
(
2
δ sy = δˆsy = δ . For this case, h ± = hˆ ±
2 N − 1 × (1 + 2 R) + R2 N
[
+(1 + h + R)
(ω ≠ 0,ωˆ = ω ,ω
N 1 R+ N −1 N −1 1 N N − 2 N − 1
Rˆ can
manipulation, the mean and variance of still be evaluated as
variance of Rˆ for a variety of special cases
where
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E{Λn } =
Γ ( N + n)Γ ( N − n) Γ 2(N)
obtained from (14)
it by
× 1F1 ( − n; N ; − Nh R)1 F1 (n; N ; − Nh R) For small frequency error, i.e., ξˆ small, (13) +
−
[
(ω ≠ 0,ωˆ ≠ ω ,ω = 0 ⇒ δ ,δˆ ≠ 0,δ = δ , δˆ = δˆ ) . Here, h = W (δ )[1 ± W (δ )] / 2 , hˆ = W (δˆ )[1 ± W (δˆ )] and Rˆ is given by the sy
]
E Rˆ = −1 + E{Λ} + ξˆ E{Λ2 } − E{Λ}
{} (
)
var Rˆ = 1 − 2ξˆ × var{Λ}
[
±
]
0
(15)
{}
E Rˆ = R + O(1 / N ) and thus, for this case,
{} (
δˆ ≠ 0, δ sy = δˆsy = 0 . Here h + = hˆ + = W (δ ), h − = hˆ − = 0 and Rˆ = (U + − U − ) / U − / hˆ + .
]
[
( )
2
)
sy
= ωˆ ⇒
)]
Here
h± =
()
[
]
Rˆ = (U + − U − ) / U − / hˆ + .
Hence, by analogy with Case 2b, the mean and variance of the SNR estimator are also given by (16) where now, however, one must make
{} {}
(
and once again
1 N + 1 E Rˆ = + h R+ ˆ ( N − 1) h N −1 2
1 / hˆ + and
W (δ ) 1 ± W0 δ − δˆ / 2, hˆ + = W δˆ , hˆ − = 0
hˆ + Rˆ + 1 = U + / U − , the
1 N N − 2 N − 1
by
δ , δˆ ≠ 0, δ sy = δ − δˆ, δˆsy = 0 .
hˆ + Rˆ can be directly obtained ˆ of Case 0 by replacing from the moments of R + R by h R . Thus,
1 hˆ +
2
(ω ≠ 0,ωˆ ≠ ω ,ω
Compensation
moments of
var Rˆ =
)
{}
E Rˆ
Case 3b: Frequency Uncertainty, Imperfect Frequency Estimation, Half-Symbol Phase
=ω ⇒ δ =
Recognizing then that
( )
ξˆ = tan 2 δˆ / 4 , the results are obtained from
var Rˆ by 1 / hˆ + .
Case 2b: Frequency Uncertainty, Perfect Frequency Estimation, Half-Symbol Phase Compensation
[
generic form of (8). The method used to obtain the moments of the SNR estimator is analogous to that used for Case 2a. In particular, noting that for this case
(13) by multiplying
the estimator is asymptotically (large N) unbiased.
)
0
sy
Although not obvious from (13), it can be shown [6] that the mean of the SNR estimator can be written in the form
sy
sy
±
+2ξˆ E{Λ3 } − E{Λ}E{Λ2 }
(ω ≠ 0,ωˆ = ω ,ω
[( N − 1) / N ] . 2
Case 3a: Frequency Uncertainty, Imperfect Frequency Estimation, No Phase Compensation
can be simply approximated by
{}
{}
var Rˆ of (16) by multiplying
the distinction between are not equal. (16)
h + and hˆ + since they
IV. Numerical Results and Comparisons To compare the performances of the estimator corresponding to the various cases just discussed, we first define a parameter
2 2 N − 1 × (1 + 2h + R) + ( h + R) N
{}
Nˆ = N var Rˆ / R2 (or in the cases where a
where for this case, as noted above, we can
h = hˆ . Once this is done in (16), + then since hˆ is known, we can once again
bias-removed
completely remove the bias from the estimator by defining the bias-removed estimator
number of symbols that are needed to achieve a fractional mean-square estimation error of 100% using that estimator. Then, if one wishes to achieve a smaller fractional mean-square
further set
+
+
( )
Rˆ 0 = [( N − 1) / N ]R − 1 / Nhˆ + whose mean
is given by
{ }
{ }
estimator
is
possible,
Nˆ 0 = N var Rˆ 0 / R2 ) that measures the
E Rˆ 0 = R and whose variance is
estimation error, say
{ }
{}
var Rˆ / R2 = ε 2 (or
var Rˆ 0 / R = ε ), then the required number IEEE Communications Society Globecom 2004
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2
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of symbols to achieve this level of performance would simply be
Nreq (ε 2 ) = Nˆ / ε 2 (or
Nreq (ε 2 ) = Nˆ 0 / ε 2 ). As an example, consider
the bias-removed SNR estimator for Case 2b for which
Nˆ 0 can be determined from (16) as
2 4 1 − 1 +1 + 2 N ( h + R) 2 h + R Nˆ 0 = 2 1− N
( )
(17)
Clearly, the above interpretation of the
ˆ is a bit circular in that Nˆ of meaning of N 0 0 (17) depends on N . However, this dependence is mild for reasonable values of N . Thus, to a good approximation one can
2
( h R)
2
+
4 +1 h+ R
(20) 2 4 h = + + 1 Nˆ * = lim 2 + ˆ+ 2 ˆ N →∞ + h hˆ R R h R Thus, by comparison with (18) we see that the asymptotic penalty for imperfect frequency estimation and thus the inability to remove the
{}
N var Rˆ
[( N − 1) / N ]2 ) by
R2 and equating the result 2 to ε results in a quadratic equation in N whose solution can be exactly expressed as
ˆ* Nˆ 0* − 1 2ε 2 Nˆ * < 2 + N0 Nreq (ε 2 ) = 1 + 02 1 + 1 − 2 ε2 2ε Nˆ 0* + 2ε 2 (19) Thus we see that the exact number of requisite symbols is not more than two extra symbols beyond the number that would be obtained from the approximation in (18).
)
ˆ versus R in dB Fig. 1 is a plot of N 0 with N as a parameter for the biased-removed estimator of Case 0 where the results are determined from (10). We observe that a value of N = 50 is virtually sufficient to approach 2
h + / hˆ + . ˆ versus R in Figs. 2 and 3 are plots of E R dB for fixed δ / 2π = fT and fractional frequency estimation error η = δ − δˆ / δ as
(
)
a parameter varying between 5% and 20% in steps of 5%. We observe that for a relative frequency uncertainty of half a cycle ( δ / 2π = .5 ), the amount of bias is quite small over the range of frequency estimation errors considered. When the relative frequency error increases to a full cycle ( δ / 2π = 1.0 ), then sensitivity of the bias to frequency estimation error becomes more pronounced. Fig. 4 is a
ˆ versus R in dB for a fixed fractional plot of N estimation error η = 5% and δ / 2π = fT as a parameter varying between 0.5 and 0.9. These curves are the analogous ones to Fig. 1 with the purpose of demonstrating the sensitivity of the number of symbols required for a given level of mean-square error performance to frequency uncertainty and estimation error.
We have extended the work of [6] by defining a suitable SSME when the split-symbol observables consist of the outputs of the two I&D filters per symbol, and we have computed exact expressions for the mean and variance of
+
ˆ versus h R in dB would be plot of N 0 identical to Fig. 1 in accordance with (16) and IEEE Communications Society Globecom 2004
( )
V. Conclusions
ˆ = 1 + 2(1 + 2 R) / R . the asymptotic value N For the biased-removed estimator of Case 2b, a * 0
2
{}
(18)
( )
)
+
bias is reflected entirely in the ratio
Alternatively, for this case one use the exact expression for the fractional mean-square 2 estimation error to solve directly for Nreq ε . In particular, dividing (16) (multiplied by
( (
)
{}
required number of symbols to achieve a 2 fractional mean-square estimation of ε would approximately be given by +
(
h+ E Rˆ = + R hˆ
ˆ by its limiting value Nˆ * replace N 0 0 corresponding to N = ∞ , in which case the
Nreq (ε 2 ) ≅ Nˆ 0* / ε 2 , Nˆ 0* =
the comments below this equation. Thus, the degradation in performance when frequency uncertainty is present but is perfectly estimated and fully compensated for is reflected in a horizontal shift of the curves in Fig. 1 to the right by an amount equal to h + = W δ = sinc 2 δ / 4 . Equivalently, a larger number of symbols is now required to achieve the same SNR estimation accuracy as for the case of no frequency uncertainty. For Case 3b where the frequency uncertainty estimate is imperfect but is still used for compensation, the asymptotic ( N large) behavior is obtained from (16). In particular, for N → ∞
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this SSME under various scenarios of different amounts of knowledge of the carrier frequency uncertainty and to what extent this knowledge is used in calculating the SSME. In the case of perfect knowledge and full compensation of the frequency uncertainty, the SSME is asymptotically unbiased and is easily converted to a bias-removed SSME with precisely zero bias for any number of symbols N. In the case of nonzero frequency estimation error but compensation using the best available frequency estimate, the resulting nonzero bias is a purely multiplicative factor determined by the frequency estimation error and independent of the true SNR.
10 7
{ }
Nˆ 0 = N var Rˆ0 / R2
10 50
3
( )
∞ Nˆ 0*
2
1.5 1 0
2
4 R, dB
6
8
10
Fig. 1 Case 0: No Frequency Offset, Perfect Frequency Estimate, No Phase Compensation
10 7
References
5
{}
E Rˆ
{}
E Rˆ = R
3
η = 5% − 20%
2 1.5 1 0
2
4
6
8
10
R, dB Fig. 2. Case 3b: Frequency Offset, Imperfect Frequency Estimate, Half-Symbol Phase Compensation; Relative Frequency Uncertainty δ/2π=fT=.5
10 7 5
{}
E Rˆ
3
η = 5% − 20%
{}
E Rˆ = R
2 1.5 1
0
2
4
R, dB
6
8
10
Fig. 3. Case 3b: Frequency Offset, Imperfect Frequency Estimate, Half-Symbol Phase Compensation; Relative Frequency Uncertainty δ/2π=fT=1.0
15 10
.9 .8 .7 .6 .5
7
{}
Nˆ = N var Rˆ / R 2
1. D. R. Pauluzzi and N. C. Beaulieu, “A comparison of SNR estimation techniques for the AWGN channel,” IEEE Trans. Commun., vol. 48, no. 10, October 2000, pp. 1681–1691. 2. M. K. Simon and A. Mileant, “SNR estimation for the baseband assembly,” Jet Propulsion Lab., Pasadena, CA, Telecommunications and Data Acquisition Prog. Rep. 42-85, May 15, 1986. 3. B. Shah and S. Hinedi, “The split symbol moments SNR estimator in narrow-band channels,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-26, September 1990, pp. 737–747. 4. S. Dolinar, “Exact closed-form expressions for the performance of the split-symbol moments estimator of signal-to-noise ratio,” Jet Propulsion Lab., Pasadena, CA, Telecommunications and Data Acquisition Prog. Rep. 42-100, Feb 15, 1990. 5. Y. Feria, “A complex symbol signal-to-noise ratio estiimator and its performance,” Jet Propulsion Lab., Pasadena, CA, Telecommunications and Data Acquisition Prog. Rep. 42-116, February 15, 1994. 6. S. Dolinar, Y. Feria and M. K. Simon, “A split-symbol signal-to-noise ratio estimator for autonomous receivers of M-PSK signals,” Jet Propulsion Lab., Pasadena, CA, to appear in an upcoming issue of Telecommunications and Data Acquisition Prog. Rep. 7. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. New York: Dover Press, 1972.
N =5
5
δ / 2π = 0
5 3 2
1.5 1 0
2
4
6
8
10
R, dB Fig. 4. Case 3b: Frequency Offset, Imperfect Frequency Estimate, Half-symbol Phase Compensation; Fractional Frequency Estimation Error η = 5%
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