Precise comparison of experimental and theoretical SNRs in CO2 laser heterodyne systems: comments

Jeffrey H. Shapiro MIT Department of Electrical Engineering & Computer Science, Cambridge, Massachusetts 02139. Received 12 December 1984. 0003-6935/85/091245-03$02.00/0. © 1985 Optical Society of America. In the recent article by Foord et al.} excellent agreement

is reported between calculated and measured signal-to-noise ratios (SNRs) in a coherent laser radar. As the authors cor­rectly point out, this agreement is a novel result, in that pre­vious published studies have had SNRcalc exceed SNRmeas by 5-10 dB. It will be shown here, however, that there is a dis­crepancy in the data processing technique described in Sect. IV.A of Ref. 1. Moreover, the effect of correcting this dis­crepancy is to decrease SNRmeas by 5.63 dB, placing the Foord et al. experiment in the same situation found in earlier studies.

The problem with Sec. IV.A of Ref. 1 concerns the use of a linear rather than a square-law detector in the spectrum an­alyzer, which impacts how the authors compare their mea-

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sured voltage signal-to-noise ratio (SNRvolt)meas to their cal­culated power signal-to-noise ratio (SNRpower)calc for the speckle target case of interest. In what follows, familiarity with the Foord et al. experiment will be assumed. Suppose that

Now, if we use the fact that {|n(m;n)|2} are exponentially distributed, we can rewrite Eq. (7) as follows:

is the signal entering their spectrum analyzer when the radar optics are open and that and use

is the signal entering their spectrum analyzer when the radar optics are blocked. In these expressions, s(t) is the complex envelope of the speckle target return, n(t) is the complex en­velope of the local-oscillator shot noise, and ΩD is the target-return Doppler shift (effectively an intermediate frequency). The spectrum analyzer produces two collections of outputs, signal plus noise:

in lieu of (8), where

and noise only:

where

and still obtain (9). At last we come to the difficulty,

appears to generate Section IV.A of Ref. 1

Here, w(t;td) is the window function of the dwell-time td spectrum analyzer, T is the intersample time of the spectrum analyzer, ƒm is the mth center frequency (arranged in in­creasing order) of the spectrum analyzer, and K > N. Per explicit and implicit results from Ref. 1 it follows that

with

where angular brackets denote ensemble average. With the foregoing notation we have that

Via the ergodic hypothesis for N » 1 [compare Eq. (9)], q. (13) should approach

Because {|n(m;n)|} and {|s(m;n) + n(m;n)|} are Rayleigh distributed,2 Eq. (15) reduces to

where (ƒm0,ƒm1-1) is the frequency interval corresponding to the signal bandwidth B in the standard formula (7). Were a square-law detector employed, the following measured SNR could be obtained: where the approximation is to assume σ2

s(m) » σ2n for m0 ≤

m < m1. So, according to Eqs. (13) and (16),

It is easily shown, using the assumed statistics, that as N → ∞, with σ2s(m) » σ2

n over m0 ≤ m < m1. If we use

with probability one as N → ∞; in practice a large but finite N suffices.

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in (17), we get [from (7) and (17)]

In other words, assuming (18) is an equality, Foord et al. should have derived (SNRp o w e r)m e a s from (SNRv o l t)m e a s via

A quick calculation gives

for the correction factor. In summary, Sect. IV.A of Foord et al.1 mistreats the

Rayleigh distributed outputs of the linear detector system they employ by assuming [compare their Eq. (5)] a result valid for the exponentially distributed outputs of the square-law detector system beloved by theorists.

Preparation of this paper was supported by U.S. Army Research Office contract DAAG29-84-K-0095.

References 1. R. Foord, R. Jones, J. M. Vaughan, and D. V. Willetts, "Precise

Comparison of Experimental and Theoretical SNRs in CO2 Laser Heterodyne Systems," Appl. Opt. 22, 3787 (1983).

2. Squares of Rayleigh variates are exponentially distributed, so that this statement is consistent with the remarks preceding Eq. (10).

1 May 1985 / Vol. 24, No. 9 / APPLIED OPTICS 1247