To: Don Campbell & Mike Davis file: 9level

From: Jon Hagen

Date: 5-10-96, edited 9-1-98

Subject: Optimum signal levels for a. decoding , b. low- correlation, and c. =1 correlation (power estimation) when using coarse quantization.

Summary: For any quantization scheme, the optimum signal level for estimating the low- correlation function is also the optimum signal level for weak signal decoding. For the particular case of 9-level processing vs. analog processing, decoding requires a relative integration time of at least 1.032 for equivalent statistics. Low- correlation requires a relative integration time of 1.064 while =1 correlation (power measurement via the zero lag) requires a relative integration time of at least 1.103. In both situations the 9-level quantization effectively multiplies the input signal-to-noise ratio by 1/1.032, a reduction of 0.14 dB. This best 9-level performance is obtained when the rms input voltage is equal to 1/(2 0.27) or 1.85 times the spacing between the digitizer decision levels. (This compares with 1/(2 0.6) for 3-level sampling). For 9-level sampling, the optimum signal level for power measurement (zero lag) nearly the same as the optimum signal level for small correlation.

Quantization schemes

Figure 1 shows the digitizer transfer function, g(V) for nine-level scheme. The digitizer decision levels are all separated by an amount 2 while the output codes are all separated by unity. This symmetric arrangement can be made for any odd number of levels. Figure 2 shows an 8 level scheme to illustrate digitizers with an even number of levels. When every output code is used, as in Figure 2, the output step at zero volts is twice the other output steps. It can be made symmetric by using only odd numbers as output codes. (The choice of output codes, however, is not usually arbitrary but is dictated by the design of the arithmetic unit used to multiply (in the case of correlation) or to add (in the case of decoding).

Figure 3 shows the familiar three-level scheme, which is the simplest scheme with an odd number of levels. Figure 4 shows the general scheme; the special cases in Figures 1, 2, and 3 can all be represented by Figure 4. The three-level scheme, for example is represented by Figure 4 if m0=0, m1=1, V1 = , and V2= ∞.

Analysis

We can analyze the most general case (Figure 4) and then carry out numerical calculations for any particular case. We first consider decoding, where the variance for any quantization scheme is a function of a sum and an integral. (The variance for correlation will turn out to be proportional to the square of the variance for decoding).

I. Decoding

For decoding we use a transversal filter implemented as a set of arithmetic units, each made of a sign-flipper and an accumulator. The arithmetic unit for a given range flips signs in just the right sequence to undo the sign reversals on an encoded signal arriving from a point target at that particular range. The input signal v(t) is the sum of a signal voltage, ±vs, and a random noise voltage, vn. The digitizer output is g(v(t)). To calculate the VanVleck correction and to find the efficiency with respect to analog processing we can consider the decoding operation to be the averaging of a non-coded signal, i.e. a dc component, vs, in the presence of Gaussian noise. The signal, vs , can be measured precisely even with coarse quantization because it is dithered by the noise. In practice, noise is unavoidable so the averaging required with dithering, must be done in any event. All the quantization schemes considered are symmetric about zero so when no signal is present, the noise averages to zero. When a small signal is present the average of the digitizer output will be proportional to the signal voltage, vs. The average output can therefore be written as <g> = vs d<g>/dvs. Assuming that vs is much smaller than the rms voltage, , we can write the input distribution function as

The average (time average) output value, <g>, from the digitizer is the integral of g(v)F(v) where g(v) is the digitizer transfer function of Figure 4. Carrying out this integral we find

where the voltage distribution function is given by

To find the variance in the estimator of vs we must find the mean square of the digitizer output. Since we have assumed a low signal-to-noise ratio we can simply find the variance of the digitizer output when the input is noise alone. Since <g>=0, we can write:

where N is the number of samples accumulated to form the estimate. The variance in the estimate of vs is given by

This variance can be calculated by using the expression d<g>/dvs from Equation 2.

II. Correlation (when the correlation coefficient is small)

Price's Theorem is a differential equation relating <g(v)g(v')> to the true correlation function <vv'> [In the shorthand used here v denotes v(t) and v' denotes v(t+ ) Likewise, g denotes g(t) while g' denotes g(t+ ) ]. Price's theorem can be written as

where is the normalized correlation coefficient, = <vv'>/ <v2>. Using the general quantization scheme of Figure 4 and considering the small case, we find from Equation 6 that

which is just 2 times the square of d<g>/dvs as given by Equation 2. The variance of the estimator of gg', for the small- case, is given by

When the correlation is low, the expectation of g(v)g(v') is proportional to the correlation function <vv'>. In this case the variance in the estimate of vv' is given by

which, for small , becomes

Note that, except for the 1/N factor, the variance of vv' (correlation) is just the square of the variance of vs, the variance for decoding, given by Equation 5.

III. Correlation (zero lag)

The zero lag provides a measurement of power, i.e. of 2. The relative variance in power as calculated from an estimate of the zero lag is just

where variance of g2 and the derivative of <g2> with respect to 2 are easily shown to be given by

and

Equivalent signal-to-noise ratios

In practice the quantity to be measured is pulled out of the noise by subtracting "offs" from "ons." In this case, the normalized variance will depend not only on N, the number of samples used to form an "on" or an "off" but also on s/n, the ratio of signal power to noise power.

Signal-to-noise in Decoding

Here the quantity being estimated is vs . Its normalized variance is given by

Once again assuming a low signal-to-noise ratio we find that

where v is either voff or von. The normalized variance in the estimated signal voltage is equal to the normalized variance of the signal plus noise voltage divided by the signal-to-noise ratio.

Signal-to-noise in Correlation

Now the quantity being estimated is s = <vs v's>/ <vs 2> . Using the same kind of analysis as above we find that the normalized variance of the estimate of s is equal to the normalized variance of vv' divided by the square of the signal-to-noise ratio.

VanVleck Corrections

Quantization schemes with many bits approach the multi-bit (analog processing) limit; the correlation function of the digitizer output becomes exactly proportional to the true correlation function <vv'>. At the other extreme, the correlated output of a two-level sampler <g2(v)g2(v')> , is given by 2/ arcsin( ). The form of <g(v)g(v')> can be readily found using Price's Theorem. Nine-level sampling produces such a linear relation between <g9(v)g9(v')> and <(vv')> that, in practice, no correction should be needed.

Numerical calculations

Sheet I of the attached Mathcad calculations shows the variance when correlating the output of a 3,5,7, or 9 level digitizer. The variance for decoding is just the square root of the correlation variance. The dynamic range of any of these schemes can be read from the graph. For example, if an integration time of 1.6 (relative to analog processing) is allowed, the range of / for 3-level sampling is about 20log(.8/.45) = 5dB. With the same criterion, the range for 9-level sampling is about 20log(.9/.07) = 22dB. For all of these odd-numbered quantization schemes, the large signal limit (small ) is equivalent to one-bit sampling. At the small signal limit, however, the variance increases without limit. This contrasts with even- numbered schemes which become equivalent to one-bit sampling at both the large signal and small signal limits.

Sheet II shows the variance (relative to analog processing) for the zero lag (power measurement). Note that the optimum threshold level for 9-level power measurement is virtually identical to the optimum threshold for 9-level small- correlation measurement. The optimum thresholds are quite different for 3-level sampling. The 9-level scheme also provides a much greater dynamic range.

Sheet III shows the VanVleck correction for correlation. Note that the 9-level correlation function is very linear over the full range, 0< <1.

Sheet IV shows the VanVleck correction for the zero lag (the power estimator).

Sheet V shows the VanVleck correction for decoding. In both the 3 and 9 level schemes the curve is quite linear and, in practice, no correction should be necessary.