WAPP Signal Processing Page

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The Wideband Arecibo Pulsar Processor Signal Processing Page

WAPP - Signal Processing Page

Introduction
Raw Correlation
Example

Introduction

The WAPP uses custom digital hardware to create estimates of correlation functions from sampled data.
Equation Correlation

The digital hardware used to estimate correlation function uses quantization to represent input
Eqn Quantized Correlation Estimate

After creating an estimate of input correlation functions, the Wiener-Khintchine Theorem is applied to create estimates of input Power spectral density (PSD) and Stokes parameters
Eqn Wiener Khintchine

In practice, the measured correlation function proudced by the digitial hardware is a biased¹estimator of the actual correlation function. This is due to coarse quantization of the sampled data values. A correction to the raw correlation must be applied. Also, the raw correlation values written to disk by the WAPP have offset terms which must be removed. This discussion will outline these issues and discuss methods of dealing with their effects.

¹ Baised in the sense that the measured correlation has a one-to-one correspondence to the actual correlation function, but there relationship is not linear

Raw Correlation

The raw data produced by the WAPP are unsigned integers that represent estimates of correlation functions. This data requires

Counter Offset

The multiplication applied by the NAIC correlator chips produces only positive values {0,1,2}. This complication simplifies the post-multipication storage into simple ripple counters. The offset produces

Correlation Offet Correction
Mode	Separate IFs (Normal)	Summed IFs (Search)
3 Level
9 Level
	Where p(x) = Correlation Coefficient {-1<p<1} Crate = Correlation Sample Rate (microseconds) BW = Bandwidth (MHz)

Quantization Correction (Van Vleck Correction)

The custom designed CMOS chip used to measure correlation fucntions uses a very coarse quantization of the input data and subsequent multipication. The standard mode of the chip uses only 3 levels (or 1.5 bits), while an special mode can combine produces from mutliple chip to create a 9 level respresentation of the input data. The resulting correlation function has some addtional noise due to quantization that reduces SNR, hower this effect is modest and deemed a worth tradeoff. However, the

Plot of Quant Corr, 3 Level

Sensitivity

The loss associated with 3-level and 9-level correlation is fairly modest. However, it is affected by the signal level relative to the threshold voltages. Consider the extreme case where the power level is much larger than the threshold levels. In this case, virtually the only values that will be produced by the sampler are the maximum and minimum level and the sampler approaches the performance of a 2 level (single bit) correlator. Conversely, if the level drops signficantly below the

For the 3-level, we correlator SNR of the measured correlation fucntion, as compared to a sampler with inifinite quantizition is 81%, but this value only holds when the power level is ideal. The full relationship is expressed by the following formula:

Power Estimation

The total power in the input band is estimated using from the autocorrelation function at delay =0 (zerolag). If the value produced by the WAPP was a perfect measurement of input correlation, the zerolag would be directly proportional to input power. However, like the quantization correction discussed above, the zerolag must be manipulated to produce a nonbiased estiamte of input power.
Power 3 Level Correction

Power Spectra

The FFTW subroutine library (available at http://www.fftw.org) was used to implement the fourier transfer to create power spectrum. In Snap, a Hamming window is applied as a matter of course.

Example

References

[1]Weinreb, Sander, "A Digital Spectral Analysis Technique and its Application to Radio Astronomy," MIT Technical Report #412, 1963
[2] Hagen, J.B. and D.T. Farley, "Digital-correlator techniques in radio science," Radio Science, 8, p775,1973
[3] Thompson A.R. et al, "Interferometry and Synthesis in Radio Astronomy", John Wiley and Sons, New York, 1986
[4] Dowd, Andrew, "Sampler and Quanization considerations for the Onsala Hybrid Spectrometer", Onsala Space Observatory Internal Memo., Sept. 1990

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