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How to Choose a Walsh Function

How to Choose a Walsh Function. Darrel Emerson NRAO, Tucson. (1913). What’s a Walsh Function?. A set of orthogonal functions Can be made by multiplying together selected square waves of frequency 1, 2, 4, 8,16 … [i.e. Rademacher functions R(1,t), R(2,t), R(3,t) R(4,t), R(5,t) …]

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How to Choose a Walsh Function

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  1. How to Choose a Walsh Function Darrel Emerson NRAO, Tucson

  2. (1913)

  3. What’s a Walsh Function? • A set of orthogonal functions • Can be made by multiplying together selected square waves of frequency 1, 2, 4, 8,16 … [i.e. Rademacher functions R(1,t), R(2,t), R(3,t) R(4,t), R(5,t) …] • The Walsh Paley (PAL) index is formed by the sum of the square-wave indices of the Rademacher functions E.g. R(1,t)*R(2,t)*R(3,t) is a product of frequencies 1, 2 and 4=PAL(7,t)

  4. Ordering Walsh Functions • Natural or Paley order: e.g. product of square waves of frequencies 1, 2 & 4 (Rademacher functions 1,2 & 3) =PAL(7,t) • WAL(n,t): n=number of zero crossings in a period. Note PAL(7,t)=WAL(5,t) • Sequency: half the number of zero crossings in a period: • CAL or SAL. (Strong analogy with COSINE and SINE functions.) • Note WAL(5,t)=SAL(3,t), WAL(6,t)=CAL(3,t) • Mathematicians usually prefer PAL ordering. • For Communications and Signal Processing work, Sequency is usually more convenient. • For ALMA, sometimes PAL, sometimes WAL is most convenient

  5. WAL12,t) From Beauchamp, “Walsh Functions and their Applications”

  6. Dicke Switching or Beam Switching ON source OFF source PAL(1,T) Rejects DC term off – on – off – on – off – on – off – on - PAL(3,T) Rejects DC + linear drift off – on – on – off – off – on – on – off - PAL(7,T) Rejects DC + linear + quadratic drifts off – on – on – off – on – off – off – on - PAL index (2N-1) rejects orders of drift up to (t N - 1)

  7. ALMA WALSH MODULATION First mixer 1st LO 180 180 Walsh generators Spur reject 90 90 Sideband separation Dig. Dig. DTS DTS Correlator + - Antenna #1 Antenna #2

  8. TIMING ERRORS • If there is a timing offset between Walsh modulation and demodulation, there is both a loss of signal amplitude and a loss of orthogonality. Timing offsets at some level are inevitable, & can arise from: • Electronic propagation delays, PLL time constants, & software latency • Differential delays giving spectral resolution in any correlator (XF or FX) Mitigation of effect of Walsh timing errors is the subject of the remainder of this talk.

  9. Sensitivity loss If a Walsh-modulated signal is demodulated correctly, there is no loss of signal (Left) If a Walsh-modulated signal is demodulated with a timing error, there is loss of signal (loss of “coherence”) (Right) Product  Correct demodulation Timing error

  10. Loss of Sensitivity for a timing offset of 1% of the shortest Walsh bit length

  11. Crosstalk, or Immunity to Correlated Spurious Signals WAL(5,t)*[WAL(6,t) shifted] Crosstalk. Spurious signals not suppressed WAL(5,t)*WAL(6,t) No Crosstalk Product  Product averages to zero Product does not average to zero

  12. A matrix of cross-product amplitudes For 128-element Walsh function set. In WAL order Amplitudes are shown as 0 dB, 0 dB to -20 dB, -20 to -30 dB, with 1% timing offset. Weaker than -30 dB is left blank. NOT ALL CROSS-PRODUCTS WITH A TIMING ERROR GIVE CROSS-TALK ODD * EVEN always orthogonal ODD * ODD never EVEN * EVEN sometimes

  13. Crosstalk: The RSS Cross-talk amplitude of a given Walsh function, when that function is multiplied in turn by all other different functions in a 128-function Walsh set.

  14. Finding a good set of functions • It is not feasible to try all possibilities. The number of ways of choosing r separate items from a set of N, where order is not important, is given by: For N=128, r=64, this is Optimization strategy • Choose r functions at random from N, with no duplicates. Typically for ALMA: N=128, r= # antennas = 64 • Vary each of the r functions within that chosen set, one by one, to optimize the property of the complete set. • Repeat, with a different starting seed. 10 6 to 10 7 tries. • Look at the statistics of the optimized sets of r functions.

  15. From sets of 64 functions selected from 128 to give the maximum count (=1621/2016) of zero cross-products. The relative occurrence of a given level of RSS crosstalk between all cross-products of that set, with 1% timing offset Most likely level of RSS cross-talk 3.79%. Lowest 3.4%.

  16. A possible choice of functions for 50, or 64 antennas, from a 128-function set, chosen to: • Maximize number of zero cross-products (1621/2016) • Then minimize the RSS cross-product amplitude (3.4%) ( For the best 50 functions, omit those given in bold font.) However, maximizing the number of zero cross-products does not lead to the best result

  17. Preselected for max # zero cross-products Chosen randomly From different sets of 64 functions, chosen at random from the original 128-function Walsh set, relative occurrence of the value of cumulative RSS of crosstalk summed over all possible cross-products of each set.

  18. WAL indices 0-31, 47-63, 113-127 The magic set of Walsh functions for 64 ALMA antennas: Thanks for listening. T H E E N D

  19. (1913)

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