#### Transcript Slide 1

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) …] 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) Product of Rade mache r Functions produce a Walsh function 16 12 Amplitude • • R(1,t) 8 R(2,t) R(3,t) 4 PAL(7,t) 0 -4 Tim e (1 period) 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 WAL12,t) From Beauchamp, “Walsh Functions and their Applications” Dicke Switching or Beam Switching ON source OFF source PAL(1,T) PAL(3,T) PAL(7,T) off – on – off – on – off – on – off – on - Rejects DC term off – on – on – off – off – on – on – off - Rejects DC + linear drift off – on – on – off – on – off – off – on - Rejects DC + linear + quadratic drifts PAL index (2N-1) rejects orders of drift up to (t N - 1) ALMA WALSH MODULATION 1st First mixer LO 180 180 Walsh generators Spur reject 90 90 Sideband separation Dig. Dig. + - Antenna #1 DTS Correlator DTS Antenna #2 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. Sensitivity loss If a Walsh-modulated signal is demodulated correctly, the no loss of signal (Left) If a Walsh-modulated signal is demodulated with a timing error, there is loss of signal (loss of “coherence”) (Ri Self product of WAL(5,t) w ith a tim e slip Self product of WAL(5,t) with itself, no time slip 12 12 8 WAL(5,t) WAL(5,t) 4 WAL(5,t) Product w ithout slip Amplitude Amplitude 8 WAL(5,t) with a time delay 4 Product without slip Product with slip Product 0 0 -4 -4 Tim e (1 period) Correct demodulation Tim e (1 period) Timing error Loss of sensitivity, % , for timing offset of 1% of shortest bit length 2.50% Sensitivity loss (%) 2.00% 1.50% 1.00% 0.50% 0.00% 0 20 40 60 80 100 120 140 WAL(N,t) (N~SEQUENCY * 2) Loss of Sensitivity for a timing offset of 1% of the shortest Walsh bit length Crosstalk, or Immunity to Correlated Spurious Signals WAL(5,t)*[WAL(6,t) shifted] WAL(5,t)*WAL(6,t) No Crosstalk Crosstalk. Spurious signals not Product of WAL(5,t) with WAL(6,t) shifted suppressed Product 12 12 8 8 WAL(5,t) 4 WAL(6,t) Product Amplitude Amplitude Product of WAL(5,t) with WAL(6,t) WAL(5,t) 4 WAL(6,t) shifted Product 0 0 -4 -4 Tim e (1 period) Product averages to zero Time (1 period) Product does not average to zero 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. NOT ALLthan CROSS-PRODUCTS Weaker -30 dB is left blank. WITH A TIMING ERROR GIVE CROSS-TALK ODD * EVEN always orthogonal ODD * ODD never EVEN * EVEN sometimes Loss of sensitivity, % , for timing offset of 1% of shortest bit length 2.50% RSS crosstalk power Sensitivity loss (%) 2.00% 1.20% 1.50% 1.00% 0.50% 1.00% 0.00% 0 20 40 60 80 RSS Crosstalk powers (%) WAL(N,t) (N~SEQUENCY * 2) 0.80% 0.60% 0.40% 0.20% 0.00% 0 20 40 60 80 100 120 WAL index 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. 140 100 120 140 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: N ( N r) r For N=128, r=64, this is 37 2 .39 5 1 0 Optimization strategy 1. Choose r functions at random from N, with no duplicates. Typically for ALMA: N=128, r= # antennas = 64 2. Vary each of the r functions within that chosen set, one by one, to optimize the property of the complete set. 3. Repeat, with a different starting seed. 10 6 to 10 7 tries. 4. Look at the statistics of the optimized sets of r functions. Relative probability 64 Antennas: Relative probability of given level of Xtalk occurring 1.2 1 0.8 0.6 0.4 0.2 0 Rel. Probability of given RSS crosstalk value Gauss fit 3.2 3.7 4.2 RSS crosstalk, % , for 1% timing shift 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%. A possible choice of functions for 50, or 64 antennas, from a 128-function set, chosen to: 1. Maximize number of zero cross-products (1621/2016) 2. Then minimize the RSS cross-product amplitude (3.4%) 0 1 2 3 4 7 8 11 12 15 16 22 23 24 31 32 34 35 37 39 40 44 47 48 51 52 55 56 59 61 62 63 64 67 69 71 72 79 80 81 84 87 88 89 91 94 95 96 103 104 111 112 114 115 116 119 120 121 122 123 124 125 126 127 - - ( 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 64 Antennas: Relative probability of given level of Xtalk occurring Relative probability of given level of Xtalk occurring 1.2 1 Relative probability Relative probability Preselected for max # zero cross-products Rel. Probability of Chosen randomly 0.8 0.6 1.2 1 0.8 0.6 0.4 0.2 0 given RSS crosstalk value Rel. Probability ofGauss fit given RSS crosstalk value 3.2 3.7 RSS crosstalk, % , for 1% timing shift 0.4 0.2 0 2.00 4.2 Gauss fit 2.50 3.00 3.50 4.00 4.50 RSS crosstalk, % , for 1% timing shift 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. Criteria for choosing the subset of 64 functions from the total set of 128 Walsh functions Randomly chosen, no optimization, most probable result RSS Crosstalk Level (1% time slip) Number of zero products Total # crossproducts (excluding self-products) Total Sensitivity Loss (1% time slip) The set of functions: WAL indices 3.25% 1362 2016 1% Most subsets of 64 functions randomly chosen from 0-127 Random seed, selecting only sets having the maximum number of zero cross-products 3.79% 1621 2016 1% (Not useful) Random seed, then optimize for max number of zero products, then minimize RSS crosstalk 3.41% 1621 2016 1% See Table 1 Random seed, then optimize only for max number of zero products. Worst crosstalk could be: 4.3% 1621 2016 1% (Not useful) Lowest possible sensitivity loss, ignoring crosstalk 2.31% 1365 2016 0.50% WAL 0-63 Worst possible sensitivity loss, ignoring crosstalk 2.31% 1365 2016 1.50% WAL 64-127 Random seed, then optimize for minimum RSS crosstalk, then minimize sensitivity loss 1.82% 1366 2016 0.80% WAL indices 0-31,47-63,113-127 The magic set of Walsh functions for 64 ALMA antennas: WAL indices 0-31, 47-63, 113-127 Thanks for listening. THE END (1913)