How to Choose a Walsh Function Darrel Emerson NRAO, Tucson

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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 R(1,t)*R(2,t)*R(3,t) is a product of frequencies 1, 2 and 4 =PAL(7,t) E.g.

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

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

What can go wrong? If the signal amplitude varies during the time T of a complete period of a function PAL(n,T), there is a loss of orthogonality. –Possible Solution: Choose functions with the largest number of component Rademacher products. E.g. PAL(13,T), =R(1,T)*R(3,T)*R(4,T), a product of 3 square waves, is immune to drifts of the form A.t 0 + B.t 1 + C. t 2. (t=time, A, B & C constants.) Higher order (i.e. faster) Rademacher terms are better. The max order of polynomial drift eliminated is equal to the number of component products of Rademacher functions, -1. We want signals, modulated by different functions, to remain orthogonal. Possible problems: 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.

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) TIMING ERRORS 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, 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) Correct demodulationTiming error Product 

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

Crosstalk, or Immunity to Correlated Spurious Signals Note the symmetries: Second halves of WAL(5,t) & (6,t) are negative copies of the first halves Second half of the product is a copy of the first half. WAL(5,t) & (6,t) are PAL(7,t) & (5,t). Both include R(1,t) as a factor.

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

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.

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 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 to 10 7 tries. 4.Look at the statistics of the optimized sets of r functions.

A function with an odd PAL index remains orthogonal to a function with an even PAL index, even in the presence of a relative time slip. A pair of functions that both have odd PAL indices never remain orthogonal after a relative time slip A pair of functions that both have even PAL indices, may or may not remain orthogonal with a time slip. –With any even PAL function, the first half is a repeat of the second half. –Look at one half of onto even functions, to see if the half-function itself is odd or even. If the half-functions remain orthogonal, then the entire function remains orthogonal. –If the two half-functions are both even, then sub-divide them again. –Continue subdividing until we’ve found whether the functions remain orthogonal. Crosstalk, or Immunity to Correlated Spurious Signals ODD * EVEN always orthogonal ODD * ODD never EVEN * EVEN sometimes

Half of all products are a mix of odd & even indices. These remain orthogonal. So, N 2 /2 products accounted for. Odd-odd pairs are never orthogonal: N 2 /4 of all possible products can be rejected. Even-even pairs may or may not remain orthogonal. (N 2 /4 not yet accounted for.) Divide these into half-functions, and apply above crteria to the half-functions One half of the N 2 /4 will satisfy the odd-even criterion so remain orthogonal. One quarter of the N 2 /4 can be rejected by the odd-odd rule, leaving one quarter of the N 2 /4 to be considered further. From these N 2 /16 remaining functions, apply the above steps. From successive steps, the # of functions found to remain orthogonal: N 2 /2 + N 2 /8 + N 2 /32 + N 2 /128 … = 2. N 2 /3. Crosstalk, or Immunity to Correlated Spurious Signals What fraction of all Walsh function products remain orthogonal in the presence of a time shift?

The relative occurrence of a given count of zero-crosstalk products, in ~10 5 randomly chosen sets of 64 functions, taken from a full set of 128 Walsh functions. The most likely count of zero-crosstalk products is 1362 (out of 2016 possible products) The best set found has 1621 zero cross-products. The worst set found has <1000 zero cross-products.

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%.

( For the best 50 functions, omit those given in bold font.) 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%) However, maximizing the number of zero cross-products does not lead to the best result

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. Preselected for max # zero cross-products Chosen randomly

Relative occurrence of individual Walsh functions in the best 0.25% of 64-element sets originally chosen at random from 128 functions

Graphical representation of all choices of 64 sets of functions blocked into groups each of 8 functions, which give the minimum value of RSS crosstalk. Note the universal appearance of the lowest, and of the highest, sequencies. The relative occurrence of a given Walsh sequency in the 64 sets shown above.

Criteria for choosing the subset of 64 functions from the total set of 128 Walsh functions RSS Crosstalk Level (1% time slip) Number of zero products Total # cross- products (excluding self-products) Total Sensitivity Loss (1% time slip) The set of functions: WAL indices Randomly chosen, no optimization, most probable result 3.25% % Most subsets of 64 functions randomly chosen from Random seed, selecting only sets having the maximum number of zero cross-products 3.79% %(Not useful) Random seed, then optimize for max number of zero products, then minimize RSS crosstalk 3.41% %See Table 1 Random seed, then optimize only for max number of zero products. Worst crosstalk could be: 4.3% %(Not useful) Lowest possible sensitivity loss, ignoring crosstalk 2.31% %WAL 0-63 Worst possible sensitivity loss, ignoring crosstalk 2.31% %WAL Random seed, then optimize for minimum RSS crosstalk, then minimize sensitivity loss 1.82% % WAL indices 0-31,47-63,

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

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Criteria for choosing the subset of 64 functions from the total set of 128 Walsh functions RSS Crosstalk level Number of zero products Total number of products (excluding self-products) Total Sensitivity Loss The set of functions: WAL indices Note Randomly chosen, no optimization, most probable result 3.25% % Most subsets of 64 functions randomly chosen from Random seed, selecting only sets having the maximum number of zero cross-products 3.79% %(Not useful) 1,2 Random seed, then optimize for max number of zero products, then minimize RSS crosstalk 3.41% %See Table 1 3 Random seed, then optimize for max number of zero products. Worst crosstalk then could be: 4.3% %(Not useful) 4 Lowest possible sensitivity loss, ignoring crosstalk 2.31% %WAL Worst possible sensitivity loss, ignoring crosstalk 2.31% %WAL Random seed, then optimize for minimum RSS crosstalk, then minimize sensitivity loss 1.82% % WAL indices 0-31,47-63, Table 2: Summary of Results Notes to Table 2: Summary of Results 1.Results given are the most probable result of a random choice of functions. See Figure 4 & Figure 6. 2.See Figure 5. RSS crosstalk is the most probable value within the given criteria. Sensitivity loss is the most probable value. 3.See Figure 5. RSS crosstalk is the lowest found according to the given criteria after ~10 7 tries 4.See Figure 5. Worst RSS crosstalk found in ~10 7 tries. 5.See Figure 6 Sensitivity loss calculated according to section 2. 6.This is the recommended set of functions for M=64 antennas chosen from the full set of N=128 functions. See Section 5.

Crosstalk, or Immunity to Correlated Spurious Signals Note the symmetries: Second halves of WAL(5,t) & (6,t) are negative copies of the first halves Second half of the product is a copy of the first half. WAL(5,t) & (6,t) are PAL(7,t) & (5,t). Both include R(1,t) as a factor.

Crosstalk, or Immunity to Correlated Spurious Signals Product of WAL(5,t) with WAL(7,t) [or PAL(7,t) with PAL(4,t)] Symmetries: WAL(5,t) [PAL(7,t)] second half reverses sign WAL(7,t) [PAL(4,t)] second half repeats. Product: second half reverses sign Product averages to zero: Crosstalk Suppressed even with time slip

Crosstalk, or Immunity to Correlated Spurious Signals Note the symmetries: Second halves of WAL(5,t) & (6,t) are negative copies of the first halves Second half of the product is a copy of the first half. WAL(5,t) & (6,t) are PAL(7,t) & (5,t). Both include R(1,t) as a factor.

What can go wrong? If the signal amplitude varies during the time T of a complete period of a function PAL(n,T), there is a loss of orthogonality. –Possible Solution: Choose functions with the largest number of component Rademacher products. E.g. PAL(13,T), =R(1,T)*R(3,T)*R(4,T), a product of 3 square waves, is immune to drifts of the form A.t 0 + B.t 1 + C. t 2. (t=time, A, B & C constants.) –The max order of polynomial drift eliminated is equal to the number of component products of Rademacher functions, -1. 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 can arise from: –Propagation delays through analog or digital electronics and cables, & software latency –Differential delays are fundamental to spectral resolution in any correlator (XF or FX) Solution : is more complicated, and is the subject of this talk. We want signals, modulated by different functions, to remain orthogonal. Possible problems:

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