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

Product of Rademacher Functions produce a Walsh function

-4

0

4

8

12

16

Time (1 period)

Am

plitu

de R(1,t)

R(2,t)

R(3,t)

PAL(7,t)

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 (2N-1) rejects orders of drift up to (t N - 1)

Dig.

DTS

Dig.

DTS

90

180

First mixer1st LO

Correlator+-

ALMA WALSH MODULATION

Walshgenerators

180

90

Sidebandseparation

Spurreject

Antenna #1 Antenna #2

• 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.

Self product of WAL(5,t) w ith a time slip

-4

0

4

8

12

Time (1 period)

Am

plit

ud

e

WAL(5,t)

WAL(5,t) with a timedelay

Product without slip

Product with slip

Self product of WAL(5,t) with itself, no time slip

-4

0

4

8

12

Time (1 period)

Am

plit

ud

e WAL(5,t)

WAL(5,t)

Product w ithout slip

Sensitivity loss

If a Walsh-modulated signal is demodulated correctly, there isno 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 demodulation Timing error

Product

Loss of sensitivity, %, for timing offset of 1% of shortest bit length

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

0 20 40 60 80 100 120 140

WAL(N,t) (N~SEQUENCY * 2)

Sen

siti

vity

lo

ss (

%)

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

Crosstalk, or Immunity to Correlated Spurious Signals

Product of WAL(5,t) with WAL(6,t)

-4

0

4

8

12

Time (1 period)

Am

plit

ud

e WAL(5,t)

WAL(6,t)

Product

Product of WAL(5,t) with WAL(6,t) shifted

-4

0

4

8

12

Time (1 period)

Am

plitu

de

WAL(5,t)

WAL(6,t) shifted

Product

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

A matrix of cross-product amplitudesFor 128-element Walsh function set.

In WAL order

Amplitudes are shown as0 dB, 0 dB to -20 dB, -20 to -30 dB,with 1% timing offset.

Weaker than -30 dB is left blank.

NOT ALL CROSS-PRODUCTSWITH A TIMING ERRORGIVE CROSS-TALK

ODD * EVEN always orthogonalODD * ODD neverEVEN * EVEN sometimes

RSS crosstalk power

0.00%

0.20%

0.40%

0.60%

0.80%

1.00%

1.20%

0 20 40 60 80 100 120 140

WAL index

RS

S C

rossta

lk p

ow

ers

(%

)

Crosstalk: The RSS Cross-talk amplitude of a given Walsh function,when that function is multiplied in turn by all other different functions in a128-function Walsh set.

Loss of sensitivity, %, for timing offset of 1% of shortest bit length

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

0 20 40 60 80 100 120 140

WAL(N,t) (N~SEQUENCY * 2)

Sen

siti

vity

lo

ss (

%)

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:

NN r( ) r

For N=128, r=64, this is 2.395 1037

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.

64 Antennas: Relative probability of given level of Xtalk occurring

00.20.40.60.8

11.2

3.2 3.7 4.2

RSS crosstalk, %, for 1% timing shift

Re

lati

ve

pro

ba

bili

tyRel. Probability ofgiven RSScrosstalk value

Gauss fit

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%.

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.)

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

Relative probability of given level of Xtalk occurring

0

0.2

0.4

0.6

0.8

1

1.2

2.00 2.50 3.00 3.50 4.00 4.50RSS crosstalk, %, for 1% timing shift

Re

lati

ve

pro

ba

bili

ty Rel. Probability ofgiven RSScrosstalk value

Gauss fit

From different sets of 64 functions, chosen at random from the original 128-functionWalsh set, relative occurrence of the value of cumulative RSS of crosstalk summed over all possible cross-products of each set.

64 Antennas: Relative probability of given level of Xtalk occurring

00.20.40.60.8

11.2

3.2 3.7 4.2

RSS crosstalk, %, for 1% timing shift

Re

lati

ve

pro

ba

bili

ty

Rel. Probability ofgiven RSScrosstalk value

Gauss fit

Preselected for max # zero cross-products

Chosen randomly

Criteria for choosing the subset of 64 functions from the total set of 128 Walsh

functions

RSSCrosstalkLevel (1% time slip)

Number of

zero products

Total # cross-products

(excludingself-products)

Total Sensitivity Loss (1% time slip)

The set of functions:

WAL indices

Randomly chosen, no optimization, most probable result

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

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

(1913)