#### 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
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%
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
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
crosstalk value
Rel. Probability ofGauss fit
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
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)
```