– 1 –

Microprocessor-based

Decca Navigator

Hyperbolic Radionavigation Receivers

A.J. Fisher

Department of Computer Science,

University of York,

Heslington, York YO1 5DD, U.K.

fisher@minster.york.ac.uk

ABSTRACT

Hardware and software aspects of the design of various microprocessor-based Decca

Navigator radionavigation receivers are discussed.

Two new hardware designs are presented. The Mark 5 receiver uses the “orange”

channel of the Decca Navigator transmissions, and is based on a design published by

J.D. Last; but unlike Last’s design, it computes a position fix by directly measuring

the phase of the signal, using phase-locked-loop techniques, with software correction

for interrupt latency. The Mark 6 receiver dispenses with the phase-locked loop, and

uses instead digital signal processing techniques to ascertain the phase of the

received Decca Navigator carriers.

Details of several algorithms used in the Mark 5 and Mark 6 receivers are

presented, and their implementation discussed. A phase-determination algorithm is

presented which compensates for local oscillator drift and for skew error due to

movement of the vehicle during the measurement cycle, and which appears to be

particularly resistant to errors due to skywave propagation effects. Automatic

frequency control and phase estimation and adjustment algorithms are presented

which allow the bandwidth of the signal channel to be made very small (± 1.5 Hz);

this improves the performance of the receiver in the presence of noise. The conversion

from hyperbolic to Cartesian coordinates is discussed, and a suitable algorithm is

described.

A further algorithm, based on number-theoretic techniques, is presented for

performing zone identification by phase comparison using the master (6f) and orange

(8.2f) transmissions only.

Test results are presented and analysed, and the performance of both the Mark 5

and Mark 6 receivers is found to compare well with that of older and more

conventional designs.

– 2 –

1. Introduction

Many radionavigation systems rely on the measurement, by a mobile receiver, of

the relative phase of continuous-wave signals broadcast by a number of fixed

transmitters. In systems which determine the location of the mobile receiver in a two-

dimensional Cartesian coordinate space, the loci of points of constant phase difference

are hyperbolae, and systems of this type are consequently known as hyperbolic

radionavigation systems. An introduction to the principles of hyperbolic navigation

systems is given in many text-books on navigation, e.g. Tetley and Calcutt [1], and

many detailed descriptions of various systems have appeared in the literature [2–5].

It is sometimes asserted that the satellite-based Global Positioning System (GPS)

[6] will soon make terrestrial radionavigation systems redundant. Receivers for

terrestrial radionavigation systems are currently much more common, much cheaper,

and above all much simpler than GPS receivers, however, and new applications,

especially for the (terrestrial) Decca Navigator system [2], are continually being

found. Given the large “user base”, and above all the simplicity of the new computer-

based Decca Navigator receivers compared to their GPS counterparts, it is likely that

the Decca Navigator system will remain in widespread use for many years.

Consequently, it is believed that a description of new techniques for making use of

these signals will be of interest.

Since the mobile receiver has in general no absolute time reference, it is necessary

to receive three transmissions in order to obtain a fix in two dimensions. The Decca

Navigator system uses four transmitters: one is designated the master, and the other

three are called the red, green and purple slaves. This arrangement provides a degree

of redundancy. A receiver is able in principle to choose any three of the four

transmitters to obtain a two-dimensional fix, although in practice the choice is limited

to the master and any two of the three slaves.

The transmissions must be multiplexed in some way, so that they can be

distinguished at the receiver. Two forms of multiplexing are in common use in

hyperbolic radionavigation systems: time-division (TDM) and frequency-division

(FDM). The Decca Navigator system was designed as an FDM system. The basic

system transmits signals from each of the four transmitters at different, harmonically-

related, frequencies, viz. 5f, 6f, 8f and 9f, where f is the so-called fundamental

frequency of about 14 kHz (figure 1). A fifth frequency, the so-called orange frequency

(8.2f), is also transmitted. The use of the orange frequency is discussed below.

Most of the older types of receiver are of the type known as the multiplying

receiver. A multiplying receiver receives the transmitted signals simultaneously, and

for each master–slave pair it synthesizes the frequency which is the lowest common

multiple of the master and slave frequencies, by multiplying the frequency of each

signal by an appropriate constant. Phase comparison is performed at the common mul-

tiple frequency. A block diagram of a typical multiplying receiver is shown in figure 2.

The multiplying receiver is quite complex. It has four parallel channels of tuned RF

amplification. The pass-band frequency of each channel must be changed when the

receiver is switched to receive a different Decca “chain” transmitting on a different set

of frequencies (i.e. with a different value of f). More importantly, the phase delay of

each channel must be identical to within a close tolerance, despite possibly large

variations in signal strength between signals received on the different channels.

– 3 –

Master

RedSlave

GreenSlave

PurpleSlave

Rx

6f

..........................................

8f

. . . . . . . . . . . . . . . . . . . . . . . . . .

9f............................................................................................

5f

....

....

....

....

....

....

....

....

....

....

....

....

....

....

....

...

Figure 1. A chain of Decca Navigator transmitters and a mobile receiver

5f × 6

6f × 4

8f × 3

9f × 2

× 5

× 3

30fdiscrim-inator

24fdiscrim-inator

18fdiscrim-inator

purple

red

green

displays

frequencymultipliers

Figure 2. A conventional multiplying receiver for Decca Navigator transmissions

– 4 –

2. The single-channel Decca Navigator receiversof Last et al.

An alternative design by Last and Linsdall [7] uses a single “frequency-agile” RF

channel which is switched as necessary to cover the various frequencies of a

particular chain, under control of a microprocessor. A single-channel receiver has

many advantages over a four-channel receiver; in particular, the matter of the

equalization of phase delays does not arise, and the complexity and cost are much

reduced. The new designs described in this report, with the exception of the zone

identification algorithm presented in section 5, follow Last and Linsdall in using a

single channel.

As has been pointed out by Last et al. [8–10], the design of a Decca Navigator

receiver can be simplified by considering the Decca transmissions as a TDM signal,

rather than as an FDM signal. This is made possible by the so-called multipulse

transmissions. At regular intervals, the five frequencies (5f, 6f, 8f, 8.2f, and 9f) are re-

grouped, and each station in turn transmits all five frequencies at once, for a short

interval (450 ms), during which the other three stations are silent. The cycle of

transmissions repeats in a 20 s cycle. A single-channel receiver tuned to (for example)

the master frequency will therefore receive a signal at 6f in turn from each of the four

transmitters, if it listens for the whole of a 20 s cycle. By performing phase

comparison directly at the master frequency, a fix can be obtained. Frequency

multiplication is not required.

Figure 3 describes the pattern of transmissions from a chain of Decca stations. The

letters M, R, G and P indicate that a frequency is being broadcast by the master, red

slave, green slave, or purple slave, respectively.

Last points out [10] that either the master frequency (6f) or the orange frequency

(8.2f) can be used as the basis of a TDM receiver. There are gaps in the cycle of

transmissions at these two frequencies during which no station is transmitting; this

facilitates the acquisition of the cycle timing. Figure 4 shows a block diagram of a

single-channel 8.2f receiver based on Last’s design. The frequencies shown relate to

the Northumbrian chain, whose fundamental frequency is f = 14075.833 Hz, so the

orange frequency is 8.2f = 115421.83 Hz [11]. A broadly-tuned amplifier passes the

whole band of orange frequencies from all of the Decca chains. A mixer shifts the

band so that the desired orange frequency of 115.422 kHz appears at 2 kHz. A narrow-

band filter, with a bandwidth of ± 12 Hz, selects the desired signal, which is then

amplitude-limited. The 2 kHz square wave output of the limiter is filtered again to

recover the fundamental sine wave component, and fed to a phase comparator. The

reference input of the phase comparator is a square wave at 2 kHz (nominally). The

output of the phase comparator is a train of pulses, whose width is proportional to the

phase difference between the two inputs. These pulses are used to gate a square wave

at 2 MHz into a counter, which is set to count up during the master transmission, and

down during a slave transmission, for equal periods. If the counter starts at zero, at

the end of counting it will contain the phase difference between the master and the

slave signals, in units of thousandths of a cycle. This information is made available to

a microprocessor which controls the instrument. Software processing is required to

compensate for minor constant and long-term frequency error in the 2 kHz local

oscillator.

– 5 –

M M R M G M P M

M R R R G R P R

M G R G G G P G

M P R P G P P P

M M R M G M P supervisory................................................................................... M

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Mastermulti-pulse

Redmulti-pulse

Greenmulti-pulse

Purplemulti-pulse

seconds

6f(master)

8f(red)

9f(green)

5f(purple)

8.2f(orange)

Figure 3. Timing of Decca Navigator transmissions

115.422 kHztuned

amplifier2 kHz limiter 2 kHz

×

phase

comparator

and

gate

counter

U/D

/to micro-

processor

117.422 kHz2 kHz 2 MHz

from micro-

processor

Frequency values shown relate to the Northumbrian chain

Figure 4. Last’s design for a single-channel Decca Navigator receiver

The critical element of the single-channel receiver just described is the narrow-band

filter which precedes the limiter. The filter used by Last was constructed from passive

components, to a 3rd-order maximally-flat design. The –3 dB bandwidth was ± 12 Hz,

and the response in the time domain allowed amplitude settling to –100 dB within

100 ms, a time short compared with the 450 ms duration of the multipulse. The

attenuation at the adjacent channel frequency of 2 kHz ± 102.5 Hz was –49 dB.

This is a stringent specification. To meet the requirements, the filter must be

constructed from close-tolerance components, and it is likely to be both bulky and

expensive. It is possible that, in certain places where the coverage of Decca chains

overlaps, the stop-band attenuation afforded by a passive filter would be inadequate.

This is most likely to be the case if the receiver is close to one or more of the

transmitters of a certain Decca chain, but it is tuned to a different, more distant,

chain; perhaps because one wishes to test the receiver, or to obtain a second fix to

reduce the error uncertainty. In these circumstances, Last prescribes a single-

channel, narrow-band RF filter. But this increases the complexity of a receiver which

is designed to be tunable to different Decca channels, since both the tuned RF

amplifier and the local oscillator must be re-tuned when selecting a different channel.

Since the channels are so closely spaced, it would be difficult to achieve adequate

tracking between the RF filter and the local oscillator, and it would probably be

necessary to resort to a switched set of RF filters, as well as a switched set of local

oscillator crystals or a synthesized local oscillator. This would add further to the

complexity and cost of the design.

– 6 –

3. The Fisher Mark 5 receiver

3.1. Hardware

It seems natural to consider substituting an active filter for the 2 kHz IF filter, and

the most appropriate type appears to be the phase-locked loop (PLL). A PLL in place

of the second 2 kHz filter in figure 4 would permit the bandwidth of the filter to be

made arbitrarily narrow, and the effective stop-band attenuation to be made

arbitrarily large, by careful choice of the loop characteristics. A clean square wave

could be taken from the output of the voltage-controlled oscillator (VCO) of the PLL,

which would be suitable for feeding directly to the phase comparator. The rest of the

design would work unchanged.

Once one considers replacing the IF filter with a PLL, however, other simplifica-

tions suggest themselves. The counter and the phase comparator can be moved inside

the loop (figure 5). The value in the counter at any instant now reflects the

instantaneous phase of the IF signal, and hence of the signal received at the aerial.

By reading the counter, the microprocessor can determine the phase of the received

signal at any instant. There is no need for separate reference signals at 2 kHz and

2 MHz, which are required in the design of figure 4. The counter is a simple up-

counter (or, alternatively, a simple down-counter), which runs continuously; no gating

is required.

115.422 kHztuned

amplifier2.0 kHz limiter

∫ VCOcounter

÷ 3521

∼ 4 MHz/

to micro-

processor

L.O.

114.286 kHz

(4 MHz ÷ 35)

I.F.

1136.1190 Hzf

D= 0 Hz

fC

= 1136.1190 Hz

90°3.3 Hz threshold

lock indication

to micro-

processor

Frequency values shown are for the case when the loop is locked

and the receiver is tuned to the Northumbrian chain

Figure 5. The Fisher Mark 5 receiver

During each cycle of the IF signal, the counter counts from zero up to some

maximum count P–1. The resolution to which the phase can be measured is ± 1 count

in P counts. It is therefore an advantage for P to be as large as possible, up to a limit

imposed by the number of bits in the counter register. This implies that the IF should

be as low as possible.

It is important that the phase noise (jitter) in the local oscillator be kept as low as

possible, since jitter directly affects the accuracy to which the phase of the received

signal can be measured. Consequently, the local oscillator signal is derived by

frequency division from the system’s stable 4 MHz master clock. The local oscillator

frequency is therefore constrained to be 4 MHz / n, where n is an integer. Now the

orange frequencies all lie between f35

and f34

, where f35

= 4 MHz / 35 and

f34

= 4 MHz / 34. A choice of n = 35 for aerial signal frequencies less than (f35

+ f34

) / 2,

– 7 –

and n = 34 for frequencies above this value, therefore maximizes the counter modulus

P. A computer simulation confirmed this fact, and also confirmed that in no case does

P exceed 216 – 1. A 16-bit counter can therefore be used.

The IF filter has a wide bandwidth which passes all of the orange channels, now

shifted in frequency to the range 120–1679 Hz. A two-pole RC low-pass filter is

adequate. A Decca channel is selected by setting the appropriate division modulus in

the counter and in the local oscillator; there is no need to re-tune any filters.

The output from the filter is amplitude-limited. The output from the limiter is a

noisy square wave. Because of the wide bandwidths of the RF and IF filters, the

Decca signal is not visible above the noise when the limited signal is viewed on an

oscilloscope, except for the very strongest signals; but the PLL which follows the

limiter has a narrow bandwidth, and consequently has no difficulty in extracting the

signal.

A quadrature phase detector generates a reliable indication of lock, which can be

read by the microprocessor. The program uses the lock indication to determine the

timing of the 20 s cycle of transmissions, by detecting the breaks in the received

signal.

3.2. Design of the phase-locked loop

All of the equations in this section of the report are taken from Gardner [12].

The performance of the receiver depends critically on the design of the phase-locked

loop (PLL). A PLL has many parameters. In any given application, some parameters

can be optimized only at the expense of other parameters. The final choice of

parameter values is therefore always a compromise, and there is no indisputably and

unambiguously “best” choice of values.

A second-order loop with active filter was chosen, since this is the simplest type of

loop which has the property that, when in lock, there is a negligible phase error

between the input signal and the oscillator signal. This is essential, since the final

position fix depends on the measured oscillator phase, which is taken to be the same

as the phase of the IF signal. For the same reason, the phase jitter in the VCO must

be as small as possible. Furthermore, the loop does not have to accommodate an input

whose frequency varies with time (if one neglects the Doppler shift of the input

signal, which is negligible at these low frequencies). These considerations argue in

favour of making the (single-sided) noise bandwidth BL

of the loop as small as

possible, especially when the input SNR is low.

Gardner [12] distinguishes between two types of acquisition behaviour: “pull-in” and

“lock-in”. If the input frequency is within the pull-in bandwidth of the loop, the VCO

frequency will be gradually pulled towards the input frequency, and the loop will

eventually lock. During acquisition, the loop will in general “skip” several cycles. On

the other hand, if the input frequency is within the narrower lock-in bandwidth,

acquisition will take place within a single cycle, without “skipping”.

In this application, the input frequency is known and constant; so, if the VCO

centre frequency is chosen appropriately, the lock-in bandwidth can be made almost

arbitrarily narrow, and the loop will exhibit lock-in rather than pull-in acquisition.

– 8 –

The duration of the lock-in transient is approximately 1 / ωn

seconds, where ωn

is

the natural frequency of the loop (in rad s–1). In the worst case, the loop must lock

within 350 ms after the signal is applied (see figure 6). This fixes a lower bound for

ωn:

ωn ≥ 1 / 0.35 = 2.857 (1)

On the other hand, the requirement for a small noise bandwidth BL

means that ωn

cannot be made arbitrarily large. The quantities ζ (the loop damping factor), ωn

and

BL

are related by

2 BL = ω

n (ζ + 1 / 4ζ) (2)

A choice of ζ = 0.707, ωn

= 5.66 yields a value of BL

= 3 Hz, which is a good compro-

mise between fast acquisition and narrow noise bandwidth.

The lock-in bandwidth ωL

is given by

ωL = 2ζω

n (3)

which, with the chosen values of ζ and ωn, is 8 rad s–1, or 1.27 Hz, which is 756 ppm

of the highest IF (1679 Hz). The VCO centre frequency must be accurate to within

this tolerance, i.e. 4 MHz ± 3026 Hz in the worst case. This is not difficult to achieve,

provided that variations in ambient temperature are small. The accuracy and stability

of the VCO are considered further in section 3.8.

When the loop is locked, the phase counter output is phase-locked to the IF, and

the difference frequency fD

is zero (figure 5). When the loop is free-running in the

absence of a signal, fD

is not, in general, zero. It is necessary that fD

< ωL

rad s–1, or

else the loop will not lock. The value of fD

when the loop is not in lock was computed

for all possible Decca frequencies, and was found to be less than 0.332 Hz in every

case. This is well within ωL.

The remaining loop parameters are easily chosen. The time constants of the active

loop filter are given by

τ1 = αK

OK

D / ω

n

2 (4)

τ2 = 2ζ / ω

n (5)

where KO

is the VCO gain constant (in units of rad s–1 v–1), KD

is the phase detector

gain factor (in units of v rad–1), and α is a dimensional quantity called the signal

suppression factor, where 0 ≤ α ≤ 1.

The inclusion of a limiter between the IF filter and the phase comparator has the

effect of making the receiver adapt itself to varying signal-to-noise input conditions.

Davenport [13] (quoted by Gardner [12]) shows that the presence of the limiter causes

the effective gain of the loop to be reduced when the input SNR is low. This reduction

is modelled by the signal suppression factor α, which is close to 1 when the SNR is

high, and close to 0 when the SNR is low. Typical values of α were determined by

measuring the response of the quadrature lock detector under actual signal con-

ditions; the minimum value of α encountered in practice was found to be about 0.1.

Using this value of α, and the measured values KO

= 23.57 rad s–1 v–1, KD

= 3.183

– 9 –

v rad–1, gives values of τ1

= 0.2342 s, τ2

= 0.2499 s, which are easily realizable.

When the input SNR is high and α approaches 1, ωn, ω

L, ζ and B

Lwill all increase

to maximum values of ωn

= 17.90, ωL

= 80.2, ζ = 2.24, BL

= 21 Hz. This maximum

value of BL

is well within the channel separation, which is 102.5 Hz [10].

3.3. Correction for interrupt latency

Nine phase readings are taken at intervals of 1.25 s, starting at t = 18.0 s and finish-

ing at t = 8.0 s in the next cycle (figure 6). A complete set of readings comprises six

from the master station, and one from each of the slave stations. The phase values

are obtained by reading the counter “on the fly” at 1.25 s intervals, under interrupt

control.

M M R M G M P supervisory................................................................................... M

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

18.0(0)

19.25(1)

0.5(2)

1.75(3)

3.0(4)

4.25(5)

5.5(6)

6.75(7)

8.0(8)

seconds

8.2f(orange)

phasesamples

Figure 6. Timing of phase samples

A timer generates an interrupt at precise 1.25 s intervals, but there is an unknown

delay before the processor is able to acknowledge and service the interrupt. The

interval between the occurrence of the timer “tick” and the moment when the

processor executes the first instruction of the interrupt routine is called the latency

time. The latency time is variable, but the design of the hardware and software

associated with the microprocessor ensures that it will not exceed 200 processor clock

cycles (25 µs at a clock rate of 8 MHz). A typical value is 50 clock cycles (6.25 µs). The

first step in the processing of the “raw” phase readings is a correction to eliminate the

effect of the variable interrupt latency time.

Since the VCO frequency and the interrupt timer frequency are nearly identical, it

is possible to correct for variable interrupt latency by reading the “exact” time from

the interrupt timer and subtracting this “exact” time from the phase counter reading.

The start of the interrupt routine looks like this. P is the division modulus of the

phase counter.

Interrupt routine:

read timer value t;

read phase counter value p (0 ≤ p < P);

record phase ϕ = (p – t) mod P;

etc

It is assumed that the interrupt timer counts from zero up to 1.25 × 4 × 106 – 1,

driven by a 4 MHz clock derived from the processor clock; that it “rolls over” to zero

after reaching its maximum count; and that an interrupt is generated when the zero

state is entered. The value of t will therefore be a small integer, which represents the

interrupt latency; and, under the assumption that the VCO frequency is exactly equal

to 4 MHz, the value of ϕ will be the phase of the IF signal at the exact moment when

– 10 –

the interrupt was generated, plus a constant which represents the interval between

the execution of the two “read” operations, plus or minus a quantization error of 1

count.

The actual VCO frequency was computed for all possible Decca frequencies, and

was found to lie in the range 4 MHz ± 807 Hz. At the extremes of the range, the error

introduced by assuming that the VCO frequency is exactly 4 MHz will not exceed

807 × 2.5 × 10–5 = 0.02 counts in the maximum possible latency time of 25 µs. This

represents a time interval of 5 ns, which is insignificant when compared with the

other sources of error in the system.

3.4. Normalization of phase readings

We now have a set of nine phase readings which represent the phase of the

received signal at exact 1.25 s intervals. We require to calculate a set of phase

differences, comprising the phase of each slave signal at the instant when the phase

of the master signal is zero.

Let O+ and O– denote addition modulo 1.0 and subtraction modulo 1.0, respectively;

i.e. a O+ b is the fractional part of a + b, and a O– b is the fractional part of a – b. The

fractional part of a real number x is the unique real number y in the range 0 ≤ y < 1

such that x = n + y for some integer n. Note that y is always zero or positive.

For 0 ≤ x < 1, define

sext(x) = x if 0 ≤ x < 0.5,

x – 1 if 0.5 ≤ x < 1. (6)

It is intended that fractional values be stored in fixed-point form, typically in 16

bits, with the implied binary point at the “left-hand” end of the word. Fractional

arithmetic can therefore be implemented as ordinary addition and subtraction of

unsigned integers, and sext(x) is simply the “sign-extended” value of x, i.e. x

interpreted as a twos-complement number.

The phase readings ai

(for i = 0, …, 8) are first scaled by dividing by P, so that they

lie in the range 0 ≤ ai< 1. Let the desired phase differences be ϕ

R, ϕ

G, ϕ

P.

Since the phase readings are taken at equal time intervals, the following

relationships hold among these quantities:

ai = a

0 O+ ib O+ ε

i for i = 1, 2, 3, 5, 7 (7)

a4 = a

0 O+ 4b O+ ε

4 O+ ϕ

R (8)

a6 = a

0 O+ 6b O+ ε

6 O+ ϕ

G (9)

a8 = a

0 O+ 8b O+ ε

8 O+ ϕ

P (10)

where b is a constant which represents the frequency of the signal, i.e. the constant

phase advance from ai

to ai+1

, and the εi

(i = 1, …, 8) are error terms, which in an

ideal world would be zero. This system of equations must be solved to determine ϕR,

ϕG

and ϕP. An arbitrary threshold e = 0.05 is chosen, and a solution is found, if poss-

ible, which satisfies |ηi| < e for each i, where each η

iis a linear combination of the ε

i.

– 11 –

The following algorithm is used.

Phase determination algorithm:

1. d := a0; for i := 0 to 8 do a

i:= a

iO– d od;

2. for j := 1, 3, 5, 7 do

3. d := sext(aj) / j;

4. for i := 1 to 8 do ai:= a

iO– id od

5. od;

6. d := a2; for i := 0 to 8 do a

i:= a

iO– d od;

7. if |sext(ai)| < e for each i = 1, 3, 5, 7 then

8. ϕR

:= a4; ϕ

G:= a

6; ϕ

P:= a

8

9. else error

10. fi

Line 1 first eliminates the arbitrary starting phase, by subtracting a0

from each ai.

At this point, a0

= 0, and a1

= b O+ ε1; in other words a

1= b + ε

1– n, where n is an

unknown integer. We do not know how many complete cycles elapsed between the

measurement of a0

and a1; all we know is the fractional part of the phase difference,

i.e. the phase difference modulo 1.0.

Next, in lines 3–4 (with j = 1), ia1

is subtracted from each ai. At this point,

a0 = a

1 = 0 (11)

ai = ε

i O– iε

1 for i = 2, 3, 5, 7 (12)

a4 = ε

4 O– 4ε

1 O+ ϕ

R (13)

a6 = ε

6 O– 6ε

1 O+ ϕ

G (14)

a8 = ε

8 O– 8ε

1 O+ ϕ

P. (15)

Note that the fact that n is unknown does not matter, since the subtraction in line 4

is modulo 1.0, and the subtraction of an unknown integer has no effect.

In the absence of errors, ai

= 0 for i = 0, 1, 2, 3, 5, 7, and a4

= ϕR, a

6= ϕ

G, and

a8

= ϕP, which are the desired phase differences. The value of ϕ

Pin particular is likely

to be inaccurate, however, since its error term has a contribution from 8ε1; so further

corrections are made by executing lines 3–4 with j successively taking the values 3, 5,

7. For each value of j, a correction d is computed by d := sext(aj) / j. d is evaluated as

a signed (twos-complement) fraction, lying in the range –0.5 / j ≤ d < +0.5 / j. The

magnitude of the correction decreases with increasing j; each successive iteration

applies a finer correction to the ai. After the final iteration, at line 6,

a0 = a

7 = 0 (16)

ai = ζ

i for i = 1, 2, 3, 5 (17)

a4 = ζ

4 O+ ϕ

R (18)

a6 = ζ

6 O+ ϕ

G (19)

a8 = ζ

8 O+ ϕ

P, (20)

where ζi= ε

iO– i (ε

7/ 7).

Note that the only error terms which enter into the value of ai

at this point are the

error εi

belonging to that reading, and ε7. The other error terms can be shown (by a

tedious but straightforward analysis) to cancel out. It is not possible to use the value

of a7

directly, however, by replacing line 2 of the algorithm by for j := 7 do, since the

– 12 –

number of complete cycles in a7

– a0

is unknown; the resulting values of ϕR, ϕ

G, ϕ

P

would each be in error by an unknown integral multiple of 1/7 of a cycle. The correct

result could be obtained by basing the correction on a0, a

1and a

7, by replacing line 2

by for j := 1, 7 do, but the algorithm would then be less tolerant of error, since it is

more likely that the value of id in line 4 will at some point exceed 0.5.

Next, in line 6, a2

is subtracted from all nine phase values, to normalize the phase

of the master multipulse to zero. This is done to simplify further processing. After

this is done, the resulting error ηiin the calculated phase of each multipulse is

ηi = ζ

i O– ζ

2 = ε

i O– ε

2 O– (i–2) ε

7 / 7 (21)

where i = 2, 4, 6, 8 for the master, red, green and purple multipulse (respectively).

Finally, in lines 7–10, each ai

for i = 1, 3, 5, 7 is checked to see that |sext(ai)| < e.

(Note that a0

= a7, and a

2= 0.) If each a

iis within the tolerance, the final values of a

4,

a6, a

8are taken as the desired phase differences.

The threshold of ± 0.05 of a cycle represents a time uncertainty of about 433 ns, or

a distance error of about 65 m on the baseline. If the initial phase readings are totally

uncorrelated, the chance that random phase readings will be mistakenly accepted is 1

in more than 1200. This probability was determined by a computer simulation. The

accuracy of the system is disussed further in section 3.8.

The algorithm will fail if, at any stage, the correction id applied in line 4 exceeds

0.5, in which case “cycle slip” will occur and the results will be incorrect. This will

happen only if |εi| is large for some i, and this mode of failure will be detected by the

threshold test in line 7, since some of the aiwill have large residual errors.

3.5. Coordinate conversion

The final logical step in the operation of the software is the conversion from a

hyperbolic coordinate system (ϕR, ϕ

G, ϕ

P) to the Ordnance Survey coordinate system

[14], which is a Cartesian system (xV, y

V). (The suffix V stands for “vehicle”, i.e. the

Decca receiver.) The software ignores the phase reading which pertains to the slave

which is furthest away, and which can be assumed to be the least accurate; in York

this is the reading from the green slave (which is at Peterhead). The problem

therefore reduces to that of converting from hyperbolic coordinates (ϕ1, ϕ

2) to

Cartesian coordinates (xV, y

V), where ϕ

1, ϕ

2are any pair of ϕ

R, ϕ

G, ϕ

P.

Most location finders give their location in polar coordinates, i.e. latitude and

longitude. Methods for transforming from hyperbolic to polar coordinates are well

known [15]. Conversion from hyperbolic to Cartesian coordinates is in fact

considerably easier, since no account need be taken of the curvature of the Earth. The

basic equations to be solved are [2]:

Φ1 = d

M1 + d

VM – d

V1 (22)

Φ2 = d

M2 + d

VM – d

V2 (23)

where (for n = 1, 2):

– 13 –

dMn

is the distance (in Decca “lanes”*) from the master to the nth slave;

dVn

is the distance from the vehicle to the nth slave;

dVM

is the distance from the vehicle to the master; and

Φn = ϕ

n+ N

n

where Nn

is an integer number of cycles which represents the inherent lane

ambiguity of Decca Navigator systems.

—————————

* A “lane” is the distance which corresponds to a phase shift of one cycle.

From (22), by partial differentiation w.r.t. xV:

∂Φ1 / ∂x

V = ∂d

VM / ∂x

V – ∂d

V1 / ∂x

V (24)

Now

dVM

= √(xVM

2 + yVM

2) (25)

so

2xVM∂d

VM / ∂x

V = ———————– (26)

2√(xVM

2 + yVM

2)

= xVM

/ dVM

(27)

and similarly

∂dV1

/ ∂xV = x

V1 / d

V1 (28)

∂dV2

/ ∂xV = x

V2 / d

V2 (29)

So

∂Φ1 / ∂x

V = x

VM / d

VM – x

V1 / d

V1 = A, say (30)

∂Φ1 / ∂y

V = y

VM / d

VM – y

V1 / d

V1 = B (31)

∂Φ2 / ∂x

V = x

VM / d

VM – x

V2 / d

V2 = C (32)

∂Φ2 / ∂y

V = y

VM / d

VM – y

V2 / d

V2 = D (33)

Now consider how a small change (∆Φ1, ∆Φ

2) in (Φ

1, Φ

2) is related to a

corresponding small change (∆xV, ∆y

V) in (x

V, y

V):

∆Φ1 = A ∆x

V + B ∆y

V (34)

∆Φ2 = C ∆x

V + D ∆y

V (35)

and this pair of simultaneous equations has the solution

∆xV = (D ∆Φ

1 – B ∆Φ

2) / Z (36)

∆yV = (A ∆Φ

2 – C ∆Φ

1) / Z (37)

where

– 14 –

Z = AD – BC. (38)

The following iterative algorithm is used. Given an initial estimate of position

(xV, y

V) and measured phase differences (ϕ

1, ϕ

2), the algorithm computes an updated

(more accurate) position (xV, y

V). All distances are assumed to be in Decca lanes. e is

an arbitrary threshold which in the current implementation is set to 5 / 216.

Coordinate conversion algorithm:

1. while

2. xVM

:= xV – x

M; y

VM:= y

V– y

M;

3. xV1

:= xV – x

1; y

V1:= y

V– y

1;

4. xV2

:= xV – x

2; y

V2:= y

V– y

2;

5. dVM

:= √(xVM

2 + yVM

2);

6. dV1

:= √(xV1

2 + yV1

2);

7. dV2

:= √(xV2

2 + yV2

2);

8. Φ1 := d

M1 + d

VM – d

V1; Φ

2:= d

M2+ d

VM– d

V2;

9. ∆Φ1 := sext (Φ

1 O– ϕ

1); ∆Φ

2:= sext (Φ

2O– ϕ

2);

10. A := xVM

/ dVM

– xV1

/ dV1

; B := yVM

/ dVM

– yV1

/ dV1

;

11. C := xVM

/ dVM

– xV2

/ dV2

; D := yVM

/ dVM

– yV2

/ dV2

;

12. Z := AD – BC;

13. ∆xV

:= (D ∆Φ1

– B ∆Φ2) / Z;

14. ∆yV

:= (A ∆Φ2

– C ∆Φ1) / Z;

15. |∆xV| > e or |∆y

V| > e

16. do

17. xV := x

V – ∆x

V; y

V:= y

V– ∆y

V

18. od

3.6. Use in fast-moving vehicles

Previous Decca Navigator designs have not worked well if the vehicle is moving

quickly, since the effective time at which the multipulse phase readings are taken is

separated by 2.5 s, which corresponds to a large distance when travelling at high

speed, and the resulting “skew” error is significant. For example, a vehicle travelling

at 100 km h–1 travels about 208 m in the 7.5 s between the master multipulse and the

purple multipulse. Assuming (as a first-order approximation) that the velocity is

constant, or (more precisely) that the rate of change of measured phase with time is

constant, the phase-determination algorithm (section 3.4) will reference all phase

readings to the same instant, which is the instant when the first reading a0

is taken.

This is because a constant rate of change of phase with time is identical to a constant

frequency shift—in fact, a Doppler shift. The frequency of the signal does not appear

explicitly anywhere in the algorithm. The algorithm will function even if the input

frequency is a constant offset ∆f removed from the nominal frequency f of the

transmitted signal. The “skew” error is therefore eliminated. Of course, in practice,

the rate of change of phase with time is not constant, so the error will not be

eliminated; but it will be significantly reduced.

3.7. Implementation

A prototype Mark 5 receiver was constructed, which uses a Motorola 68000

microprocessor. The iterative coordinate conversion routine described in section 3.5

converts Decca coordinates into Ordnance Survey coordinates, which are displayed to

a resolution of 10 m on a liquid crystal display. A keypad provides a user interface.

– 15 –

The program is written in the programming language C, and occupies about 9 k bytes

of memory, with a data area of about 4 k bytes. Fixed-point arithmetic is used

throughout: 16-bit for the phase determination algorithm, and 32-bit for the

coordinate conversion algorithm.

The processor clock runs at 8 MHz. However, the computation of a position fix takes

only a small fraction of a second, out of the 20 s available, so it is evident that a

slower clock could be used, which would reduce power consumption. Also, a less

capable, cheaper processor would be adequate. An implementation of the coordinate

conversion algorithm on a Z80, in assembly language using fixed-point arithmetic,

was found to take 6 s per conversion. A 68000 was chosen in preference to a Z80,

however, because it could be easily programmed in C.

3.8. Performance

The Mark 5 receiver was subjected to preliminary tests in and around York

(England), using transmissions from the Northumbrian chain. The performance of the

system was certainly comparable to that of other Decca Navigator systems. In

particular, the algorithm for solving the phase difference equations (section 3.4)

worked flawlessly in practice.

The accuracy of the system is, however, compromised by two error terms. The first

is a random error due to uncertainty of propagation conditions, errors in the phase

measurement process, and slight variations in the phase of the transmitted signals.

On top of this, there is a larger error, of up to ± 1 km, which arises from uncertainty

in the known positions of the transmitting stations. The exact coordinates of the

transmitters are known in principle, but this information is proprietary to Racal-

Decca Marine Navigation Ltd. The program uses the publicly-available coordinates

given in the Admiralty List of Radio Signals [11] to an accuracy of ± 30″ of longitude

and latitude, which over most of the British Isles corresponds to about ± 500 m of

easting and ± 1000 m of northing. The existence of this fixed error has no theoretical

significance, and the error would of course be removed in a “production” receiver.

Random errors in Decca Navigator systems have been shown to have a Gaussian

distribution [2], and it is therefore customary to express the error in a set of fixes as

the radius of a circle, whose centre is the mean position, in which 68.26% of the fixes

lie. For a Gaussian distribution, this value is the standard deviation (SD) of the set.

The stability of the phase of the slave signals relative to the phase of the master

signals is maintained to within a SD of 0.005 cycles (expressed in [2] as 0.01 of a

“mean lane”).

The SD in metres of the errors attributable to transmitter phase instability varies

from location to location, depending on the lane width at the location and the angle of

intersection of the position lines. For a two-slave fix, it is given by [2]

σ = w cosec β √ [(σ1 cosec (γ

1/ 2))2 + (σ

2 cosec (γ

2/ 2))2] (39)

where w is the lane width on the baseline, β is the angle of cut between the position

lines, σ1

(resp. σ2) is the SD of the phase of the first (resp. second) slave signal in

“mean lanes”, and γ1

(resp. γ2) is the angle subtended by the first (resp. second) slave

baseline at the point of observation (see figure 7). The values of these quantities at

the observation site (at York) are: w = 1298.68 m, β = 52˚, σ1

= σ2

= 0.01, γ1

= 20˚,

– 16 –

γ2

= 84˚. Substituting these values into equation (39) gives a predicted SD of 98.05 m

for fixes obtained in the vicinity of York using the red and purple slaves. This figure

represents the best possible performance of the device. The actual performance will be

worse than this, because of propagation anomalies and errors in the receiver.

The measured SD of a large number of readings taken at a fixed point in York was

207.4 m. The readings were all taken during “summer night” conditions, in which the

adverse effects of skywave propagation (unwanted reflections from the ionosphere) are

moderately severe. It is estimated [2] that skywave propagation increases the SD of

the received phase from 0.01 to 0.095 mean lanes at a distance of 300 km from the

transmitter. This is the distance from York to the red slave (which is at Stirling). The

purple slave is much nearer (at Burton Fleming, about 50 km from York), and the

effect of skywave propagation can be neglected. Setting σ1

= 0.095 in equation (39)

gives a predicted SD of 902.0 m, much larger than the observed figure of 207.4 m.

It is conjectured that the superior performance of the prototype receiver is due to

the behaviour of the phase-determination algorithm. Ionospheric reflections tend to

produce phase disturbances which are short compared to the duration of a cycle of

measurements (20 s). The presence of a skywave component will cause errors in the

master phase samples a0–3

, a5

and a7, causing them to differ, and it is likely that one

or more of them will lie outside the threshold of ± 0.05 of a cycle. So, the readings

which are subject to skywave error tend to be rejected, and the readings which are

taken during the intervals when skywave is largely absent tend to be accepted.

Conventional receivers do not have this check, so erroneous phase readings are

simply accepted, and contribute to the error of the fix. There is no conclusive evidence

to support this conjecture; but it is certainly evident from tests that the number of

sets of phase readings which are rejected because they fail the threshold test

increases markedly during “summer night” conditions, and especially during twilight.

A further major practical problem was found to be the instability of the voltage-

controlled oscillator in the phase-locked loop with changes in ambient temperature.

The analysis of the PLL given in section 3.2 showed that the VCO centre frequency

must be 4 MHz ± 3026 Hz over the working temperature range, i.e. ± 756 ppm, in

order for the loop to lock. Assuming (conservatively) a working temperature range of

10 C to 20 C—which rules out, inter alia, operation at sea—the worst-case temperat-

ure dependence of the VCO is found to be ± 75 ppm C–1. By comparison, the variation

of capacitance with temperature of a typical varactor (variable-capacitance diode) is

± 300 ppm C–1 (which implies a frequency variation of ± 150 ppm C–1), and that of the

VCO in a 74HC4046 device is ± 1500 ppm C–1.

Three possible solutions to this problem were considered. An oven-stabilized VCO

would certainly solve the problem, but with the obvious disadvantages of size, weight,

and high current consumption.* A radically new VCO design, based on a 74HC4046

device connected in a self-stabilizing frequency-control loop, was tried [16]. This

approach worked, giving a frequency variation of of only ± 5 ppm C–1. At about this

time it became clear, however, that the filtering action of the PLL could be

implemented entirely digitally, thereby circumventing most problems of stability, and

simplifying the hardware. The result of this train of thought was the Mark 6 receiver,

which is described next.

—————————

* though the oven could double as a means of keeping the skipper’s coffee warm.

– 17 –

MasterAllerdean Greens

Red SlaveStirling

Purple SlaveBurton Fleming

ReceiverYork0 50 km

γR

= 20°

γP

= 84°

Purple SlaveBurton Fleming

ReceiverYork

To master

To red

..

..

..

..

..

..

..

..............

Tangent toRed P.L.

.................

.................

Tangent toPurple P.L.

β = 52°

Figure 7. Map showing the location of transmitters in the Northumbrian chain

and the test site in York

– 18 –

4. The Fisher Mark 6 receiver

4.1. Hardware

Compared with traditional Decca Navigator receivers, the hardware component of

the new design is very simple (figure 8). A broadly-tuned amplifier passes the whole

band of orange frequencies from all of the Decca chains. A mixer shifts the band; the

shifted band lies between 120–1679 Hz. A two-pole low-pass anti-aliasing filter is

followed by a limiter which delivers a noisy square wave to an input port of the

processor (a Motorola 68000) via a one-bit sampling latch. No other signal-processing

hardware is needed.

115.422 kHztuned

amplifier2 kHz limiter latch

to micro-

processor

5 kHzL.O.

114.286 kHz

(4 MHz ÷ 35)

Frequency values shown relate to the Northumbrian chain

Figure 8. The Fisher Mark 6 receiver

from

latch

6.22 Hz

4 pole

sample

1.5 (5.0) Hz

4 pole

I

sample Q

90°

1136.1190

Hz*100 Hz

....

....

....

....

....

....

....

....

...

1 bit

5000 sa/s

....

....

....

....

....

....

....

....

...

16 bit

5000 sa/s

....

....

....

....

....

....

....

....

...

16 bit

100 sa/s

....

....

....

....

....

....

....

....

...

16 bit

100 sa/s

....

....

....

....

....

....

....

....

...

1 bit

5000 sa/s

* for Northumbrian chain

Figure 9. Signal flow (software processing) in the Mark 6 receiver

– 19 –

The use of a limiter and one-bit latch, i.e. a one-bit D-to-A converter, follows the

example of Last (figure 4) and of the Mark 5 receiver (figure 5). There is little to be

gained by using a D-to-A converter with more than one bit. In the absence of noise,

the output of the latch is a square wave at the IF (typically 1.1 kHz). The one-bit

output of the latch is effectively bandpass-filtered in software, to recover the

fundamental component of the IF. The output of this filter, in the absence of noise, is

a sine wave whose precision depends on the number of bits used to perform the

digital filtering—16 in the present implementation. This precision does not depend on

the precision to which the signal is quantized.

If noise is present, this is no longer always true: precision can be lost, i.e. the SNR

at the output of the digital bandpass filter can be reduced, by quantizing to only one

bit. However, Davenport* [13] (quoted by Gardner [12]) shows that, under the

important assumption that the noise is Gaussian and uncorrelated with the signal,

the reduction in output SNR is at most 1.06 dB. Secondly, in conditions of moderate

input SNR, the output SNR is increased by the presence of the limiter. Thirdly,

quantizing to one bit provides a form of automatic gain control: the system is able to

cope with almost arbitrarily high input signal levels, and it is easy to guarantee (by

design) that numerical overflow in the DSP software will not occur. Finally, the fact

that the signal is one bit wide simplifies the programming of the mixers and of the

“local oscillator” quite significantly.

—————————

* writing in a pre-digital age!

4.2. Reduction to baseband

The output from the limiter is sampled by the processor at a rate of 5000 sa s–1.

Samples are exclusive-OR’d with a pair of digitally-synthesized quadrature square

waves whose nominal frequency is equal to the IF. This process mixes the IF to

baseband, in the conventional fashion (see, e.g., [17]). The resulting I and Q

components are filtered individually by a simple recursive digital filter comprising

four cascaded single-pole Butterworth stages, each with a cut-off frequency of 6.22 Hz.

The purpose of this filter is to enable the sample rate to be reduced, in order to

facilitate further processing. The characteristics of the filter were chosen in order to

be efficiently implementable on the chosen processor; the critical filtering is done in

subsequent stages.

The filtered I and Q signals are then re-sampled at a rate of 100 sa s–1, and filtered

individually by a four-pole Bessel low-pass filter with cut-off frequency 1.5 Hz. The

overall effect is that the signal frequency of around 115 kHz has been effectively

filtered by a bandpass filter with a bandwidth of ± 1.5 Hz; see figure 9. A Bessel filter

is used because the phase shift through such a filter is approximately proportional to

frequency, so that a signal passing through a Bessel filter experiences a roughly

constant time delay, regardless of its frequency (see, e.g., Cunningham [18], section

2.4.6). This, in turn, minimizes the distortion of the waveform in the I and Q

channels, and minimizes the effect of small variations in frequency (from the 0 Hz

ideal) on the accuracy of the subsequent phase measurements.

The quadrature square waves are generated by a “software local oscillator” whose

action can be described as the repetitive execution of the statement

– 20 –

Φ := Φ + ∆Φ (40)

under interrupt control at the sampling rate, which is 5000 times each second. Φ is a

32-bit phase accumulator, and ∆Φ is a 32-bit phase increment which is repeatedly

added to Φ. The two mixers in figure 9 are implemented by exclusive-OR’ing the one-

bit input samples with the top (most significant) bit of Φ and Φ + 230, respectively.

The LO frequency is therefore given by

F = 5000 × (∆Φ / 232) (41)

provided that ∆Φ ≤ 231. (Above this limit aliasing occurs.) Because the transitions of

the LO signal are synchronized to discrete 200 µs “clock ticks”, however, there will be

strong spurious components at sub-multiples of the sampling rate. Fortunately, these

spurii do not coincide with signal frequencies in any case which has been

investigated; and if this problem were to arise, it could be solved by changing the

frequency of the first LO (shown as 4 MHz ÷ 35 in figure 8).

4.3. Signal acquisition

The filter bandwidth of ± 1.5 Hz is much narrower than that which is commonly

used in Decca Navigator receivers. Last (section 2) for example uses a ± 12 Hz band-

width, and the Fisher Mark 5 receiver (section 3) uses a bandwidth of ± 21 Hz, falling

to ± 3 Hz in conditions of poor SNR. The use of such a narrow bandwidth imposes

severe constraints on the stability of the local oscillator: a tolerance of ± 1.5 Hz in

114.286 kHz equates to only ± 13.125 ppm, which is difficult to achieve in the sort of

physical conditions in which a Decca Navigator receiver is expected to be able to

operate, e.g. at sea.

To overcome this problem, the receiver enters an acquisition phase when it is first

switched on. In this mode of operation, the bandwidth of the Bessel filter is changed

to ± 5 Hz (from its normal value of ± 1.5 Hz). A sequence of 256 readings of the output

from this filter is taken, at intervals of 78.125 ms, spanning one complete 20 s Decca

cycle. Each value in the sequence is a complex number whose real part denotes the

instantaneous value of the I component and whose imaginary part that of the Q

component.

A 256-point complex fast Fourier transform (FFT) of the sequence is then taken.

Since the input is complex, the output of the transform represents the spectrum of the

input in a band from –6.4 Hz to +6.4 Hz. The largest peak in this spectrum (more

precisely, the complex value with the largest magnitude) is found; this is presumed to

belong to the desired signal. The frequency (positive or negative) of this peak is then

subtracted from the phase increment which controls the frequency of the digitally

synthesized software “local oscillator”.

Since the length of the sequence is 20 s, spectral lines in the FFT are separated by

0.05 Hz. Under the assumption that the largest peak coincides exactly with the

maximum energy of the received signal, therefore, the acquisition phase reduces the

frequency error in the I and Q components to within ± 0.05 Hz.

Figure 10 shows an actual spectrum obtained during an acquisition phase, with the

filter bandwidth set to ± 5 Hz. Figure 11 shows, for comparison, the spectrum after

acquisition has been completed and the software LO adjusted; the filter bandwidth is

– 21 –

± 1.5 Hz. Observe how an initial frequency error of about –1.8 Hz, which would have

caused the signal to fall outside the narrow filter bandwidth, has been reduced to a

small fraction of a Hz.

Frequency (Hz)

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6

Figure 10. Spectrum of Decca Navigator signal taken during acquisition phase

(filter bandwidth ± 5 Hz; linear voltage scale on y axis)

Frequency (Hz)

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6

Figure 11. Spectrum of Decca Navigator signal after acquisition

(filter bandwidth ± 1.5 Hz; linear voltage scale on y axis)

– 22 –

4.4. Phase measurement

After acquisition, the Decca signal has been reduced to baseband, i.e. to within a

fraction of a Hz of 0 Hz. To illustrate this, figure 12 shows the baseband I and Q

components over a complete 20 s cycle of transmissions. Figure 13 shows the same

data expressed in polar coordinates:

Time (s)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1617 18 1920

0

Figure 12. I and Q components over a 20 s cycle

(filter bandwidth ± 1.5 Hz, linear voltage scale on y axis)

Time (s)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Phase

(rad)

− π

0

+ π

Magni-

tude

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

station

trans-

mitting M M M R M G M P

Figure 13. ϕ and r components over a 20 s cycle

(phase represented by solid line, magnitude by dashed line)

– 23 –

Q = r cos ϕ (42)

I = r sin ϕ (43)

Inspection of figure 13 shows that the computed phase ϕ remains almost constant

at between about 0.2 π and 0.4 π whenever the master station is transmitting, with

excursions to about 0.4 π during the red multipulse and –0.6 π during the purple

multipulse. The signal strength during the green multipulse is too poor to compute an

accurate value for the signal phase; and the computed value of ϕ is of course

meaningless when no station is transmitting, i.e. when r is close to zero. The value of

0.4 π for the initial phase of the master signal is arbitrary, but the drift of 0.2 π in

10 s, i.e. a frequency of 0.01 Hz, is typical.

As in the Mark 5 receiver, nine phase readings a0, …, a

8are taken at 1.25 s

intervals (figure 6). Since the computed phase changes so slowly with time, correction

for interrupt latency, necessary in the Mark 5 receiver and described in section 3.3, is

not required. The algorithm of section 3.4 is used to compute the phase differences ϕR,

ϕG, ϕ

Pfrom the a

i.

4.5. Automatic frequency control

The acquisition phase adjusts the software “local oscillator” frequency to bring the

signal, now converted to baseband, within a small fraction of a Hz of 0 Hz. Further

continual adjustment of the LO frequency is required to compensate for small drifts

which occur, mainly because of temperature changes, in the master clock frequency of

the system. Rather than repeat the acquisition phase from time to time—which,

although giving the correct result, takes a significant time to perform—a more

efficient method of automatic frequency control is employed.

Consider the first two phase samples a0, a

1taken during the first master

transmission (labelled (0) and (1) in figure 6). If the software local oscillator frequency

is correct, a0

= a1, because frequency is the time-derivative of phase, and the LO

frequency is correct when the IF is zero. The automatic frequency control (AFC)

technique is to wait until the end of a Decca cycle, i.e. after a full set of nine phase

readings has been acquired, and then to execute

∆Φ := ∆Φ – β (a1 – a

0) (44)

where ∆Φ is the software LO phase increment (equation (40)) and β is a positive

constant. Under the assumption that a1

– a0

is much less than one cycle—an

assumption justified by the initial acquisition phase and the continual application of

the AFC procedure—this technique will adjust the LO frequency to keep the down-

converted received signal centred on 0 Hz.

The time interval between samples is 1.25 s, and samples lie in the range 0 ≤ ai< 1

(section 3.4). If the frequency F is too high by x Hz, then a1

– a0

= 1.25 x (cycles), so

∆Φ is decreased by 1.25 βx (by equation (44)). This results in a corrective change in F

of

∆F = – (5000 × 1.25 / 232) βx = –1.45519 × 10–6 βx Hz (45)

by equation (41). The gain of the AFC loop is therefore

– 24 –

K = –1.45519 × 10–6 β (46)

Hz / Hz. At present β is set to 3 × 216, so K = –0.2861 Hz / Hz.

The loop is a first-order frequency-lock (not phase-lock) loop, with transfer function

–KH(z) = ——————– (47)

1 – (1+K) z–1

and will be stable provided –2 < K ≤ 0 (see, e.g., chapter 3 of [18]). The loop is over-

damped if K > –1, and under-damped if K < –1. A “sanity check” is applied: no

correction is made if F is found to differ from its expected value by more than 5 Hz.

4.6. Phase estimation and adjustment

A compromise must be reached between, on the one hand, a narrow channel

bandwidth, which improves overall performance in the presence of noise but makes

the timing of the phase readings critical, and on the other hand a wide bandwidth,

which worsens the noise performance but facilitates the timing of phase readings.

The abrubt change of phase encountered by the receiver at the boundary between

the transmissions from different stations is seen by the receiver as a signal which

contains a high frequency component, which can lie outside the filter bandwidth. For

example, in the worst case, the phase can change by π radians in the 150 ms between

transmissions, which corresponds to a frequency of 3.33 Hz. Now the filters in

conventional receivers are wider than this; but the automatic frequency control (AFC)

technique presented in this paper enables the filter bandwidth to be made much

smaller. One consequence of a narrow filter bandwidth is that, if there is a large,

abrupt phase change, the signal appears to disappear for a period of the order of the

reciprocal of the filter bandwidth. Consequently, it is difficult to acquire the signal,

i.e. to determine the exact instant at which each transmission in the cycle begins.

Furthermore, the measurement of the phase of the signal is made less reliable,

because the perceived signal strength at the instant at which a measurement must be

taken is poor.

A related problem concerns the need for accurate timing of phase samples.

Although the multipulses last for 450 ms, the bandwidth of the digital low-pass filters

in the I and Q channels (± 1.5 Hz) is sufficiently small that the “window” within the

multipulse during which a phase reading may be taken is short. This may readily be

seen from figure 13: a sample must be taken within a few tens of ms of the centre of

the “hump” which marks (for example) the red multipulse; if the window is missed, an

incorrect phase reading will be obtained. The “window” is more critical if the relative

phase of the slave transmission is large; e.g. in figure 13, a timing error of 0.1 s will

give rise to a large inaccuracy in the measured value of ϕP, but a smaller inaccuracy

in the measured value of ϕR. The larger the gradient dϕ / dt at the instant when the

phase reading is taken, the more critical is the accurate timing of the phase sample.

Maximum accuracy is attained when there is no “hump”, i.e. when dϕ / dt = 0.

It is clear then that the overall accuracy of the system will be improved if it can be

arranged that the measured phase of the slave multipulse (relative to the master

phase) is small. To achieve this result, the phase of the software LO is adjusted

– 25 –

during the quiet period between transmissions in such a way that, so far as possible,

there is no abrupt change of phase of the baseband signal.

Figure 14 shows the times at which these phase adjustments are made. The

duration of a multipulse is 450 ms, and a gap (or silent period) of 150 ms precedes

and follows each multipulse. In the middle of the gap which precedes each slave

multipulse, the phase of the LO is advanced by the estimated phase of the slave

multipulse signal relative to that of the master. In the middle of the gap which

follows the multipulse, the LO phase is retarded by the same amount.

δR, δ

Gand δ

Pare estimates of the current values of the phase of the red, green and

purple slave signals (resp.) relative to that of the master. On the assumption that

these estimates are exact, the low-pass filters in the I and Q channels (figure 9) will

see a constant-frequency signal with no abrupt phase changes, and the algorithm of

section 3.4 will compute values for ϕR, ϕ

G, ϕ

Pwhich are close to zero.

The phase adjustments are in fact made 375 ms earlier than shown in figure 14;

e.g. the +δR

adjustment is made at t = 2.200 s, and the –δR

adjustment at t = 2.800 s.

This is because the signal takes 375 ms to propagate through the digital filters (figure

9). The cycle timing is derived from information on the I and Q channels; by the time

the multipulse appears at this point it has already been present at the aerial, and at

the mixers (figure 9), for 375 ms.

The value of the delay (375 ms) was determined by experiment. Since linear-phase

Bessel filters are used in the I and Q channels, the delay will be almost constant (see

section 4.2). The precise timing is not critical, a variation of ± 100 ms causing no

appreciable change in performance; but it is essential to take the delay into account.

If no allowance at all is made for the delay, the phase of the LO at the inputs to the

mixers will change instantaneously at an instant when there is also a large signal

component present, and large disturbances of amplitude and phase (“glitches”) on the

I and Q channels will occur.

M M R M G M P supervisory................................................................................... M

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

2.575

+δR

3.175

−δR

5.075

+δG

5.675

−δG

7.575

+δP

8.175

−δP

seconds

8.2f(orange)

phaseadjust-ments

Figure 14. Adjustment of LO phase

To calculate δR, a sequence θ

Rof N phase estimates is maintained, where N is a

constant which in the present implementation is 15. Successive values in the

sequence will be denoted by θR[0], …, θ

R[N–1]. Successive values of θ

Rrepresent

successive values of the calculated phase of the signal from the red slave relative to

that from the master. Similarly, θG

and θP

represent successive estimates of green and

purple phase; these sequences are used to calculate δG

and δP. The desired phase

values, from which a position will ultimately be computed, are the most recent

estimates θR[N–1], θ

G[N–1], θ

P[N–1].

– 26 –

The following algorithm is applied once every 20 s cycle, when a complete set of raw

phase data a0, …, a

8has been collected. First (step 1) ϕ

Ris calculated using the phase-

determination algorithm of section 3.4; this value will generally be close to zero. A

new phase estimate is then calculated by taking the sum, modulo 1.0 cycle, of the

current value of δR

and the newly-computed ϕR

(step 3); this new estimate is stored in

θR[N–1], items in the sequence being “pushed down” to make room for the new

reading (step 2). Finally (step 4) a new value for δR

is calculated from the mean of the

most recent N estimates. The green and purple components are treated in an

identical way.

Phase estimation algorithm:

1. Given a0, …, a

8, calculate ϕ

R, ϕ

G, ϕ

P

using the algorithm of section 3.4;

2. for i := 0 to N–2 do

θR[i] := θ

R[i+1]; θ

G[i] := θ

G[i+1]; θ

P[i] := θ

P[i+1]

od;

3. θR[N–1] := δ

RO+ ϕ

R; θ

G[N–1] := δ

GO+ ϕ

G; θ

P[N–1] := δ

PO+ ϕ

P;

4. δR

:= (1 / N) Σ sext(θR[i]);

δG

:= (1 / N) Σ sext(θG[i]);

δP

:= (1 / N) Σ sext(θP[i]);

(In step 4, the summations are over i = 0, …, N–1.)

Figure 15 illustrates the effect of the phase-adjustment algorithm. Compare this

figure with figure 13, which shows the same information but with the phase-

adjustment algorithm excluded. Observe that the phase in figure 15 changes almost

linearly with time. Small ripples corresponding to the red and purple multipulses can

be distinguished, and a larger ripple corresponding to the green multipulse; however

the signal strength of the green multipulse is too poor for a reading to be taken, and

the green phase is not used in the final computation of position.

Note that δR, δ

G, δ

Pare derived from the phase estimates θ

R, θ

G, θ

Pby a moving-

average filter [18]. This type of filter was chosen because it was considered desirable

that the currently-reported position should not depend on indefinitely many previous

readings. With the current choice of N = 15, and assuming that phase readings are

taken every 20 s (i.e. that no readings are missed), the δ values are derived from five

minutes’ worth of readings. Readings taken more than five minutes ago have no effect

on the average.

Note also that the phase estimation algorithm does not compromise the

performance of the system at high speeds. The position is calculated from the most

recent values of θR, θ

G, θ

P, and any large change in ϕ

R, ϕ

G, ϕ

Poccasioned by a large

movement of the vehicle during a 20 s cycle of readings will cause a corresponding

large change in the most recent θ values. However, at high speeds it will no longer be

the case in general that ϕR, ϕ

G, ϕ

Pare close to zero. Furthermore, it is not possible to

reduce the I and Q bandwidth much below ± 1.5 Hz, because it must always be

possible for a phase change of π radians to be detected in a multipulse which lasts

only 450 ms. The practical implication of these considerations is that the phase

estimation technique described in this section improves the accuracy of the device

when stationary, and does not worsen the accuracy when in motion; and it does not

reduce the maximum speed at which the device will operate. The performance of the

Mark 6 receiver is discussed further in section 4.9.

– 27 –

Time (s)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Phase

(rad)

− π

0

+ π

Magni-

tude

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

station

trans-

mitting M M M R M G M P

Figure 15. ϕ and r components over a 20 s cycle

(phase represented by solid line, magnitude by dashed line)

4.7. Cycle timing

The receiver synchronizes with the 20 s cycle of transmissions by detecting the gaps

in the received signal. Like many pattern-recognition tasks, this is easy for a person

to do, but deceptively difficult for a machine.

Several approaches were tried. The method which was finally adopted consists in

measuring the magnitude (the r component) of the received signal, and comparing it

against a threshold T. The signal is defined to be present if r ≥ T. When first switched

on, the receiver enters a re-sync state, in which it waits for the signal to appear (i.e.

for r ≥ T). When this happens, the long (2.5 s duration) master pulse is presumed to

have begun, and t is accordingly set to 17.50 (see figure 6). If the signal disappears

before the end of 2.5 s, the transmission cannot have been the long master pulse: a

false alarm is declared, and the re-sync state is re-entered. If no false alarm occurs,

the signal is monitored until t = 17.00 s in the next cycle, when the re-sync state is re-

entered, and the process repeats.

The re-sync state is also entered if the signal disappears during any non-multipulse

transmission. If, however, the signal disappears during a multipulse, the system does

not re-sync; instead, a flag is set to notify the processing routines that the signal from

that slave is too weak to be usable. If more than one slave is missed in this way, a

position cannot be computed on the current 20 s cycle.

The main difficulty in this technique lies in deciding on a value for T. Again,

several methods were tried, before finally settling on the following, which works very

well. The peak value p of the signal from the master between t = 19.25 s and

t = 19.90 s, i.e. towards the end of the long master transmission, is recorded, and T is

set to 0.25 p. This places the threshold at a constant position on the leading edge of

the long master pulse, even if the amplitude of the master pulse changes from cycle to

– 28 –

cycle. (T is not however allowed to fall below the noise floor of the receiver, so that

there is a negligible chance of random noise being interpreted as a signal.) Every time

r exceeds p during the specified interval, p is updated to the new peak value of r; also,

p is decayed by a factor of 0.95 at 20 s intervals. The reason for looking for peaks only

during the specified interval is that it prevents false alarms, e.g. the large-amplitude

purple slave multipulse in figure 15, from falsely updating p.

It is acknowledged that the method described is ad hoc; but so are many pattern-

recognition algorithms; and it has been compared with many alternatives, and has

been found to be the best of all that were tried. In particular, methods based on

looking for a small value of dϕ / dt [7] were tried and found to be inferior to that

described here, mainly because dϕ / dt undergoes random variations, and can

approach arbitrarily close to zero, when no signal is present.

4.8. Implementation

A prototype Mark 6 receiver was constructed, which uses a Motorola 68000 micro-

processor. The Mark 6 prototype was in fact derived from the Mark 5 prototype

(section 3.7) by the removal of the phase-locked loop—compare figures 5 and 8. The

slow sample rate (5000 sa s–1) means that a 68000 (with an 8 MHz clock) is quite fast

enough; a specialized DSP processor is definitely not required in this application.

The user interface is identical to that of the Mark 5 receiver (section 3.7).

As for the Mark 5 receiver, the program is written in the programming language C;

the software occupies about 17 k bytes, with a data area of about 12 k bytes. Fixed-

point arithmetic is used throughout: 16-bit for the phase determination and phase

estimation algorithms, and 32-bit for the coordinate conversion algorithm.

4.9. Performance

The Mark 6 receiver was subjected to preliminary tests in and around York

(England), using transmissions from the Northumbrian chain. The overall perform-

ance of the prototype was comparable to that of the Mark 5 prototype (section 3.8)

while in motion. In particular, the comments in section 3.8 concerning the uncertainty

in the known transmitter coordinates apply equally to the Mark 6 receiver.

As explained in section 3.8, random errors in position fixes are assumed to have a

Gaussian distribution. The predicted standard deviation (SD) of a fix at York using

the red and purple slaves is 98.05 m, taking into account transmitter instability but

not propagation anomalies or imperfections in the receiver. This figure represents the

best possible performance of any receiver.

The measured SD of a large number of readings taken at a fixed point in York was

103.7 m. The readings were all taken during “dusk” conditions [2]. The expected SD

during “dusk” is about half that during “summer night” [2], so the performance of the

Mark 6 receiver is about the same as that of the Mark 5 receiver, which (as noted in

section 3.8) compares favourably with that of conventional receivers.

The problems of VCO instability encountered in the Mark 5 receiver, and

mentioned in section 3.8 as one of the reasons for the development of the Mark 6

receiver, were entirely absent.

– 29 –

5. The master–orange zone identification algorithm

5.1. The problem

In traditional Decca Navigator terminology, a lane is the area between two

hyperbolic lines of equal phase difference. Lanes are narrowest on the baseline joining

the master station to a slave, where they are half a wavelength wide. The system will

fail if the vehicle travels more than half a lane-width during one 20 s cycle of

transmissions. This imposes a maximum speed of about 117 km h–1 in the worst case,

which is motion along the baseline. (The typical maximum velocity is, however, much

higher.) The most common method of reducing this source of ambiguity in Decca

Navigator receivers is to receive two transmissions at harmonically-related

frequencies nf and (n+1) f, and to measure the phase ϕy

of the (n+1) f signal and the

phase ϕx

of the nf signal at the same instant. The value of ϕy

– ϕx

then gives the

phase at that instant within a cycle of frequency f. For example, the purple and

master signals have frequencies 5f and 6f (respectively), where f is about 14 kHz. A

lane at 5f is about 2.13 km wide along the baseline, and a lane at 6f is about 1.78 km

wide. Measurement of ϕy

– ϕx

enables the position to be fixed within a lane at

frequency f, which is about 10.6 km wide.

Some sets receive signals at 8f (red) and 8.2f (orange), i.e. at frequencies 40 (f / 5)

and 41 (f / 5). Such sets are able to give a position within a lane corresponding to a

frequency of about 2.8 kHz, which is about 53.2 km wide along the baseline. This

latter technique is called zone identification.

In order to apply the zone identification technique in the conventional way, the red

(8f) signal must be received. The phase-determination algorithm of section 3.4 could

be adapted to work with the red signal, but only if signals from the red slave were

capable of being received. Now the technique of obtaining a fix from the orange

frequency requires only that signals from the master and any two of the three slaves

are receivable. In other words, the receiver is able to disregard one of the slaves

entirely. (This will normally be the slave whose signal is weakest, or which is least

favourably located.) This feature is very useful in practice, and it seems a pity to

sacrifice it. Consequently, a method was sought by which zone identification can be

performed on the orange and master (6f) frequencies.

5.2. A new solution

It does not appear to have been generally realized that zone identification is not

limited to signals at frequencies nf and (n+1) f, but can be applied to any coherent

signals at frequencies K1

f and K2

f, where K1

and K2

are integers and K1

≠ K2. For let

ϕ0

be the phase within a cycle of frequency f, and let ϕy, ϕ

xbe the phase within a cycle

of frequency K1

f, K2

f (respectively). Assume, without loss of generality, that K1

> K2,

and that K1

and K2

are coprime (i.e. have no common divisor except 1; this can always

be achieved by choosing f appropriately). Let the phases ϕ0, ϕ

y, ϕ

xbe normalized to lie

in the range 0 ≤ ϕ < 1. Then we have

ϕy = (K

1ϕ

0) mod 1.0 (48)

ϕx = (K

2ϕ

0) mod 1.0 (49)

From (48) and (49) we can derive

– 30 –

K1ϕ

0 = n + ϕ

y (50)

K2ϕ

0 = m + ϕ

x (51)

for some integers n and m. Solving for ϕ0

gives

N + (ϕy

– ϕx)

ϕ0 = ——————— (52)

K1

– K2

where N = n – m.

Note that the range of values of (ϕx, ϕ

y) has only one degree of freedom. Not all

pairs (ϕx, ϕ

y) are possible. Although ϕ

xand ϕ

yare analogue quantities, they are not

independent: while ϕx

can take any value in the range 0 ≤ ϕx

< 1, and ϕy

can take any

value in the range 0 ≤ ϕy

< 1, for any given ϕx, ϕ

yis constrained to take one of K

2

discrete values. Conversely, for any given value of ϕy, ϕ

xis constrained to take one of

K1

discrete values.

This fact is illustrated in figure 16, which is a graph of ϕy

against ϕx, in the

particular case K1

= 41, K2

= 30. The slope of the line is K1

/ K2, and the permissible

values of (ϕx, ϕ

y) are those through which the line passes. Permissible values of (ϕ

x, ϕ

y)

are those for which (K1ϕ

x– K

2ϕ

y) is an integer. Since a phase of 0.0 is identical to a

phase of 1.0, the graph must be considered to “wrap around” in both the ϕx

and ϕy

directions; so there is really only one line in figure 16, which extends from the point

(0, 0) to the point (1, 1). Distance along the line from the origin represents the value

of ϕ0

which corresponds to a particular value of (ϕx, ϕ

y). It is this distance that we

wish to find.

0.0 1.00.0

1.0

ϕy

ϕx

Figure 16. Graph of ϕy

against ϕx

for the case K1

= 41, K2

= 30

Since K1

and K2

are coprime, each permissible pair of values (ϕx, ϕ

y) occurs once

only during a single cycle of frequency f. (See, e.g., Schroeder [19], chapter 2.) If the

line in figure 16 passes through a point in the (ϕx, ϕ

y) plane, it does so only once. For

example, the pair (9/41, 9/30) occurs when ϕ0

= 0.007317, and for no other value of ϕ0.

In other words, for each permissible pair (ϕx, ϕ

y), the corresponding value of ϕ

0is

– 31 –

determined uniquely. It follows that the value of N in equation (52) is determined

uniquely, given ϕx

and ϕy. For the case K

1= 41, K

2= 40 (i.e. the conventional zone-

identification case), N = 0, so (52) reduces to ϕ0

= ϕy

– ϕx, as expected.

Now consider figure 17 in which the value of N is tabulated against ϕx

and ϕy

for

the case K1

= 41, K2

= 30. The value of N is tabulated for integer values of K1ϕ

xfrom

0 to K1

– 1, and integer values of K2ϕ

yfrom 0 to K

2– 1. Each diagonal in the table

corresponds to one of the (ϕx, ϕ

y) lines in figure 16. As in the case of figure 16, these

diagonals must be considered to “wrap around” at the edges of the table. If one starts

at the bottom left-hand corner of the table, and moves one step at a time in a north-

east direction, one arrives eventually back in the bottom left-hand corner, having

visited each entry once. In following this path, the value of ϕ0

increases uniformly

from 0 to 1, and (by equation (52)) N takes on different values in the range

0 … (K1

– K2) – 1. Since in following this path ϕ

0changes continuously, i.e. without

discontinuities, a discontinuity in ϕx

or ϕy

must coincide with a change in N, and vice

versa. When ϕx

“wraps around” from 1.0 to 0.0, ϕy

– ϕx

suddenly increases by 1.0, so N

must decrease by 1 at this point. When ϕy

“wraps around”, ϕy

– ϕx

suddenly decreases

by 1.0, so N must increase by 1 at this point. Inspection of figure 17 reveals that this

is indeed the case.

6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0

2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7

9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3

5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A

1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6

8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2

4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9

0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5

7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1

3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8

A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4

6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0

2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7

9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3

5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A

1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6

8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2

4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9

0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5

7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1

3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8

A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4

6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0

2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7

9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3

5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A

1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6

8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2

4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9

0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5 1 8 4 0 7 3 A 6 2 9 5

Figure 17. Table of N (modulo (K1

– K2)) for integer values of K

1ϕ

xfrom 0 to K

1– 1,

and integer values of K2ϕ

yfrom 0 to K

2– 1, for the case K

1= 41, K

2= 30.

(N.B. “A” stands for 10.)

We now derive a formula for N in terms of ϕx

and ϕy. There is a constant increment

M (modulo K1

– K2) in N as one reads across a row of the table. By inspection, in the

case K1

= 41 and K2

= 30, M = 7. Also, the entries along a diagonal are constant, since

N changes only at the edges of the table. Along a diagonal, K1ϕ

x– K

2ϕ

yis a constant

integer; and when ϕx

= ϕy

= 0, N = 0. It follows that

N ≡ M (K1ϕ

x – K

2ϕ

y) (modulo K

1 – K

2) (53)

By equation (52), when ϕx

is increased by one whole cycle, N increases by 1 (because

ϕ0

remains unchanged). Writing ϕx

+ 1 for ϕx, and N + 1 for N, in (53) gives

– 32 –

N + 1 ≡ M (K1 (ϕ

x + 1) – K

2ϕ

y) (modulo K

1 – K

2) (54)

Subtracting (53) from (54):

M K1 ≡ 1 (modulo K

1 – K

2) (55)

By substituting instead ϕy

– 1 for ϕy

in (53), one can deduce by an identical argument

that

M K2 ≡ 1 (modulo K

1 – K

2) (56)

In fact, (56) is a consequence of (55), since the left-hand side of (56) is obtained from

the left-hand side of (55) by adding an integral multiple of (K1

– K2), and this does not

change the congruence properties.

We can easily prove that M exists. Recall that K1

and K2

are coprime, and that

K1

> K2. Then it follows that K

1and (K

1– K

2) are coprime. For assume the contrary,

viz. that k is a factor of both K1

and (K1

– K2) for some k > 1; i.e. that

K1 = mk (57)

K1 – K

2 = nk (58)

for some m, n. Then

K2 = (m – n) k (59)

and therefore k is a factor of both K1

and K2, i.e. K

1and K

2are not coprime, which is

false. By an identical argument, K2

and (K1

– K2) are coprime. Theorem 23 in chapter

1 of Eames [20] states that, if two integers a and b are coprime, then a possesses a

unique multiplicative inverse modulo b, i.e. there exists an x such that ax ≡ 1 (modulo

b). Therefore M exists, and is unique.

Zone identification can thus be performed in the following way.

1. Using a two-channel receiver, and the phase-determination algorithm given in

section 3.4, determine ϕx

and ϕy.

2. Using equation (53), with K1

= 41, K2

= 30 and M = 7, calculate N (modulo 11).

3. Using equation (52), calculate ϕ0.

To test this procedure, a computer simulation was carried out. A large number of

values of ϕ0

in the range 0 ≤ ϕ0

< 1 were taken, and corresponding values of ϕx

and ϕy

were calculated by equations (48) and (49). Then a value of ϕ0

was calculated from ϕx

and ϕy, using the procedure just described. The calculated value was found to be equal

to the known value of ϕ0

in each case, to within the normal tolerance of floating-point

arithmetic (10–15 of a cycle).

5.3. Predicted performance

The master–orange zone-identification algorithm requires that the phase be

measured to an accuracy of ± 1 / (2K1) of a cycle in ϕ

xand ± 1 / (2K

2) of a cycle in ϕ

y

– 33 –

(by equation (53)). If these limits are exceeded, a gross error in N might occur, since

the quantity K1ϕ

x– K

2ϕ

ymight be rounded to the “wrong” integer. In section 3.8 it

was shown that a distance error of 98.05 m at York corresponds to a phase error of

0.005 cycles, so the observed distance error of 207.4 m corresponds to a phase error of

δϕ = 0.005 × (207.4 / 98.05) = 0.01058 cycles (60)

(SD). The required tolerance of ± 1/82 = ± 0.01220 is therefore about 1.2σ, and zone

identification can be expected to succeed, on average, in about 77.6% of cases. This

figure was obtained by reference to tables of the normal probability function P(x) [21].

This performance is inferior to that of conventional receivers. However, an error of

1 in the rounded value of K1ϕ

x– K

2ϕ

ywill cause N to be in error by 7 (by equation

(53)). This corresponds to a phase error of 7/11 of a cycle (or, equivalently, –4/11 of a

cycle) at a frequency of 2.8 kHz. Since a lane at this frequency is 53.2 km wide, this

corresponds to a distance error of 19.3 km on the baseline, and more elsewhere. A

displacement of 19.3 km in 20 s implies that the vehicle is moving at 3485 km h–1.

Similarly, an error of 2 in the rounded value of K1ϕ

x– K

2ϕ

yimplies a phase error of

3/11 of a cycle, a distance error of 14.6 km, and a speed of 2614 km h–1. The

probability that K1ϕ

x– K

2ϕ

yis in error by 3 cycles or more (i.e. more than 3.6σ) is only

0.032% [21]. Zone identification failures can therefore almost always be detected in

practice, and erroneous results filtered out.

6. Conclusions

Decca Navigator receivers have been in use since the time of the Normandy

Landings during the Second World War; and over the years the Decca system has

become one of the most widely used, and most reliable, of position-fixing methods.

Many widely different designs of receiver have been proposed. The very earliest, of

course, did not use computer technology; but there is now a spectrum of receiver

types, ranging from the purely analogue devices of the 1940s, with their mechanical

phase detectors (“Decometers” [1]), to digital devices which have a minimum of RF

and other analogue electronics.

The two new designs described in this paper, the Fisher Mark 5 and Mark 6, lie

towards the digital end of the spectrum. There is a progression from the very earliest

designs, through the Mark 5, to the Mark 6: the hardware phase-locked loop of the

Mark 5 receiver has been superseded in the Mark 6 design by digital algorithms.

Digital techniques allow the bandwidth of the signal-processing channel to be made

very small; and the development of the Mark 6 receiver included the design of

techniques which compensate for oscillator drift, which becomes prominent when the

bandwidth is narrow.

The master–orange zone identification algorithm in section 5 is not related to either

the Mark 5 or the Mark 6 design. It assumes a different hardware configuration, viz.

a two-channel receiver. It is perhaps unusual for number theory to find an application

in the design of radio receivers.

– 34 –

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– 35 –

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