# Radar Technology - Ambiguous Measurements

Theoretical Basics: Ambiguous Measurements | Theoretical Basics: Signals and Range Resolution
How a Bat's Sensor Works | Low Probability of Intercept (LPI) Radar | Electronic Combat: Overview | Electronic Combat in Wildlife

Police Constable: So, when was that?
Witness: I know for sure that it was a Monday because, well, in the morning I said...sod it!'
Constable: Oh, I understand. But which Monday?
Witness: Sorry, I don't know. But I remember that a full moon was rising in the morning of that day
Constable: Aaa-ha. Can you tell me anything more?
Witness: (Ponders) Er, yes. It was exactly one year after our new government had been elected. On the television they commented on their achievements so far...
Constable: Thank you very much! That was Monday, Sep 24, 2001.

Taken individually, all of the witness's statements above are ambiguous: there are 52 Mondays in a year, and without further information it is impossible to tell which of these (or even which year) was meant. Knowing that it was a Monday eliminates six out of every seven possibilities for the day of the case in question. Elections are held every four years, but this statement is ambiguous too. However, this piece of information takes away three out of any four possibilities for the year in question. Finally, by combining this with the (also ambiguous) information about the moon phase, the constable is able to eliminate all possibilities but one.

Now let's rephrase this example in radar terminology. The witness uses three different measuring scales for giving the time:

• The 'week' scale has seven time slots (days) and thus rolls over from day seven (Sunday) to day one (Monday) after seven days.

• The 'election period' scale has a resolution of 1 year - ie, 365 days. There are four time slots, and it rolls over every 1460 days (4*365).

• The 'moon phase' scale has a re solution of 7.25 days and four time slots. It rolls over every 29 days.

Any given combination of day, election year status and moon phase does not reappear before 7*1460*29 = 296,380 days, or 812 years, have elapsed. That is, this set of scales yields unambiguous measurements for any timespan that a police constable could possibly be interested in. But an archaeologist confronted with the same information would most probably frown upon the ambiguity of 812 years and ask for more information to resolve it.

### So What's the Connection to Radar?

In a radar, the timescales are equivalent to pulse repetition times (PRT). The time slots are still time slots, but usually they are called range bins. This is because the round trip delay time for an echo is a direct measure of a target's range. Each of the PRTs yields an ambiguous measurement, but a proper choice of the values for a set of PRTs can eliminate all ambiguities out to distances far beyond the limits given by the receiver's sensitivity.

#### How to Choose a Set of PRTs

Imagine that the witness in the example above had used a third timescale of fortnights instead of moon phases. A fortnight is a period of two weeks - that is, 14 days. Thus, their statement would now be something like: 'It was a Monday, in the first year after an election, on the eighth day of a fortnight'.

The problem with this set of PRTs is that 'the eighth day of a fortnight' already indicates that the day was a Monday. Hence, the 'week' timescale has been rendered useless as it doesn't contribute any further information. The unambiguous range of the original PRT set was calculated as the product of its individual values (7*1460*29) but that was only correct because none of the factors shared a common denominator.

Upon closer examination of the week/election period/fortnight set it turns out that a fortnight's time is an integer multiple of a week's time, and this is where the problem is. Furthermore, the 14 days of a fortnight and the 1460 days of an election period share '2' as a common factor. Therefore, this set yields unique results within only 7*1460 days, or 28 years1.

So the conclusion is this: the ambiguity of a set of PRTs is determined by the least common multiple of its PRTs. A proper set of PRTs consists of values that are chosen such that they do not share a common denominator. Therefore, it is not uncommon to see radar engineers fiddling around with prime numbers.

### Other Entries in This Project

#### Electronic Combat

1This is calculated as: ambiguous range = LCM(7*1460*14) = LCM(7*1460*2*7) = 7*1460

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