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The Reason GPS Counts Time in 1.5-second Intervals

During Japan's Edo Period (1603–1868), people divided each day, as defined by the sunrise and sunset, into daytime and nighttime halves, and then divided each of these into six equal time blocks. Under this method, known as futei-jiho ("indefinite time" or "seasonal time system" in English), a single time subdivision block's length could vary by as much as 50 minutes between the winter and summer seasons. Clocks were not widespread at that time, so people instead synchronized time with their daily living cycles. In this article, we're going to explore another unique time-counting system—this one utilized by GPS.

GPS' Weak Point: The Week Number Rollover Phenomenon

Every navigation message sent by GNSS satellites contains all of the crucial information for positioning calculation. The time information is particularly essential among them.
In my previous articles, I introduced about the structuring of navigation messages using GPS (which was the first-ever GNSS) as an example. Despite the limited bit count, each of these messages can contain all of the information required, without any problem. However, this approach does have a certain weak point, which pops up every two decades or so. Specifically, this problem is known as "GPS Week Number Rollover"—something many readers may have heard of already.
GPS expresses time using only weeks and seconds, with the week count using a 10-bit number. Because just 10 bits are used, it's only possible to count weeks over a range from 0 to 1,023. Therefore, on August 22, 1999, which represented the 1,024th week from the cycle that began on January 6, 1986, GPS equipment reset to zero and caused problems for users everywhere. By the start of the 2,048th-week rollover on April 7, 2019, users had already experienced the phenomenon and thus applied what they learned to make improvements in receiver equipment. The result was fewer problems this time around—with the exception of some older equipment. Moreover, in the satellite systems that send the navigation messages, a new positioning signal was adopted, increasing the bit count for the week value. As a result, we won't have to worry about rollovers in the near future. The Week Number Rollover phenomenon is specific to GPS technology, which tightly packs information into a limited number of bits.

Initial Confusion Regarding the GPS 1.5-second Problem

The 10-bit week number is not the only concern with GPS. There is another related issue I have struggled with as a novice student in this field: the 1.5 second problem. This is something I have always been genuinely curious about — in the same way that one might ask why traffic signals are in the order red, yellow and green from top — and through research for my previous article, I feel that I have developed the knowledge necessary to accurately answer the question. Although this will require a rather deep dive into the subject, I'd like to explain my thought process, which was as follows.

1. GPS expresses time using only weeks and seconds. This is done using a 29-bit value called the "Z-count," whose most significant 10 bits are the Week Number (WN) count, and whose least significant 19 bits are the Time-Of-Week (TOW) count, which represents the number of seconds from the beginning of the week.

2. Explanations I had read stated that TOW is counted in 1.5-second intervals, which seemed odd to me.

3. A single week comprises 604,800 seconds, calculated as 7 days × 24 hours × 60 minutes × 60 seconds = 604,800. Therefore, when using binary numbering, a 20-bit value is surely needed (2^20 = 1,048,576).

4. If TOW is counted in 1.5-second intervals, however, the maximum value becomes 403,200 (calculated as 604,800 ÷ 1.5), for which a 19-bit number is sufficient (403,200 < 524,288, the latter being the result of 2^19). Counting in 1.5-second rather than 1-second intervals enables a 1-bit reduction.

5. A GPS navigation message, however, only assigns a 17-bit number for TOW, which is expressed in specification documentation as "TOW 17 bit truncated." Although it would normally require 19 bits to count up to 403,200, the truncated approach seems to do away with 2 bits entirely. How is this possible?

The Results of Counting by Sixes and Fours

6. Time is the most important piece of information in positioning calculation, and it seems that these satellites' systems have been designed so that eliminating parts of numbers presents no problem. But how is this accomplished?

7. In my previous article, I explained how navigation messages are operated at 50 bits per second (bps), which is just as slow a data transmission rate as the old telex services. I also explained how and where the time stamp is recorded in the previous "subframe"—which is a collection of "words"— and the length which is 300 bits.

8. At a bit rate of 50 bps, the leading edge of a 300 bit subframe arrives every 6 seconds. Stated another way, all time stamps in GPS navigation messages are expressed in multiples of six.

9. TOW counts time in 1.5-second intervals, and this number is used as the divisor when expressing time in standard seconds. When 6 is divided by 1.5 (or divided by 3 then multiplied by 2), the result is a multiple of 4, meaning that TOW always uses a multiple of four.

10. The decimal-type number 4, when expressed as a binary, becomes 100 because multiples of 4 expressed in binary always contain "00" as their final two digits. As follows, the TOW value, which is always a multiple of 4, necessarily contains "00" as its final two digits.

11. Therefore, even if the final 2 bits of a TOW value—which one would expect to require 19 bits—are thrown out in order to make a 17 bit number, there is absolutely no loss of information.