Definitions

GPS Time

• Zero at 06/01/1980 (UTC Time)
• written as weeks since 06/01/1980 + seconds in the current week
• Due to the original design GPS week time is stored as 10 bit integers so can only hold 1024 weeks
• If maximum week is reached it rolls over back to 0
• Can also be represented as absolute gps seconds which is just the seconds passed from inception
• Not adjusted for leap seconds
• Expresed with a resolution of 1.5 seconds

International Atomic Time (TAI) Time

• Time standard that is specifally for Earth’s geoid
• Based on weighted average of 450+ atomic clocks around 80 national labs worldwide
• It’s a continuous time scale

Coordinated Universal Time (UTC) Time

• Based on TAI + leap seconds
• More in accordance with the earths rotation + suns position due to leap seconds adjustment
• UTC does not have a specific starting point in terms of seconds or a zero point
• Due to various discrepancies that had accumulated over the inception of UTC time, it was decided to start with a 10 seconds offset from TAI time

Unix / Posix Time

• Zero at 01/01/1970 (UTC Time)
• No leap seconds (see FAQ for more info)

Leap seconds

• The difference between the observed solar time vs atomic clocks
• Solar times can change due to irregularities in the earth’s rotation
• Can only be observed
• Updated every 6 months by IERS (International Earth Rotation and Reference Systems Service)

Calculations

GPS Aboulte To GPS Week Seconds

week_no         = floor( gps_absolute / 60 / 60 / 24 / 7 )
seconds_in_week = gps_absolute - week_no * 60 * 60 * 24 * 7

GPS Absolute To Unix Time

# leap seconds for GPS -> Unix as of 05/07/2024 is 18 seconds
# 10 years + 2 leap year days + 5 gps days
# between 01/01/1970 ~ 06/01/1980 
leap_seconds = 18

t_between_unix_to_gps = 86400 * ((365 * 10) + 2 +  5)
t_between_unix_to_gps = 315964800 

unix_time = t_between_unix_to_gps + gps_absolute - leap_seconds

UTC Time to TAI time

# leap seconds for UTC -> TAI as of 05/07/2024 is 27 seconds
# UTC has a initial 10 seconds offset
leap_seconds = 27
inital_offset = 10

tai = utc + initial_offset + leap_seconds

FAQ

How does an atomic clock work?

• Without going into too much detail it works by observing the frequency between two energy states of atoms.
• 1 second is defined as 9,192,631,770 transitions between two states of caesium-133 atom.

Why is GPS week time represented in weeks + weeks in the second?

• Essentially to compact the message since navigation message to satellites need to be as compact as possible.
• 10 bits for week (2^10 = 1024 weeks) and 19 bits (2^19 = 524288 seconds) for seconds in a week (60 (seconds) * 60 (minutes) * 24 (hours) * 7 (days) / 1.5 (interval))
• Smaller than 32 bits (2^32 = 4294967296 seconds | 139 years) needed for seconds.

Why are we subtracting 18 seconds from the GPS absolute to Unix time conversion when both times don’t take into account leap seconds?

• While Unix time does not take into account leap seconds and every single day contains exactly 86,400 seconds it does deal with it by jumping back by 1 in the next day as highlighted by the bolded time in the table below.