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AN101
GPS Carrier-to-Noise Density
GPS Carrier-to-Noise Density
This application note is a brief explanation of some of the RF quantities and units
commonly used in discussion of GPS signal power levels. The carrier-to-noise density
ratio may not be familiar to those without experience in spread-spectrum systems.
Definitions for dBm, dBW, dB⋅Hz, C/N0 are given and related to the more familiar signalto-noise ratio.
Power in Decibels (dBm, dBW)
Power can be expressed in decibels by forming the ratio of the power to a reference
power level. Typical reference levels are one watt and one milliwatt. A power level in
decibel-milliwatts can be computed from a power expressed in milliwatts as:
PdBm = 10 ⋅ log(
Pmw
)
1mw
Similarly, power can be expressed in decibel-watts:
PdBW = 10 ⋅ log(
PW
P
P
P
1
) = 10 ⋅ log( mw ) = 10 ⋅ log(
+ mw ) = −30 + 10 ⋅ log( mw ) = PdBm − 30
1000 1mw
1mw
1W
1000 mw
A power level of 100mw can therefore be expressed as a power of 20 dBm or -10 dBW.
Signal-to-Noise Ratio (SNR)
Signal to noise ratio (SNR) is defined as signal power divided by noise power, with this
ratio usually expressed in decibels. For a signal power (S) and a noise power (N) defined
in common units of power, such as watts or milliwatts, the SNR is:
S
SNR = 10 ⋅ log 
N
If the units of power are already in common decibel units, such as dBW or dBm, then
SNR = S dBm − N dBm = S dBW − N dBW
Power Spectral Density and Thermal Noise
Since noise sources like thermal noise generate power in proportion to the bandwidth of
the system in question, a method of describing the power level independent of the
bandwidth is desirable. Power spectral density is a measure of power in each unit of
bandwidth.
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AN101
GPS Carrier-to-Noise Density
Thermal noise has a constant power spectral density. The power of thermal noise
generated is a function of the temperature and the noise bandwidth. It is independent of
the center frequency of that bandwidth. The noise power spectral density for noise is:
kT
where k is Boltzmann’s constant and T is the absolute temperature. Boltzmann’s constant
is the ratio of the energy in a molecule to its temperature:
k = 1.38 ⋅ 10 −23
J
where the units are joules per degree Kelvin.
°K
We recall the Kelvin temperature scale is zero at absolute zero, where the motion of
molecules stops. The Celsius temperature scale is shifted such that the zero-point is the
freezing point of water, but the step size of a degree Celsius and a degree Kelvin is the
same. Conversion between the two can be made from the relationship:
0°C = 273.15° K
Ambient thermal noise is typically calculated at 290K, or 16.85C. This is a reference
generally taken as the effective noise temperature of the earth:
N T = kT = 1.38 ⋅ 10 − 23
J
⋅ 290° K = 4.00 ⋅ 10 − 21 J
°K
This is an expression of the noise power spectral density in joules, a unit of energy.
Since a watt is one joule-per-second, we can also express the units for power spectral
density as watts-per-hertz. Ambient thermal noise power spectral density is then:
N T = 4.00 ⋅ 10 − 21
W
dBW
dBm
= −204
= −174
Hz
Hz
Hz
Carrier-to-Noise Density (C/N0)
The carrier-to-noise density is defined as the carrier power divided by the noise power
spectral density. To calculate the carrier-to-noise density then for a GPS receiver
operating at the thermal noise floor ( N 0 = N T ) we need the carrier power. The C/A code
GPS signal specification [1] gives us a nominal value for the carrier power received at the
surface of the Earth, specifying this to be at a power level of –160 dBW or above. Using
this carrier power and the thermal noise floor we computed above:
C
− 160dBW
=
= 44dB ⋅ Hz
N 0 − 204(dBW )
Hz
Now we’ll convert this to a signal-to-noise ratio for the same conditions, assuming a C/A
code receiver bandwidth of 4 MHz. To convert the carrier-to-noise density ratio to the
signal-to-noise ratio we divide by the bandwidth.
SNR =
AN101
1 Feb., 2003
− 160dBW
4 ⋅ 10 Hz ⋅ (−204(dBW
6
Hz
))
=
− 160dBW
= −22dB
− 138dBW
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GPS Carrier-to-Noise Density
The signal power is 22 dB below the noise power coming into the receiver. It is only
through despreading the C/A code that the signal can be detected at all. After the signal
is despread, it filtered to a narrower bandwidth. The decrease in bandwidth eliminates
most of the noise, which is spread over the entire bandwidth, but leaves the signal. This
is the mechanism by which process gain is achieved in a spread-spectrum receiver. A
more complete description of process gain can be found in Dixon [2].
References
[1] Global Positioning System Standard Positioning Service Signal Specification, 2nd
Edition, June 2, 1995. pp. 18. Available at
http://www.navcen.uscg.gov/pubs/gps/sigspec/default.htm
[2] Dixon, R.C., Spread Spectrum Systems, 2nd Edition, John Wiley & Sons, 1984. ISBN
0-471-88309-3
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