Page 230 - Phase-Locked Loops Design, Simulation, and Applications

MIXED-SIGNAL PLL APPLICATIONS PART 1: INTEGER-N FREQUENCY
SYNTHESIZERS Ronald E. Best 139

noise power is P and carrier power is P , a noise-to-carrier ratio (NCR) could be defined by
s
n

Note that the unit of NCR (W/W) is dimensionless. Usually one defines another noise-to-
carrier ratio NCR expressed in decibels:
dB

Checking Eq. (6.7) once again, we recognize that the ratio S (f )/P is not dimensionless,
nn m s
since S (f ) does not stand for power, but for power density, whose unit is W/Hz. The
nn m
−1
variable P , however, has the unit W, hence the ratio S (f )/P has the unit Hz .
s
s
nn m
Mathematically, it is not correct to build the logarithm of a ratio that is not dimensionless.
−1
[Try to find out what log (Hz ) is.] We can circumvent that dilemma by introducing new
variables S *(f ) and S *(f ) as follows:
nn m θθ m

B stands for bandwidth and is set B = 1 Hz. Due to multiplication with B, the unit of S *(f )
nn
m
becomes W (and not W/Hz). The new variable S *(f ) now signifies noise power (not
nn
m
power density) within a bandwidth B = 1 Hz, located at an offset f from the carrier
m
frequency. The new variable S *(f ) is defined by
θθπ m

2
2
and is now the ratio of two power quantities. Its unit is therefore rad and not rad /Hz. S *
θθ
(f ) therefore no longer represents the power density of phase perturbations, but rather the
m
mean square value of phase perturbation whose spectrum ranges from f < f < f + 1. It
m
m
is now mathematically correct to build a logarithmic quantity from S θθπ *(f ) by setting
m

The unit of S *(f ) now becomes dBc and not dBc/Hz. Let’s do a numerical example.
θθ
m dB
Assume that the carrier power is P = 1 mW, and that the noise power density S (f ) is
s
nn m
10 −15 W/Hz at an offset f = 10 kHz from the carrier frequency. The noise power S *(f )
m nn m
within a bandwidth of 1 Hz at f = 10 kHz is then given by
m