Introduction - Time-Frequency Transforms for Radar Imaging and Signal Analysis - page 12

There are two types of definitions of SNR: the average power SNR and the
instantaneous power SNR.
The average power SNR is defined as the ratio of the average signal
power to the average noise power. The average power is defined by

T T
0

1

s 2 ( t ) dt P S = (1.13)

where T is the time duration of the signal s ( t ). The average noise power is
defined by

P N = ¥
-¥

- mean { r n }) 2 p ( r n ) dr n

( r n (1.14)

where r n ={ n ( t )} is a random noise process and p ( r n ) is the probability
density function of the random noise process r n . Thus, the SNR becomes

P S
P N

SNR average = (1.15)

For an additive white Gaussian noise (i.e., uniform power spectrum
and Gaussian amplitude distribution) with zero-mean and variance s r n , the
average noise power is P N = s r n , and the average power SNR becomes

T T
0

1

s 2 ( t ) dt

SNR average = (1.16)

s r n

The instantaneous power SNR is defined as the ratio of the instanta-
neous signal power to the average noise power:

P instant
s r n

SNR instant = (1.17)