Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 39

p 1 ( x ), p 2 ( x ) p 1 ( x ), p 2 ( x )

σ 1 = 3
m 1 = 0

σ 1 = 3
m 1 = 0

σ 2 = 3
m 2 = 2

σ 2 = 5
m 2 = 0

x x

−5 0 5 0 2

Fig. 1.22 Gaussian pdf's with different means and variances

where m = the statistical mean, and r 2 = the variance. Plots of this pdf for dif-
ferent values of mean and variance are shown in Fig. 1.22 .

1.3.3 Signals in Noise

1.3.3.1 Gaussian Noise

Noise, n(t), that is encountered in electrical systems frequently has a Gaussian pdf
with zero mean, m n = 0. The pdf of this type of signal has the form:
p ð n Þ¼ 1
r

p e n 2

2p

2r 2

n o ¼E n 2

Note that the power of zero-mean Gaussian noise is Eð n m n Þ 2

ð since m n ¼ 0 Þ¼ noise variance ¼ r 2 . (Prove this as an exercise!)
If there are two Gaussian noise signals, n 1 and n 2 , with the variance of n 2 being
greater than the variance of n 1 , then the pdf of n 2 (pdf-2) has a wider spread around
its mean than n 1 's pdf (pdf-1) (see Fig. 1.22 , left). Furthermore, n 2 has more power
than the first.

1.3.3.2 Signals in Gaussian Noise

If s(t) is a deterministic signal and n(t) is noise, then z(t) = s(t) ? n(t) is a random
signal. Consider now the case where s(t) = a (a constant). If n(t) is Gaussian noise
with zero mean and variance = r 2 , then the random variable z(t) is also Gaussian
with mean and variance given at any time by:

z ¼ m z ¼Ef z ð t Þg¼Efð s ð t Þþ n ð t ÞÞg¼Efð a þ n ð t ÞÞg¼Ef a gþEf n ð t Þg
¼ a þ 0 ¼ a ;