If we have a zero-mean Wide Sense Stationary process
XX, it is a White Noise
Process if its ACF is a delta function at
τ=0
τ
0
, i.e. it is of the form:
r
X
X
τ=
PX
δτ
r
X
X
τ
PX
δ
τ
(1)
where
PX
PX
is a constant.
The PSD of XX is then given by
SX
ω=∫
PX
δτe−(iωτ)dτ=
PX
e−(iω0)=
PX
SX
ω
τ
PX
δ
τ
ω
τ
PX
ω
0
PX
(2)
Hence
XX is
white,
since it contains equal power at
all
frequencies, as in
white light.
PX
PX
is the PSD of XX at all
frequencies.
But:
Power of X=12π∫−∞∞
SX
ωdω=∞
Power of X
1
2
ω
SX
ω
(3)
so the White Noise Process is unrealizable in practice,
because of its infinite bandwidth.
However, it is very useful as a conceptual entity and as an
approximation to 'nearly white' processes which have finite
bandwidth, but which are 'white' over all frequencies of
practical interest. For 'nearly white' processes,
r
X
X
τ
r
X
X
τ
is a narrow pulse of non-zero width, and
SX
ω
SX
ω
is flat from zero up to some relatively high cutoff
frequency and then decays to zero above that.
Usually the above concept of whiteness is sufficient, but a
much stronger definition is as follows:
Pick a set of times
t1
t2
…
tN
t1
t2
…
tN
to sample
Xt
X
t
.
If, for any choice of
t1
t2
…
tN
t1
t2
…
tN
with NN finite, the
random variables
X
t1
X
t1
,
X
t2
X
t2
, ……
X
tN
X
tN
are jointly independent, i.e. their
joint pdf is given by
f
X
(
t1
)
,
X
(
t2
)
,
…
X
(
tN
)
x1
x2
…
xN
=∏i=1N
f
X
(
ti
)
xi
f
X
(
t1
)
,
X
(
t2
)
,
…
X
(
tN
)
x1
x2
…
xN
i
1
N
f
X
(
ti
)
xi
(4)
and the marginal pdfs are identical, i.e.
f
X
(
t1
)
=f
X
(
t2
)
=…=f
X
(
tN
)
=
fX
f
X
(
t1
)
f
X
(
t2
)
…
f
X
(
tN
)
fX
(5)
then the process is termed
Independent and Identically
Distributed (i.i.d).
If, in addition,
fX
fX
is a pdf with zero mean, we have a Strictly
White Noise Process.
An i.i.d. process is 'white' because the variables
X
ti
X
ti
and
X
tj
X
tj
are jointly independent, even when separated by an
infinitesimally small interval between
ti
ti
and
tj
tj
.
In many systems the concept of Additive White Gaussian
Noise (AWGN) is used. This simply means a process which
has a Gaussian pdf, a white PSD, and is linearly added to
whatever signal we are analysing.
Note that although 'white' and Gaussian' often go together,
this is not necessary (especially for
'nearly white' processes).
E.g. a very high speed random bit stream has an ACF which is
approximately a delta function, and hence is a nearly white
process, but its pdf is clearly not Gaussian - it is a pair of
delta functions at
+V
+
V
and
−V
V
, the two voltage levels of the bit stream.
Conversely a nearly white Gaussian process which has been
passed through a lowpass filter (see next section) will still
have a Gaussian pdf (as it is a summation of Gaussians) but
will no longer be white.
A random process whose PSD is not white or nearly white, is
often known as a coloured noise process.
We may obtain coloured noise
Yt
Y
t
with PSD
SY
ω
SY
ω
simply by passing white (or nearly white) noise
Xt
X
t
with PSD
PX
PX
through a filter with frequency response
ℋω
ℋ
ω
, such that from our discussion of
Spectral Properties of Random Signals.
SY
ω=
SX
ω|ℋω|2=
PX
|ℋω|2
SY
ω
SX
ω
ℋ
ω
2
PX
ℋ
ω
2
(6)
Hence if we design the filter such that
|ℋω|=
SY
ω
PX
ℋ
ω
SY
ω
PX
(7)
then
Yt
Y
t
will have the required coloured PSD.
For this to work,
SY
ω
SY
ω
need only be constant (white) over the passband of
the filter, so a nearly white process which
satisfies this criterion is quite satisfactory and
realizable.
From our discussion of Spectral Properties of
Random Signals and Equation 1, the ACF
of the coloured noise is given by
r
Y
Y
τ=
r
X
X
τ*h−τ*hτ=
PX
δτ*h−τ*hτ=
PX
h−τ*hτ
r
Y
Y
τ
r
X
X
τ
h
τ
h
τ
PX
δ
τ
h
τ
h
τ
PX
h
τ
h
τ
(8)
where
hτ
h
τ
is the impulse response of the filter.
