4.5 Filtering Random Signals and the Impact on PSDs
This section investigates the result of processing a random input signal through a system, and how the output PSD relates to the input.
4.5.1 Response of LTI systems to random inputs
An important result of stochastic processes theory is that when the input of a LTI system is a WSS process
|
where
Given that
| (4.38) |
Similarly, if a discrete-time WSS random process
|
the PSD of the WSS output process
| (4.39) |
where
4.5.2 Filtering continuous-time signals that have a white PSD
A signal model that is used in several applications is the white noise, previously discussed in Section 1.10.1.0.
A realization
| (4.40) |
and zero mean
Using Eq. (4.23), the bilateral PSD of a white noise is
| (4.41) |
When a unilateral representation is adopted, the white PSD is conveniently denoted as
One can write
| (4.42) |
In other words, using this nomenclature,
Note from Eq. (4.40) and recalling Eq. (1.54) that the continuous-time white noise power has
With infinite power, a white noise cannot be measured (it would damage the measuring equipment!). But white noise is a very good model for many practical applications, with an implicit assumption that limits the frequencies to some bandwidth of interest and makes measurements and simulations feasible. In other words, the noise can have a flat PSD only over a given finite bandwidth but, for the purpose of the experiment / simulation, be conveniently modeled as white noise.
Because power can be written as
If the input
| (4.43) |
i. e.,
All the previous discussion assumed the input signal has a white PSD. But the signal amplitudes could have any probability distribution, such as Laplacian, uniform, etc. A very special case of white noise is when this probability distribution is Gaussian, which corresponds to the so-called WGN, introduced earlier in Section 1.10.1.0.
4.5.3 Advanced: Filtering discrete-time signals that have a white PSD
It is important to distinguish the continuous-time white noise, which has a flat bilateral PSD with
value
Discrete-time white noise
As mentioned, a discrete-time white process
|
In other words, the constant PSD level
Therefore, a discrete-time bilateral white PSD (with a constant level) can be denoted as:
| (4.44) |
This result may be confusing because the values of power and spectral density coincide.
Denoting as
| (4.45) |
Eq. (4.45) suggests that the output PSD
Example 4.8. Generation of uniform and Gaussian white noise using Matlab/Octave.
A discrete-time white noise with i. i. d. samples is relatively easy to generate in Matlab/Octave
using functions that implement random number generators. For example, x=rand(1,1000)-0.5
creates a signal with an autocorrelation that approximates an impulse at the origin and
1,000 samples that are uniformly distributed. The subtraction of 0.5 is necessary to make
Converting white noise from continuous to discrete-time
The signals discussed in the previous section were already created in discrete-time. But in some cases the discrete-time white noise is obtained from a continuous-time version or should be interpreted as such.
For example, assume
From Eq. (4.38), the PSD at the output of
In more details: assume that
Example 4.9. Discretizing white noise. For example, given that white noise with
|
and the power of
Figure 4.25 depicts the PSD for both continuous and discrete-time signals. Note that the abscissa of
Example 4.10. Filtering white noise through systems with unitary-energy impulse
responses. Another situation of interest is when white noise is filtered by an LTI that has impulse
response
In this case, the Parseval relation of Eq. (A.51) indicates that
| (4.46) |
Therefore, the power of
Converting white noise from discrete to continuous-time
The conversion of a discrete-time white noise
Similar to the previous discussion about white noise C/D conversion in this section, if
Gaussian signal filtered by LTI system remains Gaussian
If the system is LTI and its input is a realization of a WGN process, then it is well-known from random processes textbooks that the filter output is another Gaussian process. In other words, a Gaussian signal filtered by a linear system remains Gaussian but, in the general case, with a non-white PSD that was shaped by the filter.
When the LTI is an ideal lowpass filter such as in Example 4.9, besides being Gaussian, the output is
also flat within the filter passband (or “white” within this band). Hence, when the input is WGN, and