Bandpass Random Processes
A random process is called a band-pass process if its power spectrum is zero outside a band. is called the band-width and is called the centre frequency of the band-pass process
If is very small compared to the centre frequency, then is called a narrow band process.
We can similarly define a low-pass random process as a random process if its power spectral density is zero outside the band
- In telecommunication, we often deal with random signals which have PSD concentrated in a small frequency band and negligible outside this band. Following are two examples. The information bearing signals like speech, image and video are low-pass signals. These information-bearing signals modulate a sinusoidal carrier for transmitting over the communication channel that acts as a bandpass filter. For example, the amplitude- modulated waveform received by a communication receiver is modeled as an amplitude-modulated random-phase sinusoid
where and M(t) is a WSS process independent of Here the modulation process translates the spectrum of from base-band to a band centred around
- The noise associated with communication signal undergoes band-pass filtering in the communication receiver and the band-pass filtered noise can be modeled as a band-pass process.
We can do the correlation and power spectral analysis of such signals in the usual manner. However, for analysis of nonlinear operations like the multiplication with a random process, the following trigonometric representation is useful.
Fig. Power spectrum of a band-pass random process
Rice’s representation or quadrature representation of a WSS process:
An arbitrary zero-mean WSS process can be represented in terms of the slowly varying components and as follows:
(1)
where is a center frequency arbitrary chosen in the band. and are respectively called the in-phase and the quadrature-phase components of
Let us choose a dual process such that
Then, (2)
and (3)
We require the processes and to be WSS.
Note that
As is zero mean, we require that
and
Again
As each of is zero-mean, we require that
.
Also and
and
Thus, will be independent of t if and only if
Under these conditions
and
How to find satisfying the above two conditions?
For this, consider to be the Hilbert transform of, i.e.
where and the integral is defined in the mean-square sense.
The frequency response of the Hilbert transform is given by
and
From the above two relations, we get
The Hilbert transform of is denoted as Therefore, from (2) and (3) we establish
and
The realization for the in phase and the quadrature phase components is shown in the figure below.
From the above analysis, we can summarise the following expressions for the autocorrelation functions
where
The variances and are given by
Taking the Fourier transform of we get
Similarly,
Notice that the cross power spectral density is purely imaginary. Particularly, if is locally symmetric about
implying that
Consequently, the zero-mean processes and are also uncorrelated.
Example
Suppose the band-limited white-noise processhas the PSD as shown in Fig below.
We have earlier shown that
The plot of is as shown in the Fig. Therefore,
Remark
(1)The representation of the band-pass process in terms of the in-phase and the quadrature phase components is not unique. By selecting different we can have different representations.
(2)The band-pass process can be represented as
where
and
and are respectively called the envelope and the phase of the process
(3)If is a Gaussian process, then ( being linear transform of ) is also Gaussian. Consequently, the processes and are also Gaussian.
(4)Under the condition of local symmetry of about and are uncorrelated. If and are also Gaussian. processes, then and will be independent . Using the results on the PDF of functions of RVs, we get following.
- The envelope will be Rayleigh-distributed. Thus
- The phase will be distributed