Let us now consider a single rectangular pulse f (t) with amplitude 1 and pulse width equal to T .In Figure 1.1 we have shifted the pulse —T/2 to place it sym-metrically around t = 0.
The equation for the pulse is
Fourier transform of f (t) is
The energy density spectrum of the rectangular pulse f (t) is
where N is an integer. The Fourier transform of f (n) is
A random process can be viewed as a mapping of the outcomes of a random experiment to a set of functions of time—in this context a signal X (t). Such a signal is stationary if the density functions p(X (t)) describing it are invariant under translation of time t. A random stationary process is an infinite energy signal, and therefore its Fourier transform does not exist. The spectral characteristics of a random process is obtained according to the Wiener-Khinchine theorem by computing the Fourier transform of the ACF. That is, the distribution of signal power as a function of frequency is given by
A discrete-time random process (sequence) has infinite energy but has a finite average power given byAccording to the Wiener-Khinchine theorem we obtain the spectral characteristic of the discrete-time random process by means of the Fourier transform of the autocorrelation sequence
The inverse Fourier transform is
Random Sequence of Pulses
In Section 1.1.4 we studied the characteristics of a single rectangular pulse f (t). Next we want to become familiar with the same properties for a random sequence of pulses with amplitude ±1; each pulse with duration T.
The ACF is plotted in Figure 1.5. It follows that the power spectral density is
which is plotted in Figure 1.6. The power spectral density of X (t) possesses a main lope bounded by well-defined spectral nulls. Accordingly, the null-to-null bandwidth provides a simple measure for the bandwidth of X (t).