Signals and Systems (GPS and Galileo Receiver) Part 2

Rectangular Pulse

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.

FIGURE 1.2. Top: Magnitude spectrumof rectangular pulse. Notice thathas

zeros atBottom: Phase spectrum argof rectangular pulse. Notice

that arghas jumps equal to

The equation for the pulse is

Let the frequency be f in Hz [cycle/s] and[radian/s]. Then the

Fourier transform of f (t) is

The magnitude spectrumand the phase spectrum argare depicted

in Figure 1.2. Notice that argof /(f) is linear forwith jumps equal toforbecause of the change of sign ofat these points.

From (1.4) follows that thefor a rectangular pulse has a triangular waveform; see Figure 1.3,

FIGURE 1.3. Autocorrelation functionof the rectangular pulse shown in Figure 1.1.

The energy density spectrumof f(t) is a real function because

The energy density spectrum of the rectangular pulse f (t) is

The energy density spectrumis depicted in Figure 1.4. In discrete time the rectangular pulse takes on the form

where N is an integer. The Fourier transform of f (n) is

Random Signals

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

FIGURE 1.4. Energy density spectrumof the rectangular pulse shown in Figure 1.1. Note thathas zeros at

The ACF of the stationary process X (t) is defined as

withdenoting the expectation operator andbeing the lag. The inverse Fourier transform is given by

The quantityis called the power density spectrum of X (t).

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.

FIGURE 1.5. Autocorrelation functionfor random sequence of pulses with amplitude

The ACF for the sample function x (t) of a process X (t) consisting of a random sequence of pulses with amplitudeand with equal probability for the outcome + 1 and —1 is

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).

Note that the power spectral densityof a random sequence of pulses with amplitudediffers from the energy spectral densitygiven in (1.11), of a single rectangular pulse by only a scalar factor T.