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.

Top: Magnitude spectrumof rectangular pulse. Notice thathas

FIGURE 1.2. Top: Magnitude spectrumtmp2D79_thumb[2]of rectangular pulse. Notice thattmp2D80_thumb[2]has


zeros attmp2D81_thumb[2]Bottom: Phase spectrum argtmp2D82_thumb[2]of rectangular pulse. Notice

that argtmp2D83_thumb[2]has jumps equal totmp2D84_thumb[2]

The equation for the pulse is

tmp2D97_thumb[2]

Let the frequency be f in Hz [cycle/s] andtmp2D98_thumb[2][radian/s]. Then the

Fourier transform of f (t) is

tmp2D100_thumb[2]

The magnitude spectrumtmp2D101_thumb[2]and the phase spectrum argtmp2D102_thumb[2]are depicted

in Figure 1.2. Notice that argtmp2D103_thumb[2]of /(f) is linear fortmp2D104_thumb[2]with jumps equal totmp2D105_thumb[2]fortmp2D106_thumb[2]because of the change of sign oftmp2D107_thumb[2]at these points.

From (1.4) follows that thetmp2D108_thumb[2]for a rectangular pulse has a triangular waveform; see Figure 1.3,

tmp2D117_thumb[2]

Autocorrelation functionof the rectangular pulse shown in Figure 1.1.

FIGURE 1.3. Autocorrelation functiontmp2D119_thumb[2]of the rectangular pulse shown in Figure 1.1.

The energy density spectrumtmp2D122_thumb[2]of f(t) is a real function becausetmp2D123_thumb[2] tmp2D124_thumb[2]

tmp2D128_thumb[2]

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

tmp2D129_thumb[2]

The energy density spectrumtmp2D130_thumb[2]is depicted in Figure 1.4. In discrete time the rectangular pulse takes on the form

tmp2D132_thumb[2]

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

tmp2D134_thumb[2]

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

tmp2D135_thumb[2]Energy density spectrumof the rectangular pulse shown in Figure 1.1. Note thathas zeros at

FIGURE 1.4. Energy density spectrumtmp2D137_thumb[2]of the rectangular pulse shown in Figure 1.1. Note thattmp2D138_thumb[2]has zeros attmp2D139_thumb[2]

The ACF of the stationary process X (t) is defined astmp2D143_thumb[2]

withtmp2D144_thumb[2]denoting the expectation operator andtmp2D145_thumb[2]being the lag. The inverse Fourier transform is given by

tmp2D152_thumb[2]


The quantitytmp2D153_thumb[2]is called the power density spectrum of X (t).

A discrete-time random process (sequence) has infinite energy but has a finite average power given bytmp2D154_thumb[2]According 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 sequencetmp2D155_thumb[2]

tmp2D159_thumb[2]

The inverse Fourier transform is

tmp2D160_thumb[2]

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.

Autocorrelation functionfor random sequence of pulses with amplitude

FIGURE 1.5. Autocorrelation functiontmp2D162_thumb[2]for random sequence of pulses with amplitudetmp2D163_thumb[2]

The ACF for the sample function x (t) of a process X (t) consisting of a random sequence of pulses with amplitudetmp2D166_thumb[2]and with equal probability for the outcome + 1 and —1 is tmp2D170_thumb[2]

The ACF is plotted in Figure 1.5. It follows that the power spectral density is

tmp2D171_thumb[2]

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 densitytmp2D172_thumb[2]of a random sequence of pulses with amplitudetmp2D173_thumb[2]differs from the energy spectral densitytmp2D174_thumb[2]given in (1.11), of a single rectangular pulse by only a scalar factor T.

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