Signals and Systems (GPS and Galileo Receiver) Part 4

Representation of Bandpass Signals

Signals can also be classified into lowpass, bandpass, and highpass categories depending on their spectra. We define a bandpass signal as a signal with frequency content concentrated in a band of frequencies above zero frequency. Bandpass signals arise in the GPS and Galileo systems where the information-bearing signals are transmitted over bandpass channels from the satellite to the receiver.

Let us now consider an analog signal s(t) with frequency content limited to a narrow band of total extent 2^ and centered about some frequencytmp2D263_thumb[2][2]_thumbsee Figure 1.12. We term such a signal a bandpass signal and represent it as


where a(t) is called the amplitude or envelope of the signal and (p(t) is called the phase of the signal. The frequency fc is called the carrier frequency. Equation (1.23) represents a hybrid form of amplitude modulation and angle modulation and it includes amplitude modulation, frequency modulation, and phase modulation as special cases. For more details on this topic we refer to Haykin (2000).

By expanding the cosine function in (1.23) we obtain an alternative representation of the bandpass signal:


where by definition


Since the sinusoidstmp2D269_thumb[2][2]_thumbdiffer by 90°, we say that they are in phase quadrature. We refer totmp2D270_thumb[2][2]_thumbi as the in-phase component of s (t) and totmp2D271_thumb[2][2]_thumbas the quadrature component of the bandpass signal s (t).Bothtmp2D272_thumb[2][2]_thumband tmp2D273_thumb[2][2]_thumbare real-valued lowpass signals.

Another representation of the bandpass signal s (t) is obtained by defining the complex envelope


such that


Equation (1.27) can be seen as the Cartesian form of expressing the complex envelopetmp2D281_thumb[2][2]_thumbIn polar form we may writetmp2D282_thumb[2][2]_thumbas


wheretmp2D286_thumb[2][2]_thumbare both real-valued low-pass signals. That is, the information carried in the bandpass signal s (t) is preserved intmp2D287_thumb[2][2]_thumbwhether we represent s (t) in terms of its in-phase and quadrature components as in (1.24) or in terms of its envelope and phase as in (1.23).

In the frequency domain the bandpass signal s (t) is represented by its Fourier transform


In the above derivation we usedtmp2D291_thumb[2][2]_thumbIf we denote the Fourier transform of the signalstmp2D292_thumb[2][2]_thumbrespectively, we may write (1.30) as


This relation is illustrated in Figure 1.12.

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