## 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 frequencysee 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 sinusoidsdiffer by 90°, we say that they are in phase quadrature. We refer toi as the in-phase component of s (t) and toas the quadrature component of the bandpass signal s (t).Bothand are 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 envelopeIn polar form we may writeas

whereare both real-valued low-pass signals. That is, the information carried in the bandpass signal s (t) is preserved inwhether 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 usedIf we denote the Fourier transform of the signalsrespectively, we may write (1.30) as

This relation is illustrated in Figure 1.12.