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Spectral analysis and modulation, part 5: Phase shift keying
This article introduces Phase Shift Keying (PSK) and its variants BPSK, QPSK, QAM, and GMSK. We look at complex modulation, a method encompassing all PSK methods, and the Hilber Transformer, a filter model for sideband (SSB) signals.
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Part 4 examines amplitude and frequency modulation.
5.3.3 Phase shift keying (PSK)
In a PSK system, the phase shift of the signal is controlled by the baseband signal, i.e.
The carrier frequency f(t) = fc and the amplitude a(t) = a are constant. The phase functions φn(t) are defined over a finite period of time, i.e. t0 ≤ t < t0 + T , where T is the symbol time as before. A simple example would be a system using binary symbols, i.e. M = 2 and square pulses for phase functions according to the below (see Figure 5-8). Commonly, the shape of the phase functions is chosen to be smoother than square pulses, for spectral occupancy reasons.
symbol "0":
symbol "1":
Figure 5-8. Baseband signal and modulated signal for PSK, M = 2.
Often, the use of phasors (Denbigh, 1998) can be beneficial when describing modulation processes. Starting out with a general cosine signal of a given frequency fn, amplitude an and phase shift φn we have
using Euler's formula, equation (5.76) can be rewritten as
where the complex number zn is denoted the phasor of the signal. If the carrier frequency fn = fc is given and constant, the phasor will represent the signal without any ambiguity, since
Further, the phasor, being a complex number, can be drawn as a vector starting from the origin and having magnitude an and angle φn in a complex xy-plane.
Phasors are a handy tool, used in many applications, not only in signal theory. However, a caution should be issued: when dealing with signals and modulation the magnitude of the phasor, denoted an, commonly refers to the amplitude of the signal, i.e. the peak value of the cosine signal. In literature dealing with electrical power applications the magnitude of the phasor may refer to the RMS value of the cosine (Denbigh, 1998).
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