Spectral analysis and modulation, part 5: Phase shift keying

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) = f_c and the amplitude a(t) = a are constant. The phase functions φ_n(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 f_n, amplitude a_n and phase shift φ_n we have

using Euler's formula, equation (5.76) can be rewritten as

where the complex number z_n is denoted the phasor of the signal. If the carrier frequency f_n = f_c 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 a_n and angle φ_n 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 a_n, 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).