# Phase modulation

Fig. 1: The "red" sinusoidal oscillation is delayed by a quarter of the period compared to the "blue" sinusoidal oscillation. In the complex bilateral frequency spectrum, this shows up as a 90 ° rotation of the two spectral lines that represent the carrier.

The phase modulation is a method by which an analog or a digital signal over a communication channel is transmitted. Phase modulation is closely related to frequency modulation . Both modulations belong to the group of angle modulation methods .

## Analog phase modulation

In the case of phase modulation, the modulated transmission signal can generally be represented by a transmission frequency , the frequency of which only changes to a certain extent when the useful signal frequency to be transmitted changes over time. This change in frequency results in a phase shift between the transmission signal and the original transmission frequency . If f s is constant over time, the transmission frequency is output. Mathematically, this relationship can be described as follows with any real constant k: ${\ displaystyle f_ {0}}$${\ displaystyle f_ {s}}$${\ displaystyle f_ {0} (t)}$${\ displaystyle f_ {0} (t = 0)}$${\ displaystyle f_ {0}}$

${\ displaystyle f_ {0, PM} (t) = f_ {0} + kf_ {s} '(t) \}$

${\ displaystyle k}$is a factor that indicates how much the phase of the transmitted signal should change depending on the useful signal, and is a type of phase modulation index . The expression describes the time derivative of the useful signal to be transmitted. The modulated transmission signal results in: ${\ displaystyle f_ {s} '(t)}$

${\ displaystyle m (t) = \ cos \ left (2 \ pi f_ {0} t + 2 \ pi kf_ {s} (t) \ right) \}$

The second summand can be visualized as follows: the instantaneous values at certain points in time of the useful signal virtually adjust the phase angle of the cosine function, from which the name of this type of modulation is derived. ${\ displaystyle f_ {s} (t)}$

### Practical applications

The analog phase modulation was only widely used in one area: NTSC and PAL color television signals transmit the color tone in a phase-modulated manner .

The fact that this method is otherwise rarely used is primarily due to a fundamental difficulty which, at least before the introduction of integrated circuits, could only be overcome with significant effort: The receiver must have a phase-synchronous "copy" of the unmodulated transmission frequency ; by comparing this reference signal with the received signal, it determines the phase shift. (The television standards mentioned above solve this problem by transmitting a few oscillations in each line of the picture (" color burst "), to which an oscillator in the receiver is synchronized.) ${\ displaystyle f_ {0}}$${\ displaystyle f_ {0}}$

Phase modulation therefore only became essential for practical application with digital transmission methods , where synchronization and demodulation can be solved using a Costas loop .

## Digital phase modulation

The phase shift keying ( English phase shift keying abbreviated PSK ) represents the digital form of the phase modulation. The sinusoidal carrier oscillation is switched over in discrete phase steps by the digital data stream to be transmitted. The designations for digital modulations come from their properties at the sampling times on the receiver side. Keying means (toggle) keys, derived from "Key" which is also the name for the Morse key.

The simplest form is binary phase shift keying (BPSK) with two phase states. With quadrature phase shift keying (4-PSK or QPSK) 2 bits are transmitted per symbol, with 8-PSK 3 bits per symbol. 4-PSK is used, for example, in the transmission of facsimiles over the telephone network.

If the phase shift keying is combined with the amplitude shift keying (ASK), the result is the quadrature amplitude modulation (QAM).

## Examples

Fig. 4: BPSK with soft keying, in the middle the switching time range

The sound sample is the answer from a fax when it is called. The first signal is a pure sine tone on which a cracking noise is superimposed several times. This is a phase shift of 180 °, see picture. It can transmit information of exactly one bit. Therefore it is called binary phase shift ( binary phase shift keying called).

With a phase shift of 90 °, 4 different states can be coded: 0 °, + 90 °, −90 °, and 180 ° ( quadrature phase-shift keying or quaternary phase-shift keying or QPSK ). With multiples of 45 ° there are 8 states or 3 bits ( octal phase-shift keying or OPSK). In general, one speaks of multiple phase-shift keying or MPSK.