After explaining how amplitude modulation and frequency modulation work, which are both *analog* modulation schemes, I’d now like to give an example of a *digital* modulation techique. A basic, but very relevant, example of a digital modulation scheme is *Phase-Shift Keying* (*PSK*). Let’s use the most basic form of that, *Binary Phase-Shift Keying* (*BPSK*).

## Carrier and Message

PSK shifts the *phase* of a carrier according to the message that is to be transmitted. A carrier is a simple sinusoidal wave, as illustrated in Figure 1.

In contrast to the message in the mentioned analog modulation schemes, the message is now digital. This means that it has, in the binary case, only two possible levels. These levels are either equal to the bit values 0 and 1, or are mapped to, e.g., −1 and +1. As an example, Figure 2 shows a message that corresponds to the bit sequence 1011010001.

## Binary Phase-Shift Keying (BPSK) Modulation

Mathematically, the modulation amounts to sending a version of the carrier with a different phase shift for each bit value. The carrier wave is defined as

\[c(t) = A\cos(2 \pi f_ct),\]

with \(A\) the amplitude of the wave and \(f_c\) the frequency. The term *keying* indicates that the phase is shifted suddenly, which makes BPSK a digital modulation technique. This constrasts with analog phase modulation, where the phase of the carrier is varied continuously.

Digital modulation schemes modulate discrete *symbols* of a given symbol time \(T\). For example, if the symbol frequency is 1 MHz, it means that \(T=1\,\mu s\). For *binary* PSK, the two carrier phases are \(\pi\) radians (180 degrees) apart. This results in two possible signals to be transmitted, \(s_0(t)\) for bits with a value of 0 and \(s_1(t)\) for bits with a value of 1,

\[s_0(t)=A\cos(2\pi f_c t)\]

and

\[s_1(t)=A\cos(2\pi f_c t+\pi)=-A\cos(2\pi f_c t).\]

To create the final modulated carrier, the waveform corresponding to each bit to be sent is transmitted for a duration of \(T\) seconds. If \(T\) would be exactly two cycles of the carrier, then this could look like Figure 2 for the bit sequence 1011010001. For clarity, the message is shown below the waveform.

In practice, there doesn’t have to be a integer ratio between the symbol and carrier frequencies.

## Multiple PSK

Instead of binary PSK, it is also possible to have more than two phase-shift offsets. This is *Multiple Phase-Shift Keying* (*MPSK*). The number of offsets \(M\) is typically taken to be a power of two, i.e., \(M=2^k\) for some small \(k\), so that an integer number of bits can be put in a single symbol. There is one bit for \(M=2\), two bits for \(M=4\), and three bits for \(M=8\). These last two are often referred to as QPSK and 8PSK, respectively. The “Q” comes from *Quadrature* PSK, but explaining that terminology is something for a future article.

The general waveform is given as

\[s_m(t)=A\cos\left(2\pi f_c t+\frac{\pi(2m+1)}{M}\right),\]

for \(m=0,\ldots,M-1\).

Note that, for \(M=2\), this results in \(s_0(t)=A\cos(2\pi f_c t+\pi/2)\) and \(s_1(t)=A\cos(2\pi f_c t-\pi/2)\) instead of the expressions given above. This is just a phase shift by \(\pi/2\), so it doesn’t matter in practice. However, it is the more traditional expression for \(M>2\).

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