Simulating Minimum Shift Keying Transmitter

Posted by Krishna Pillai

January 19, 2008

Minimum shift keying (MSK) is an important concept to learn in digital communications. It is a form of continuous phase frequency shift keying . In minimum phase shift keying, two key concepts are used.

(a) The frequency separation of the sinusoidals used for representing bits 1’s and 0’s are $\frac{1}{2T}$ , where $T$ is the symbol period.

(b) It is ensured that the resulting waveform is phase continuous.

Motivation of continuous phase

In a previous post (here), we have understood that the minimum frequency separation for two sinusoidals having zero phase difference to be orthogonal is $\frac{1}{2T}$ , where $T$ is the symbol period. However, it can be observed that at each symbol boundary, there is a phase discontinuity. The presence of phase discontinuities can result in large spectral side lobes outside the desired bandwidth. Hence the need for having a frequency modulated signal which is phase continuous.

Phase growth in MSK

To ensure continuous phase, the phase of the carrier of the MSK signal is
$\theta(t) = \frac{pi}{2}\left(\frac{t-nT}{T}\right)a_n + \frac{\pi}{2}\sum_{k=-\infty}^{n-1}a_k$ , where

$T$ is the symbol period,
$a_n$ corresponds to -1 for bit 0, +1 for bit 1 respectively.

The corresponding carrier signal is
$\begin{eqnarray}u(t) &=& \sqrt{\frac{2E}{T}} cos[2\pi f_c t + \theta (t) ] \\ & = & \sqrt{\frac{2E}{T}} cos[2\pi f_c t + \frac{pi}{2}\left(\frac{t-nT}{T}\right)a_n + \frac{\pi}{2}\sum_{k=-\infty}^{n-1}a_k] \\ & = & \sqrt{\frac{2E}{T}} cos[2\pi (f_c + \frac{1}{4T}a_n)t - \frac{n\pi}{2}a_n + \frac{\pi}{2}\sum_{k=-\infty}^{n-1}a_k] \end{eqnarray}$ .

Phase growth in MSK
Figure: Phase transition diagram for MSK (Ref: Fig10.22 in [COMM-PS]

Simulation Model

Simple Octave/Matlab code for simulating and plotting binary Minimum Shift Keying is kept here.

Next steps

We have observed that bit error probability of classical coherent binary frequency shift keying is 3dB poorer compared to bit error probability of binary phase shift keying. However, in minimum shift keying, using the knowledge of the phase transitions, we should be able to recover the 3dB loss associated with FSK and get a performance comparable to BPSK. We will hopefully discuss that in a future post.