The MSK symbols are sinusoids over one symbol interval $T$ with different frequencies $\omega_i$: $$g_i(t)=\sin(\omega_i t),\quad 0\le t < T\tag{1}$$ The pulses $g_i(t)$ are orthogonal if $$\int_{-\infty}^{\infty}g_i(t)g_j(t)dt = 0,\quad i\neq j\tag{2}$$ is satisfied. Using the pulses from (1) gives $$\int_{0}^{T}\sin(\omega_i t)\sin(\omega_j t)dt = 0,\quad i\neq j\tag{3}$$ Expanding (3) results in the condition $$\frac12\int_{0}^{T}cos(\omega_i-\omega_j)dt-\frac12\int_{0}^{T}cos(\omega_i+\omega_j)dt = 0,\quad i\neq j$$ or, equivalently, $$\sin[(\omega_i-\omega_j)T]=0,\quad i\neq j\\ \sin[(\omega_i+\omega_j)T]=0,\quad i\neq j$$ which finally results in the conditions $$\Delta\omega=\omega_i-\omega_j=\frac{m\pi}{T},\quad m=1,2,\ldots\\ \omega_c=(\omega_i+\omega_j)/2=\frac{n\pi}{2T},\quad n=1,2,\ldots\tag{4}$$ where $\Delta\omega$ is the frequency separation, and $\omega_c$ is the nominal carrier frequency. From (4), the minimum frequency separation guaranteeing orthogonality is obtained for $m=1$: $$\Delta\omega=\frac{\pi}{T}\tag{5}$$ This frequency separation is half the frequency separation of traditional FSK ("Sunde's FSK"). This is the reason why this technique is called

*minimum*shift keying. Due to the smaller frequency separation, MSK has a higher bandwidth efficiency than Sunde's FSK.

In the following I will restrict the discussion to binary MSK. With $\omega_c$ and $\Delta\omega$ given by (4) and (5) we can write the MSK signal as \begin{align} s(t)&=\cos[\omega_ct+b(t)\cdot\frac{\Delta\omega}{2}\cdot t+\phi(t)]\\ &= \cos[\omega_ct+b(t)\cdot\frac{\pi t}{2T}+\phi(t)]\tag{6} \end{align} where $b(t)$ assumes the values $+1$ and $-1$, depending on the bit sequence: $$b(t)=\sum_kA_kp(t-kT),\quad A_k\in\{+1,-1\}$$ with the rectangular pulse $p(t)$ being $1$ in the interval $[0,T]$ and zero everywhere else. The piecewise constant function $\phi(t)$ in (6) is necessary to guarantee that the MSK signal has a continuous phase. At $t=kT$ the information bit (and the function value of $b(t)$) changes from $A_{k-1}$ to $A_k$ and the value of the phase function $\phi(t)$ changes from $\phi_{k-1}$ to $\phi_k$. In order for the total phase of the MSK signal (6) to be continuous, the following equation must be satisfied: $$\omega_ckT+A_{k-1}\frac{\pi kT}{2T}+\phi_{k-1}=\omega_ckT+A_{k}\frac{\pi kT}{2T}+\phi_{k}$$ which results in $$\phi_k=\phi_{k-1}+(A_{k-1}-A_k)\frac{k\pi}{2}\quad(\textrm{mod }2\pi)\tag{7}$$ Since $|A_{k-1}-A_k|$ is either $0$ or $2$, according to (7) $\phi(t)$ can only change (modulo $2\pi$) when $k$ is odd and if $A_{k-1}\neq A_k$. So we can rewrite (7) as $$\phi_k=\begin{cases}\phi_{k-1},& k\textrm{ even OR }A_k=A_{k-1}\\ \phi_{k-1}\pm\pi,& k\textrm{ odd AND }A_k\neq A_{k-1}\end{cases}\tag{8}$$ One important conclusion from (8) is that the discontinuous function $\phi(t)$ defined by the values $\phi_k$ changes between the values $0$ and $\pm\pi$ (if we assume $\phi_0=0$), and, even more importantly, it can only change when $k$ is odd. This means that it changes at half the symbol rate (or, because we consider binary MSK, half the bit rate), i.e. its rate is $1/2T$.

The following figure shows a possible information signal $b(t)$ and the corresponding in-phase and quadrature components $a_I(t)$ and $a_Q(t)$, respectively. It can be seen that both $a_I(t)$ and $a_Q(t)$ have a rate of $1/2T$ and that both components never change at the same time, i.e. they are offset by one bit period.

The next figure shows the sinusoidally pulse-shaped signals $a_I(t)\cos\left(\frac{\pi t}{2T}\right)$ and $a_Q(t)\sin\left(\frac{\pi t}{2T}\right)$, and the final MSK-signal $s(t)$:

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