# Is this figure a correct (O)QPSK waveform?

I am currently reading Pasupathy, Subbarayan. "Minimum shift keying: A spectrally efficient modulation." IEEE Communications Magazine 17.4 (1979): 14-22.

The paper explains MSK as a form of OQPSK with a half sinusoid instead of rectangular weighting with a figure below:

But in my opinion, this figures seems incorrect. Because figure 5(a) shows a bit stream +1, -1, -1 and +1 inside a waveform, but the actual waveform is drawn as if its corresponding bit stream is +1, +1, -1, -1.

Am I missing something?

# Edit

Maybe I learned that the original problem is solved by taking a phase into account.

Another problem is, a waveform in the IEEE 802.15.4 standard differs from the original one.

It seems that a direction (upward/downward) of a waveform indicates a bit.

Are both acceptable? Is 802.15.4's O-QPSK a variant of the original one?

• Maybe I got it. a direction of a waveform (upward/downward) does not indicate a bit, but a phase of waveform does. – Jeon Oct 20 '17 at 8:22
• Now the question is the original O-QPSK vs 802.15.4's O-QPSK – Jeon Oct 20 '17 at 9:16

i think, for OQPSK with the convention that the even-indexed data samples go into $$\i(t)\$$ and the odd-indexed data samples go into $$\q(t)\$$ , the IQ waveforms, in terms of the data is :

\begin{align} i(t) &= \sum\limits_{m=-\infty}^{\infty} x[2m] \, p(t-2m T_c) \\ \\ q(t) &= \sum\limits_{m=-\infty}^{\infty} x[2m+1] \, p(t-(2m+1) T_c) \\ \end{align}

in the waveforms shown in the OP, the definition of the pulse shape $$\p(t)\$$ is:

$$p(t) = \begin{cases} 0 \qquad & t < 0 \\ \sin\left(\tfrac{\pi}{2 T_c} t \right) \qquad & 0 \le t < 2T_c \\ 0 \qquad & 2T_c \le t \\ \end{cases}$$

and the bipolar binary data is:

$$x[n] = 2 a[n] - 1$$

and $$\a[n] \in \{0,1\}\$$ are the raw bit data.

The nice thing about Offset-QPSK is that it naturally separates the even and odd samples because the $$\i(t)\$$ depends only on the even samples and switches at times of even multiples of $$\T_c\$$. $$\q(t)\$$ depends only on the odd samples and switches at times of odd multiples of $$\T_c\$$.

A smarter pulse shape to simply constrain the IQ signal to a bandwidth of $$\\frac{1}{T_c}\$$ is:

$$p(t) = \operatorname{sinc}\left(\tfrac{t - T_c}{2 T_c} \right) w\left(\tfrac{t-T_c}{N T_c}\right)$$

and

$$\operatorname{sinc}(u) \triangleq \begin{cases} 1 \qquad & u = 0 \\ \frac{\sin(\pi u)}{\pi u} \qquad & 0 < |u| \\ \end{cases}$$.

I didn't bother to define the window function here, but suggest a Kaiser of width $$\2N\$$ samples.