# How are the constellation point coordinates determined in digital modulation

I need to understand where the concept of the distance comes from. What I mean is, how do we know what coordinates are to assign to points in the constellation. For example here for QPSK the points are given coordinates $$\pm\sqrt(E_b),\pm\sqrt(E_b)$$

For BPSK on the other hand they are $$\pm\sqrt(E_b),0$$

This question was answered here as:

"A symbol with coordinates $$(x,y)$$ has energy $$x^2+y^2$$", and then it makes sense as for example for BPSK the energy per symbol is just $$E_b$$ and $$\sqrt(E_b)^2+0^2=E_b$$ but where does this come from? How to derive this? Thanks.

• Hey, has your question been answered? Sep 17, 2020 at 21:34

## 2 Answers

Constellation diagrams exist in what is called signal space which is an abstraction used to describe finite-energy signals. The coordinate axes, even if they are marked $$x$$ and $$y$$ as in Marcus Muller's answer, really represent unit-energy signals such as $$s_I(t) = \sqrt{\frac 2T}\cos(2\pi f_c t)\mathbf 1_{t \in [0,T)}(t),$$ $$s_Q(t) = -\sqrt{\frac 2T}\sin(2\pi f_c t)\mathbf 1_{t \in [0,T)}(t)$$ where $$T = \frac{n}{f_c}$$ and $$\mathbf 1_{t \in [0,T)}(t)$$ means a rectangular pulse of duration $$T$$. Note that both sinusoidal pulses consist of and integer number $$n$$ periods of a sinusoid of frequency $$f_c$$. Verify for yourself that $$s_I(t)$$ and $$s_Q(t)$$ are indeed unit energy signals and that they are orthogonal, that is, verify that $$\int_0^T [s_I(t)]^2 \mathrm dt = \int_0^T [s_Q(t)]^2 \mathrm dt = 1;\quad \int_0^T s_I(t)s_Q(t) \mathrm dt =0.$$ If we use $$\pm As_I(t)$$ as our PSK signals, then the signal energy is $$A^2$$(why?) and so if we use $$E_b$$ to denote the energy per bit, then with respect to the signal space with axes $$s_I(t)$$ and $$s_Q(t)$$, the two signals used in PSK can be represented by the constellation points $$(\sqrt{E_b},0)$$ and $$(-\sqrt{E_b},0)$$. Left-handed folks might prefer to use PSK signals $$\pm As_Q(t)$$ in which case the signal constellation would have points $$(0,\sqrt{E_b})$$ and $$(0,-\sqrt{E_b})$$. Mealy-mouthed ambidextrous people might want to use $$\pm A[s_I(t)\cos(\theta)+s_Q(t)\sin(\theta)] = \pm A\sqrt{\frac 2T}\cos(2\pi f_c t+\theta)\mathbf 1_{t \in [0,T)}(t)$$ as PSK signals in which case in the signal space with axes $$s_I(t)$$ and $$s_Q(t)$$, the constellation points would be $$\pm (\sqrt{E_b}\cos\theta,\sqrt{E_b}\sin\theta)$$. But if we had chosen axes $$\sqrt{\frac 2T}\cos(2\pi f_c t+\theta)$$ and $$-\sqrt{\frac 2T}\sin(2\pi f_c t+\theta)$$ instead (this just corresponds to a rotation of axes by $$\theta$$ from the previous signal space), then the constellation points would once again be $$(\sqrt{E_b},0)$$ and $$(-\sqrt{E_b},0)$$.

Thus, in signal space in which the coordinate axes represent unit-energy orthogonal signals, the energy of the signal corresponding to a constellation point equals the squared distance from the origin. Note that in (antipodal) PSK considered above, this squared distance from the origin is $$E_b$$ for each of the two constellation points. For QPSK (which I have ignored for the most part but you can read about it in this answer of mine), each of the four constellation points are at squared distance $$2E_b$$ from the origin which makes sense in that $$E_b$$ is the energy per bit and a QPSK signal carries two bits.

Finally, with regard to the importance of distance as a concept, given two constellation points in signal space, the probability of error (in an AWGN channel with two-sided power spectral density $$N_0/2$$) is $$Q\left(\frac{d}{\sqrt{2N_0}}\right)$$ where $$d$$ is the distance between the two points. For the PSK examples considered above, $$d = 2\sqrt{E_b}$$ which gives the familiar result $$P_e = Q\left(\frac{d}{\sqrt{2N_0}}\right) = Q\left(\frac{2\sqrt{E_b}}{\sqrt{2N_0}}\right) = Q\left(\sqrt{\frac{2E_b}{N_0}}\right)$$

What I mean is, how do we know what coordinates are to assign to points in the constellation.

We don't; you can rotate your PSK however you like it. You can also vary the diameter however you like it.

It's all convention. All that PSK means is: "the information is in the phase of the symbols, all other parameters (which is amplitude) are left at a constant value".

A symbol with coordinates $$(x,y)$$ has energy $$x^2+y^2$$

… but where does it come from?

Math. That's simply how power is defined: magnitude squared of an amplitude.