# Technical Term for the Sum I + Q

Most likely a very simple question, but I haven't been able to find a good answer.

When working with analytical signals in general, and software defined radio in particular, a common operation is to take the sum real(z) + imag(z). Is there a technical term for this sum?

• Must admit haven't seen that used very often! – Marcus Müller Dec 21 '18 at 12:31
• Google "weaver modulator" for one example... – Albin Stigo Dec 21 '18 at 12:33
• ah, to generate single side band, makes sense. – Marcus Müller Dec 21 '18 at 12:43
• Exactly, but what is a good name for it? Trying to come up with a name for a libvolk kernel. – Albin Stigo Dec 21 '18 at 13:14
• If this is for a VOLK kernel name: the complex numbers for these are just real and imaginary part one after another, so your operation is effectively volk_32f_sum_pair_32f, I'd say :) – Marcus Müller Dec 21 '18 at 14:24

This can be implemented, without needing to access the real and imaginary parts of $$z$$ individually, by writing

$$\operatorname{Re}(z) + \operatorname{Im}(z) = \frac{(1 + i)(z^* - iz)}{2},$$

where $$i$$ is the imaginary unit and $$z^*$$ is complex conjugation.

Factoring out a complex constant, $$\xi = 1/2 + i/2$$, turns this into a slightly tidier expression,

$$\operatorname{Re}(z) + \operatorname{Im}(z) = \xi(z^* - iz).$$

These expression can be derived by applying the trigonometric identities

$$\cos(x) = \operatorname{Re}(e^{ix}) = \frac{{e^{ix} + e^{-ix}}}{2}$$ $$\sin(x) = \operatorname{Im}(e^{ix}) = \frac{{e^{ix} - e^{-ix}}}{2i}.$$

Perhaps this will give you some inspiration for choosing a good function name.

There's no specific term, as far as I know, but it's a very common operation with quadrature mixing. It's the result of taking the real part of an analytic signal.

You have a complex baseband signal $$x(t)$$, you modulate it with a complex carrier resulting in an analytic band pass signal, and then you take the real part to obtain the real-valued band pass signal that can be transmitted over a channel:

\begin{align}x(t)&=x_R(t)+j\cdot x_I(t)&\textrm{complex baseband signal}\\ y(t)&=x(t)e^{j\omega_ct}&\textrm{analytic band pass signal}\\ s(t)&=\textrm{Re}\{y(t)\}&\textrm{real-valued band pass signal}\\&=x_R(t)\cos(\omega_ct)-x_I(t)\sin(\omega_ct)\end{align}

• Thanks Matt! Well this is what I'm doing... Just pondering what to name a SIMD optimized function I'm writing. – Albin Stigo Dec 21 '18 at 16:39
• @AlbinStigo: Does the function just add real and imaginary parts, or does it also do the modulation? – Matt L. Dec 21 '18 at 16:44
• result = real(z) + imag(z). For example f(2+5i)=7. – Albin Stigo Dec 21 '18 at 17:34
• But $s(t)$ is neither the sum of the real and imaginary parts of $x(t)$ nor the sum of the real and imaginary parts of $e^{j\omega_ct}$ which sum would be $\cos \omega_ct + \sin\omega_ct$, The $s(t)$ you show is actually the dot product of $[x_r(t), x_i(t)]$ and $[\cos\omega_ct, -\sin\omega_ct]$. – Dilip Sarwate Dec 22 '18 at 5:21
• @DilipSarwate: That's true, of course. But what appears to be the case is that the OP generates a complex signal $x_R(t)\cos(\omega_ct)-jx_I(t)\sin(\omega_ct)$, which, when adding its real and imaginary parts, results in the desired real-valued band pass signal. – Matt L. Dec 22 '18 at 13:18

This sounded as an odd concept to me, until I saw the recent Matlab code contest at MatlabCentral on this issue: Problem 2554. Sum the real and imaginary parts of a complex number (Last Solution submitted on May 12, 2020). On SE Maths, there is a question on Standard notation for sum of vector elements?.

So, complex numbers can be interpreted as vectors (but remember their algebraic structure is much richer). If a complex number $$z=z_r+iz_i$$ is written as $$(z_r,z_i)$$, and $$\mathbb{I}=(1,1)$$, then your sum writes $$(z_r,z_i)\cdot\mathbb{I}^T =(z_r,z_i)^T\cdot\mathbb{I}$$. The quantity can be translated into the complex scalar product, defined as:

$$x.y = \frac{1}{2}(\overline{x}y+x\overline{y}).$$

Then, if $$\mathbf{1}=1+i$$ (an analog of $$\mathbb{I}$$), the sum of the real and imaginary parts is $$z.\mathbf{1}=\mathbf{1}.z$$. $$\mathbf{1}=1+i$$ is known as the smallest Gaussian integer being a prime number, or Gaussian prime. I don't know whether it has another name. Until then, you could call it ZdotI.

Finally, @Andy Walls was partly right in his answer: norms usually are non-negative, but some have defined signed measures. They are cumbersome to use, but if one wants a unique pedantic code, your can try ComplexSignedDistance to define this quantity. I have seen seldom cases of Signed L1 norm, but I would not recommend it.