# Is there a 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

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