# Sending complex signal over a single wire

The G.hn protocol uses OFDM to transmit data. And it sounds like they are doing it on a single wire. OFDM symbols are a complex number with I and Q components. How are they able to send two separate data streams on a single wire?

If they are not doing it on a single wire, then my question is really how do wired protocols send complex I/Q data over a single wire? I'm looking for a fairly concrete answer (does not have to be specific to G.hn) as I'm interested in implementing something like this myself along a single channel.

• For example a cable TV distribution system is transmitting dozens (or hundreds) of TV channels ( analog, digital, video plus stereo audio etc) simultaneously over a single co-axial cable to your home... This is how the communication systems do work. Modulation, and frequency mixing. And the number of channels is fundamentally limited by the bandwidth of the transmission medium. Quadrature modulation enables further enhancement that two channels can be sent over the same frequency band, due to orthogonality, you can separate them later, which wouldn't be possible with other techniques. Commented Aug 13, 2017 at 11:17

Note that signals sent over wires (and over the air, and over any medium) are always real. What quadrature means is that, on a passband channel (wired or otherwise), you can transmit two signals at the same time. The first signal, which we'll call $s_I(t)$, is mixed with a carrier $c_I(t)=\cos(2 \pi f_c t + \phi)$; the second signal, $s_Q(t)$, is mixed with $c_Q(t)=\cos(2 \pi f_c t + \phi + \pi/2)$, where $f_c$ is the center frequency of the channel.

Note that it is important to use a passband channel. Quadrature communication is impossible on the baseband channel. Maybe this is what confused you, since one tends to associate wires with baseband. However, as long as you don't exceed the bandwidth of the wire, it's perfectly possible to transmit modulated signals over them. In fact, in telephony, the very first trunk links used single sideband modulation to multiplex several phone calls over a single wire -- this was before PCM and digital telephony were even invented.

The transmitter would then calculate the following signal: $$s(t) = s_I(t)c_I(t) - s_Q(t)c_Q(t),$$ where the negative sign is just a convention. Assuming the carrier phase $\phi=0$ for simplicity, this signal can also be written as $$s(t)=\text{Re}[(s_I(t)+js_Q(t))e^{j 2 \pi f_c t})].$$ This expression is sometimes useful in a digital implementation, since you only have to keep track of two complex signals instead of four real signals.

The channel adds noise, so the received signal is $r(t) = s(t) + n(t)$. The noise is usually modeled as Gaussian, white noise with PSD equal to $N_0/2$.

In the receiver, we have (ommitting a lot of algebra readily found in textbooks and on other answers in this site): $$\hat{s}_I(t) = s_I(t) + n_I(t) = \text{LPF}[r(t) c_I(t)]$$ and $$\hat{s}_Q(t) = s_Q(t) + n_Q(t) = \text{LPF}[r(t) c_Q(t)],$$ which means that the receiver can recover both transmitted signals (albeit with noise).

How to actually get all this to work in an actual system requires solving other problems that have not been addressed here and that can take a whole textbook to explain. I recommend that you start with this free textbook: http://sethares.engr.wisc.edu/telebreak.html (link at the end of the page).

• How do I transmit c_i and c_q along the same wire? Do I add them up? And at the receiver how do I separate the I and Q from the real-valued signal? The more concrete the answer, the better. Commented Aug 14, 2017 at 5:08
– MBaz
Commented Aug 14, 2017 at 14:48

Exactly the same way like when complex baseband signals get transmitted over the air:

Mix with a quadrature mixer onto s carrier frequency in the sender, mix down to complex baseband again at the receiver.

You can send many many many data streams on a single wire as long as you modulate the data streams so that they in different channels, or are orthogonal within a channel. Just like your AM or FM radio can tune to tons of different stations using a single antenna.

OFDM does both, using many different frequency carriers that are orthogonal over certain time windows, and within each frequency, using the fact that the sine and cosine functions are orthogonal over that same time window. One can send data by modulating a cosine carrier with one complex component (say I) and the sine carrier with the other (say Q), enabling sending both I and Q over a single wire (in a channel, within a time window).

Note that the trick is that the the I and Q data must stay constant within that time window for OFDM to work. Else orthogonality is broken. But you can use lots of time windows and lots of IQ channels to make up for that, usually via a big enough IFFT+cyclic_redundancy_header before heterodyning up.

• Clarification may be needed. If we have a set of vectors assumed to be in the 'frequency' domain, and then an IFFT is carried out on that set - then that would take us to the 'time domain', where we have a sequence of complex values in the time domain. In that case, the question is - if we transmit the real part and imaginary part of those time domain values on quadrature carriers, and clock out at a given rate, then does the time sequence relate directly to the frequency spectrum that we started with? Or would it just be time-domain complex number transmission using quadrature modulation? Commented Apr 27, 2021 at 7:20
• The final quadrature modulation shifts the spectrum of any IQ signals from baseband to some higher frequency, where the upper and lower sidebands of the baseband IQ no longer alias. So the spectrum of the IFFT is there, but shifted up. And only the real part needs transmitting, since the lower sideband no longer presents an aliasing problem requiring the imaginary component to resolve. Commented Apr 27, 2021 at 19:29
• Excellent points you made hotpaw2. Thanks very much. I didn't think about that before - where the baseband frequency components (the vectors) is kind of like a virtual one-sided spectrum (positive frequencies only). Very interesting. So after IFFT is done, and after choosing a clocking-out rate for the real and imaginary parts of each time-domain complex number (and also using quadrature carriers to send each part), I can see what you mean about the frequency shift of the baseband spectral components, due to the time vector components modulating the quadrature carriers. Thanks hotpaw2. Commented Apr 28, 2021 at 1:03