Does non-orthogonal multicarrier modulation use pulse shaping?

It is well established that OFDM does not use pulse shaping as it destroys the orthogonality of the subcarriers, however, suppose we instead use a form of multi-carrier modulation with non orthogonal subcarriers then should pulse shaping be employed since orthogonality is not a worry? I did some investigating and this is my results

First, we consider a single carrier (SC) QAM-16 system modulated onto a 1khz carrier, which uses pulse shaping and has a bandwidth of ~10Hz as shown below

If we then transform this to a multi-carrier system my basic idea would be to have, say, 4 subcarriers transmitting in parallel, each with a symbol period (T_s) 4 times of the SC system. If in my code I do not use pulse shaping, and simply transmit the sampled QAM-16 symbol for 4T_s. Since the original BW is 10Hz, I chose a 10Hz spacing of subcarriers, This yields the following PSD:

However, the BW was only 10Hz when pulse shaping was used, and in this case pulse shaping wasnt used so its unclear what bandwidth they should be space at.

We then consider the same scenario, but instead of interpolating the sampled value by repeating it for 4 sample periods, we interpolate it for 4T_s by using a pulse shaping filter with upsampling. We can plot the PSD of such a system, with subcarrier spaced at 10Hz intervals as before.

However, what we can see is that because the symbol period is now longer for each subcarrier, the signal switched less frequently, and so the bandwidth of the pulse shaped signal is less that 10Hz, so a 10Hz subcarrier spacing no longer makes sense. So I ask:

Should pulse shaping be used with non-orthogonal multicarrier modulation, and if so, what subcarrier spacing should be used?

Thanks!

• A parenthetical comment on your 4 sub-carrier case that I didn't want to cloud in my answer-- this would not be the way to create 4 sub-carriers as you have 1 complex carrier and 3 real carriers (I assume, unless you destroyed the one in the center if this all was zero). To create 4 sub-carriers maintain a complex IQ waveform and frequency translate each to the subcarrier location using $e^{j\omega t}$ not $\cos(\omega t)$ as you have. Commented Apr 14, 2022 at 20:02
• For this example, I have only modulated my signal onto a cosine carrier, as you have stated. With $e^j(\omega t)$ carrier modulation I have the PSD as imgur.com/a/vKh5TJS. Could you quickly comment on how you could tell my signal has 1 complex and 3 real carriers? I dont think I understand the difference - I thought multiplication by $\cos(\omega t)$ would simply shift my baseband signal up to $\omega$ in the frequency domain? Commented Apr 14, 2022 at 21:05
• I don't know if the center one was complex or not, but the other three were clearly real in the baseband equivalent signal which you get if you remove your carreir at 1000 and make it 0. Multiplying the baseband signal by cosine shifts your signal to +/-w (consider that cosine is made up of two the the $e^{j\omega t}$ terms I mentioned- look up Euler's Formula. The DFT is just showing you the carriers at $e^{j \omega t}$ and not cosines, hence the positive and negative frequency that you have relative to a carrier at 1000 Hz. Commented Apr 14, 2022 at 21:16
• In general there is no need to simulate an actual carrier for a linear system, you can have them all centered on 0 and use complex signals (that is typically the approach and significantly simplifies the processing). "Baseband I and Q" and then frequency translated to the subcarriers using the way I suggested with an exponential. Commented Apr 14, 2022 at 21:18

Yes, if we are not using orthogonal frequency multiplexing and concerned about interference in adjacent channels (specifically if we want to constrain the total spectral occupancy for one channel) then pulse shaping is a good solution. In fact, this is the only reason I am aware of for doing any pulse shaping at all- minimizing the spectral occupation of the signal.

The criteria for both the shaping and proximity of adjacent channels would be the degree of interference between the two as measured at a receiver. This interference may be with our own waveforms coming from the same transmitter, in which case we needn't be concerned with "near/far" issues- or in a worst case, may be concerned with our interference to other users in which case consideration is made with how close our transmitter is to a victim receiver. Typically the limits are set in the standards and applied as a "spectral mask". If we were to establish the spectral mask, or just want to evaluation the cross channel interference, an EVM measurement in the receiver and confirming an excellent result with all other channels off and then measuring the EVM degradation (or equivalently SNR) with the other channels on would be a good approach. This should be with a receiver using a matched filter that provides adequate rejection of the out of channel signals such that it is only the in channel leakage that effects the result.

A reasonable separation would be given by the roll-off factor used in the pulse shaping filter (assuming RRC pulse shaping as commonly done). The occupied bandwidth is given as $$R(1+\alpha)$$ where $$R$$ is the symbol rate and $$\alpha$$ is the roll-off factor (typically a value between 0.1 and .3). Thus the carriers could safely be spaced by the bandwidth. Using the detailed methodology above and the EVM requirements of the waveform, it is possible to space them even closer.

As a side suspicion: For the case of multiple channels coming from the same receiver, it would be difficult (I never like to say impossible, but tempted to) to provide a solution of multiple pulse shaped carriers in a non MIMO solution that would provide a higher spectral efficiency than OFDM since this takes advantage of the synchronization in the transmitter to ensure orthogonality and thus provide the highest density of modulated waveforms occupying a given bandwidth: meaning you can't space the carriers as close as you can with OFDM if you aren't essentially doing OFDM, with no pulse shaping.

• This makes sense - particularly the bit about RRC bandwidth really helped. The method you described for even closer spacing is interesting, but for now I just want to appreciate how non-orthogonal MC modulation compares to OFDM in terms of spectral efficiency so maybe i'll come back to that part in the future. That last suspicion is interesting - perhaps if you could, for some reason, permit a large amount of ICI you could squish the non orthogonal subcarriers close enough that it has a higher spectral efficiency than OFDM? Thanks again. Commented Apr 14, 2022 at 21:09
• @JonahF Yes exactly, your question and my answer made me think about that and I posted a question specific to that, as there are time domain analogies. Glad this makes sense to you! Commented Apr 14, 2022 at 21:13
• one further question - In terms of bandwidth where does this leave us? if we use Multi carrier modulation and space the subcarrier by $1/T_n$ (orthogonally), where $T_n = T_s/N$ is the symbol rate of the MC scheme, then the bandwidth is $(N+\alpha)/T_n$. However, when we consider OFDM the bandwidth is typically quoted as $N/T_n$ which is less that that of orthogonal MC modulation. So, does OFDM truly provide bandwidth savings? the shape of the two spectrums are different which makes me doubt myself, or are people typically assuming $N+\alpha \approx N$ without stating it? Commented Apr 15, 2022 at 21:37
• As $N$ grows very large, the occupied bandwidth approaches $N/T_n$. In your example here, where you said you spaced the sub-carriers by $1/T_n$, but then imply pulse shaping is used as you mention a bandwidth (applicable to one subcarrier) of $(1+\alpha)/T_n$- then you wouldn't have orthogonality AND the bandwidth would be $(1+\alpha)N/T_n$ which also approaches $N/T_n$ as $N$ gets large. Note too that even if you did space the subcarriers by $(1+\alpha)/T_n$ that you still wouldn't have orthogonality! Commented Apr 15, 2022 at 21:54
• Excuse my crude drawing, but look at the following diagram of the spectrum of $N$ overlapping subcarriers - imgur.com/amwomVg - each subcarrier is space by $1/T_n$ and the bandwidth of each subchannel individually is $R(1+\alpha)/T_n$. So we have bandwidth as $B = (N-1)(1+\alpha)/T_n + 2((1+\alpha)/2T_n) = (N+\alpha)/T_n$ from the $(N-1)$ spacings of $1/T_n$ and the two length $(1+\alpha)/2T_n$ lobes at the start and end. But this assumes orthogonality and pulse shaping? As you say those two contradict each other so is this not realistic? Commented Apr 15, 2022 at 22:11