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?

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?

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.

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?

It truly Is. I've been looking at OFDM for a 3rd year uni project for a while now, and its only now that i am appreciating what it means to say that the QAM symbols are the spectrum. This is aided by contrasting the spectrum of single carrier QAM vs multi carrier QAM with non-orthogonal subcarriers vs OFDM-QAM. You are right, with 32 subcarriers, a much tighter PSD is seen (imgur.com/a/CE8hYSU). I now want to understand why the zero padding is as it is, it seems strange but im sure is mathematically consistent.

Will do, thanks a lot

Right right, Ive had a look at that. From what Ive deduced then, we would split the [1 x N] vector into a [4 x N/4] array and then zero pad the length 4 column vectors to with the scheme described in your original answer to length 4*(M/N), then apply an IFFT of the same length to give, a size [4(M/N) x N/4] array over the N/4 sample periods. This is then parallel-to-serial and the resulting row vector is length [1 x 4(M/N)(N/4)] = [1 x M], so one value for each timestep! I think this is correct! Using that method I produced imgur.com/ZtErksz (1khz carrier) - looks good :-)

I thought I did, but now i'm not so sure. Put it this way, if I had a time vector of 10,000 samples over 100s, and then 800 QAM symbols of a sampled signal regularly spaced over the same 100s, how exactly would I map the 800 symbols to the input of the (I)FFT so that I am transmitting the data in parallel over 8 subcarriers for 100s? I assume with 8 subcarriers we'd need a length 8 (I)FFT - this would produce 8 output samples which id add together to get the symbol for that time step ... but then what about all the timesteps where the input sample is zero? I hope this Q makes sense. thanks

Thanks, this helps! What I don't understand about your answer is how this applies to multiple subcarriers. Lets say im trying to take a single carrier system and instead use OFDM and transmit my data over 4 subcarriers. Now if I originally have an length [1 x N] vector of QAM symbols I'd think to reshape this to a [4 x N/4] sized array which matches the 4 subcarriers. I now need to upsample this to match a time vector of length M, would I therefore have to zero-pad each row of the [4 x N/4] sized array to make it size [4 x M] and then apply a length M FFT to each row? doesn't seem right!

Thank you Tim - it does make intuitive sense, I was just wondering if there was a particular equation I was missing, guess not! I love playing fast and loose with the dirac delta, brings a bit of fire to your life.