# How are subcarrier symbols modulated into an analog OFDM signal?

I am currently researching about OFDM models (formulae) that show how the individual data symbols (eg., QAM symbols) are converted to OFDM symbols and how the OFDM symbols are turned into an analog baseband signal. Below there is a list of StackExchange questions that I found to be related. But neither of these did really make it clear to me how to construct a continuous time signal (in baseband) given the output of the IDFT.

I came up with my own model but I do not know whether it is correct.

quantity type explanation
$$N_\mathrm C$$ $$\mathbb N$$ number of subcarriers
$$N_\mathrm S$$ $$\mathbb N$$ number of OFDM symbols
$$T_\mathrm D$$ $$\mathbb R$$ duration of a IDFT bin
$$T_\mathrm G$$ $$\mathbb R$$ duration of cyclic prefix
$$T_\mathrm O$$ $$\mathbb R$$ duration of a whole OFDM symbol including cyclic prefix: $$T_\mathrm O := N_\mathrm C T_\mathrm D + T_\mathrm G$$
$$\mathbf s_n$$ $${\mathbb C}^{N_\mathrm C \times 1}$$ $$n$$-th complex symbol vector
$$\mathbf W_n$$ $${\mathbb C}^{N_\mathrm C \times N_\mathrm C}$$ $$n$$-th weight matrix for waterfilling (diagonal matrix)
$$\mathbf F$$ $${\mathbb C}^{N_\mathrm C \times N_\mathrm C}$$ DFT matrix

$$\mathbf x_{n_\mathrm S}$$ denotes the $$n_\mathrm S$$-th OFDM symbol including waterfilling. $$\mathbf x_{n_\mathrm S} := \mathbf F^\mathrm H \mathbf W_{n_\mathrm S} \mathbf s_{n_\mathrm S}$$ The analog baseband signal of the $$n_\mathrm S$$-th OFDM symbol (without CP) shall be: $$x_{n_\mathrm S}(t) := \sum_{n_\mathrm C = 0}^{N_\mathrm C - 1} \left[ \mathbf x_{n_\mathrm S} \right]_{n_\mathrm C} \cdot \mathrm{rect}\left( \frac{t - (n_\mathrm C + 0.5) T_\mathrm D}{T_\mathrm D} \right)$$

The signal $$z_{n_\mathrm S}(t)$$ shall be the same as $$x_{n_\mathrm S}(t)$$ but with a cyclic prefix prepended: $$z_{n_\mathrm S}(t) := \underbrace{ x_{n_\mathrm S}(t + N_\mathrm C T_\mathrm D - T_\mathrm G) \cdot \mathrm{rect}\left( \frac{t - 0.5 T_\mathrm G}{T_\mathrm G} \right) }_{\text{cyclic prefix}} + x_{n_\mathrm S}(t - T_\mathrm G)$$

The final baseband signal of all OFDM symbols shall be $$y(t)$$: $$y(t) := \sum_{n_\mathrm S = 0}^{N_\mathrm S - 1} z_{n_\mathrm S}(t - n_\mathrm S T_\mathrm O) \cdot %\mathrm{rect}\left( \frac{t - n_\mathrm S T_\mathrm O}{T_\mathrm G} \right)$$

Questions

1. Is the model shown above somehow reasonable? It assumes that each IDFT bin is some kind of symbol that gets modulated for a certain amount of time.
2. In case it is correct, is it done this way in reality?
3. Do I have to add zeros to the center of the (QAM) symbol vector $$\mathbf s_n$$ to produce more samples in time domain?
4. In this publication they use a model that looks like this $$x'(t) = \sum_{n_\mathrm S = 0}^{N_\mathrm S} \sum_{n_\mathrm C =0}^{N_\mathrm C} \left[ \mathbf s_{n_\mathrm S} \right]_{n_\mathrm C} \cdot \left[\mathbf W \right]_{n_\mathrm S, n_\mathrm C} \cdot \exp \left(\jmath 2 \pi f_{n_\mathrm C} t\right) \cdot \mathrm{rect} \left( \frac{t - n_\mathrm S T_\mathrm O}{T_\mathrm O} \right)$$ Unfortunately, I am not able to map my model onto this model or the other way round. Especially the complex exponential in this formula is a problem. How is it related to the discrete Fourier transform? And in case my model is correct, how is this model related to mine?

But neither of these did really make it clear to me how to construct a continuous time signal (in baseband) given the output of the IDFT

The same as with any other digital baseband signal: With a dual-channel DAC (for real (I) and imaginary (Q) part) and appropriate analog anti-imaging filtering.

There's really nothing special about OFDM here.

So, signal theoretically here, the way to go from a discrete-time signal $$x[n]$$ to a continuous-time $$x(t)$$ is that use every sample from $$x[n]$$ to weigh a separate time-domain sinc whose zeros are $$T_s=1/f_s$$-spaced, and shift the them to position $$nT_s$$, each. See the Whittaker–Shannon interpolation formula.

Re 1. & 2.:

Your model doesn't really reflect how OFDM signals are constructed: the cyclic prefixing still happens on the discrete-time signal, not on the analog signal (that would both be incredibly hard to implement, as well as bad, you want the CP to be exactly the repetition of the discrete-time samples from the end).

So, I'd recommend you stay with discrete time for as long as possible, i.e. work on $$x[n]$$ and move to continuous-time notation only when that actually happens – namely, in your DAC.

Re 3.: No? Wouldn't know where that would become necessary!

Re 4.: Ah, I know both authors :) Christian Sturm worked on OFDM radar. Cool topic. I don't have fulltext access to the document right now. so, just taking a guess, that complex exponential might just be an average Doppler shift . But I'd really have to read the paper.

If you're looking to understand OFDM radar, I'd recommend the relatively seminal thesis by Martin Braun on the topic OFDM Radar Algorithms in Mobile Communication Networks, which put Christian's work into more rigorous DSP terms, answered questions on estimation of properties from the OFDM signal etc. Wiesbeck was professor (is emeritus) at the Institute for High Frequency Electronics, with a focus on Radar, so naturally, that paper is might be more "physical device" oriented, whereas Martin was a PhD candidate at the Communications Engineering Lab, and naturally more affine to OFDM as mapping between discrete signals – which is the perspective you might be missing here.

• 1. & 2.: Okay, thanks. That implies that the length of the cyclic prefix is always an integer multiple of the IDFT bin length ($T_\mathrm D$). Oct 16, 2023 at 7:51
• Re 1. & 2.: So, the digital domain should be $$\mathbf z_{n_\mathrm s} = \mathbf J \mathbf F^\mathrm H \mathbf W_{n_\mathrm s} \mathbf s_{n_\mathrm s}$$ Where $\mathbf J$ is a selection matrix. $$\mathbf J_\mathrm{CP} := \begin{bmatrix} \mathbf 0_{N_\mathrm G, (N_\mathrm C - N_\mathrm G)} & & \mathbf I_{N_\mathrm G} \\ & \mathbf I_{N_\mathrm C} & \end{bmatrix}$$ $N_\mathrm G$ is the number of IDFT bins to be included in the cyclic prefix. Correct? Oct 16, 2023 at 8:08
• Yep, we build cyclic prefixes on integer sample duration. "Bin length" is a bit of a misleading term, because bins exist in the frequency domain, but we're in time. Anyway , you mean the right thing because a (I)DFT just maps between complex vectors of the same length. Thus, your matrix looks correct. Oct 16, 2023 at 8:19
• Thank you! And thank you for the link to the book! :-) My final goal would be to model synchronization errors (regarding carrier frequency and regarding sampling) when using OFDM for radar. Maybe this is already in the book. Oct 16, 2023 at 8:22
• CFO: pretty usual problem for ofdm receivers, typically solved by Schmidt&Cox. I'm not sure that's already in the book. SFO: if your radar TX and RX use the same reference clock to drive both the sampling frequency and the carrier/local oscillator frequency, detecting the carrier offset might be sufficient to detect the sampling frequency offset. Oct 16, 2023 at 8:28