Frequency (OFDM) Domain

To implement the channel response assuming an OFDM waveform, it is assumed that the power delay profiles are invariant over the duration of an OFDM symbol. Moreover, it is assumed that the duration of the cyclic prefix (CP) equals at least the maximum delay spread. These assumptions are common in the literature, as they enable modeling of the channel transfer function in the frequency domain as a single-tap channel.

For every link \((u, k, v, l)\) and resource element \((s,n)\), the frequency channel response is obtained by computing the Fourier transform of the channel response at the subcarrier frequencies, i.e.,

\[\begin{split}\begin{aligned} \widehat{h}_{u, k, v, l, s, n} &= \int_{-\infty}^{+\infty} h_{u, k, v, l}(s,\tau) e^{-j2\pi n \Delta_f \tau} d\tau\\ &= \sum_{m=0}^{M-1} a_{u, k, v, l, m}(s) e^{-j2\pi n \Delta_f \tau_{u, k, v, l, m}} \end{aligned}\end{split}\]

where \(s\) is used as time step to indicate that the channel response can change from one OFDM symbol to the next in the event of mobility, even if it is assumed static over the duration of an OFDM symbol.

For every receive antenna \(l\) of every receiver \(v\), the received signal \(y_{v, l, s, n}`\) for resource element \((s, n)\) is computed by

\[y_{v, l, s, n} = \sum_{u=0}^{N_{T}-1}\sum_{k=0}^{N_{TA}-1} \widehat{h}_{u, k, v, l, s, n} x_{u, k, s, n} + w_{v, l, s, n}\]

where \(x_{u, k, s, n}\) is the baseband symbol transmitted by transmitter \(u`\) on antenna \(k\) and resource element \((s, n)\), and \(w_{v, l, s, n} \sim \mathcal{CN}\left(0,N_0\right)\) the additive white Gaussian noise.

Note

This model does not account for intersymbol interference (ISI) nor intercarrier interference (ICI). To model the ICI due to channel aging over the duration of an OFDM symbol or the ISI due to a delay spread exceeding the CP duration, one would need to simulate the channel in the time domain. This can be achieved by using the OFDMModulator and OFDMDemodulator layers, and the time domain channel model. By doing so, one performs inverse discrete Fourier transform (IDFT) on the transmitter side and discrete Fourier transform (DFT) on the receiver side on top of a single-carrier sinc-shaped waveform. This is equivalent to simulating the channel in the frequency domain if no ISI nor ICI is assumed, but allows the simulation of these effects in the event of a non-stationary channel or long delay spreads. Note that simulating the channel in the time domain is typically significantly more computationally demanding that simulating the channel in the frequency domain.

OFDMChannel(channel_model, resource_grid[, ...])	Generate channel frequency responses and apply them to channel inputs assuming an OFDM waveform with no ICI nor ISI
GenerateOFDMChannel(channel_model, resource_grid)	Generates channel frequency responses.
ApplyOFDMChannel([precision, device])	Apply single-tap channel frequency responses to channel inputs
cir_to_ofdm_channel(frequencies, a, tau[, ...])	Compute the frequency response of the channel at frequencies