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Orthogonal Time Frequency Space (OTFS)


In OTFS (Orthogonal Time Frequency Space) modulation — a scheme designed for high-Doppler and time-varying wireless channels — the terms ISFFT and SFFT are key mathematical transformations used to move between different representation domains.






Figure: OTFS block diagram


1. ISFFT — Inverse Symplectic Finite Fourier Transform

Purpose: Transforms data symbols from the delay-Doppler domain to the time-frequency domain.

\[ X[n, m] = \frac{1}{\sqrt{NM}} \sum_{k=0}^{N-1} \sum_{l=0}^{M-1} x[k, l] \, e^{j2\pi \left( \frac{nk}{N} - \frac{ml}{M} \right)} \]

Here, \( N \) is the number of Doppler bins (time slots), and \( M \) is the number of delay bins (subcarriers). The ISFFT maps each data symbol from the delay-Doppler grid (where the channel is sparse and easier to equalize) to the time-frequency grid (where standard multicarrier modulation like OFDM can be applied).


2. SFFT — Symplectic Finite Fourier Transform

Purpose: Performs the reverse operation — it converts the received signal from the time-frequency domain back to the delay-Doppler domain.

\[ x[k, l] = \frac{1}{\sqrt{NM}} \sum_{n=0}^{N-1} \sum_{m=0}^{M-1} X[n, m] \, e^{-j2\pi \left( \frac{nk}{N} - \frac{ml}{M} \right)} \]

3. Summary Table

Transform From → To Mathematical Type Purpose
ISFFT Delay-Doppler → Time-Frequency Inverse Transform Used at transmitter to map data symbols
SFFT Time-Frequency → Delay-Doppler Forward Transform Used at receiver to recover data symbols

4. Intuitive View

  • Delay-Doppler domain: Represents the signal in terms of physical channel parameters (delays and Doppler shifts). Sparse and stable.
  • Time-Frequency domain: Represents how the signal occupies time and frequency. Suitable for modulation schemes like OFDM.

In summary, OTFS uses ISFFT before modulation and SFFT after demodulation to exploit both delay and Doppler diversity efficiently.


5. Similarity with the OFDM Process

In OFDM (Orthogonal Frequency Division Multiplexing), the main goal is to transmit orthogonal subcarriers to mitigate intersymbol interference (ISI). One effective way to achieve this is by transmitting orthogonal signals in the frequency domain.

However, instead of directly generating those frequency-domain signals, we apply an Inverse Fast Fourier Transform (IFFT) to the modulated symbols at the transmitter. This operation automatically ensures orthogonality among subcarriers in the frequency domain, while producing a time-domain signal suitable for transmission.

At the receiver side, we perform the Fast Fourier Transform (FFT) to convert the received time-domain signal back to the frequency domain, where the transmitted data symbols can be recovered.


Further Reading

  1. MATLAB Code for OFTS
  2. OFDM (Theory)

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