---
title: SC-IFDM–FMCW Orthogonal Waveform
url: https://www.emergentmind.com/topics/sc-ifdm-fmcw-orthogonal-waveform
type: topic
---

# SC-IFDM–FMCW Orthogonal Waveform

The SC-IFDM–FMCW orthogonal waveform is a joint waveform for integrated sensing and communication (ISAC) that combines single-carrier interleaved frequency division multiplexing (SC-IFDM) and frequency modulated continuous wave (FMCW) chirps using DFT-domain orthogonalization. This construction enables radar-grade range–Doppler sensing and high-throughput, low-PAPR communication in a single hardware pulse, achieving strictly mutual orthogonality between data and chirp components and eliminating cross-interference. The result is a class of waveforms that unify sensing and data multiplexing within a mathematically rigorous resource grid, with precise resource partitioning and channel estimation properties, making them central to JSAC designs in 5G/6G networks [2503.12638][2503.12676][2602.02248].

## 1. Technical Construction of SC-IFDM and FMCW Coexistence

Consider \(M\) DFT blocks on \(N\) subcarriers, forming a two-dimensional DFT-domain grid. SC-IFDM data symbols \(X^{\mathrm{SC}}(k,\ell)\) are packed and transformed into time domain via:

\[
s^{\mathrm{SC}}[p] = \frac{1}{\sqrt N} \sum_{k=0}^{N-1} X^{\mathrm{SC}}\bigl(k,[p]_M\bigr) e^{j2\pi\frac{k}{MN}p},~~~p=0,\dots,MN-1
\]

A discrete-time FMCW chirp is of the form:

\[
s^{\mathrm{FMCW}}[p]=\exp\!\left(j\pi \frac{p^2}{MN}\right),\quad p=0,\dots,MN-1
\]

Its DFT-domain representation is highly sparse: for each DFT block index \(\ell\), only the DFT bin \(k = (\ell + M/2) \bmod N\) is nonzero. The composite DFT-domain grid is populated as:

\[
X^{\mathrm{comb}}(k,l) = 
\begin{cases}
\sqrt{\psi}\;\omega^l_k\;s^{\mathrm{FMCW}}[l] & [M/2 + l - k]_N = 0 \\
X^{\mathrm{SC}}(k, l) & \text{otherwise}
\end{cases}
\]
where \(\psi\) is the chirp-to-data power split and \(\omega_k^l=e^{-j2\pi k l/(MN)}\).

The composite time-domain pulse is then constructed by an \(MN\)-point IDFT. A cyclic prefix (CP) is inserted to manage inter-symbol interference and ensure circular continuity, with the FMCW chirp coefficients shifted appropriately to maintain analog continuity across CP boundaries [2503.12638][2503.12676].

## 2. Orthogonality in the DFT Domain and Mutual Interference Suppression

Orthogonality arises from strict disjoint support of data and chirp elements in the DFT domain:
- The FMCW chirp occupies \(\Omega_{\rm chirp} = \{(k,\ell): k = \ell + M/2 \bmod N\}\),
- SC-IFDM data symbols use the complementary set \(\Omega_{\rm data}\).

This guarantees that
\[
\sum_{k=0}^{MN-1} X^{\mathrm{SC}}(k,l) \left(X^{\mathrm{FMCW}}(k,l)\right)^{*}=0
\]
for all \(l\), and in the time domain,
\[
\langle s^{\mathrm{SC}}(t), c(t)\rangle = 0
\]
due to non-overlapping frequency components [2503.12638][2503.12676]. Any resource trade-off (e.g., between sensing SNR and communication throughput) can be formulated as resource-set partitioning and power allocation within the DFT grid.

## 3. Enhanced Channel and Sensing Estimation Mechanisms

SC-IFDM–FMCW enables chirp-based, pilot-free channel estimation. At the receiver, the DFT output at “pilot” bins (i.e., chirp support) contains the chirp response and possible data leakage, enabling the following chirp-matched transformation:

\[
\widehat Y_1(\beta_1,\alpha_1) = \frac{1}{M} \sum_{l=0}^{M-1} Y_{dp}(k, l) e^{-j\pi\frac{(l-\alpha_1 N-\beta_1)^2}{MN}}
\]
where \(\alpha_1N+\beta_1\) parameterizes integer delay and Doppler bins. Matching both up-chirp and down-chirp blocks allows unique recovery of per-tap delay, Doppler, and amplitude without dedicated pilot symbols.

This aligns with recent DD-multiplexing generalizations (e.g., ODDM-FMCW, where a ZCA chirp sequence is allocated on complementary DD grid positions, cyclically correlated for delay extraction) and extends naturally to schemes employing square-root Nyquist pulse shaping for ISI control [2602.02248]. The matched filter followed by soft linear detection comprises the front end of both communication and sensing receivers.

## 4. Resource Partitioning, Spectrum, and PAPR Optimization

The SC-IFDM framework admits precise resource allocation:
- The set \(\Omega_{\rm chirp}\) is of size \(M\) (one per DFT block), with remaining grid locations supporting arbitrary QAM data.
- The underlying IDFT remains unchanged, permitting transparent integration into existing hardware.

Spectrum shaping is determined by the underlying pulse shape; for instance, using SRRC or similar square-root Nyquist pulses produces low out-of-band emission (OOBE) and tight time–frequency occupancy, closely paralleling advances in ODDM-FMCW waveform design [2602.02248]. By adjusting the power split between chirp and data (\(\psi\)), one can optimize for desired PAPR and sensing SNR.

The PAPR of the resulting waveform

Source: https://www.emergentmind.com/topics/sc-ifdm-fmcw-orthogonal-waveform