Wideband receivers normally learn the channel by distributing known symbols across time and frequency. That approach is dependable, but every pilot occupies a resource that cannot carry payload. This release explores a different observation primitive: reserve one full-band OCDM chirp, protect the part of its beat spectrum where the channel will appear, and use that observation to drive an otherwise conventional cyclic-prefix frequency-domain equalizer.
The central result is exact under an integer circular-delay model and remains useful in fractional-delay link simulations. A delayed OCDM chirp, once mixed with the conjugate of the known reference chirp, becomes a tone. Its beat-bin position records delay. A single reference can therefore reveal the effective impulse response across the processed bandwidth without constructing a dense pilot grid.
This is not a claim that one reference is free, or that OCDM universally outperforms OFDM. The reference itself costs one index. More importantly, a protected region must separate the channel observation from payload chirps. The useful engineering question is therefore not simply how many reference symbols are transmitted, but how much support must be protected and what happens on either side of that boundary.
01The observation problem
Integrated sensing and communication receivers need two kinds of knowledge from the same waveform. Communication needs a sufficiently accurate channel response for equalization. Sensing needs access to delay-domain structure. Conventional multicarrier systems usually acquire the first through pilots and then reconstruct the channel between them. Increasing pilot density improves observation but consumes time-frequency resources and reference energy.
OCDM represents a block on an orthogonal chirp basis generated by the discrete Fresnel transform. It remains compatible with FFT implementation and cyclic-prefix processing, but the chirp geometry creates another way to look at propagation. Instead of sampling the channel at many pilot positions, the receiver can ask what a known chirp becomes after travelling through delayed paths.
The transmitter places a known symbol at chirp index zero, reserves a guard on both sides of the usable index interval, and maps payload onto the remaining chirps. The receiver processes the same CP-valid block twice. One branch follows the normal data path. The other multiplies the received samples by the conjugate reference chirp and takes a beat FFT. The first branch carries the data; the second exposes the channel.
02Why delay becomes a beat bin
Let \(\psi_k[n]\) denote chirp \(k\) of an \(N\)-point OCDM basis. If that chirp is delayed by an integer \(\ell\) and de-chirped with \(\psi_0[n]\), the quadratic phase cancels into a sampled complex exponential:
The engineering DFT of this expression is concentrated at \((k+\ell)\bmod N\). Delay has become a beat-bin shift. For the known reference, \(k=0\), so a channel path at delay \(\ell\) appears directly in beat bin \(\ell\), apart from a deterministic phase rotation.
This identity gives the receiver a precise observation rule. If the effective channel support lies within the protected low-index band, payload chirps cannot enter those bins. The receiver reads the complex beat values, removes the known phase, divides by the reference symbol and obtains an estimate of the impulse response:
Real propagation is not aligned perfectly to sample positions. Fractional delays spread energy across nearby beat bins, just as an off-grid tone leaks across an FFT. The discrete identity still explains the dominant structure, but the protected band must include the effective leakage support rather than only the nominal path delays.
03From observation to equalization
Once the beat branch has produced \(\hat h[\ell]\), the rest of the receiver is deliberately familiar. An FFT converts the impulse response into \(\hat H[m]\), and a regularized frequency-domain equalizer processes the received block:
The equalized block then returns to the Fresnel domain and only payload indices are demapped. The proposed observation is therefore an additional branch around a conventional CP/FDE chain, not a replacement for the full receiver. Its online additions are one de-chirp multiplication, one beat FFT, protected-bin scaling and one transform from the estimated impulse response to the channel frequency response.
The allocation also separates two costs that are often combined. The explicit reference cost is \(1/N\). The protected-region cost is \(2G/N\), because both low-index isolation and circular wraparound must be controlled. With \(N=4096\) and \(G=64\), the reference occupies 0.024 percent of the grid while the complete protected region occupies 3.13 percent.
These numbers do not establish an absolute advantage across every receiver design. They show that reference count, reserved bandwidth and reference energy are different quantities. A fair comparison must keep all three visible.
04What the simulations show
The numerical campaign uses an NR-scale processed grid with 30 kHz subcarrier spacing, a 33.33 microsecond useful block, a 288-sample cyclic prefix and independent block-static fractional-delay TDL-B realizations. The principal comparison uses QPSK, a 100 ns RMS delay-spread case and the nominal CP boundary, without oracle timing selection.
At lower SNR, the tested OFDM comb-pilot LMMSE receiver performs better. The single-reference receiver closes the gap around 20 dB, crosses the tested comb-pilot baseline between 20 and 25 dB, and then approaches the block-training and oracle controls. At 25 dB, the audited campaign records a BER of approximately \(1.3\times10^{-4}\) for the main single-reference curve, compared with approximately \(2.1\times10^{-3}\) for the 128-pilot OFDM LMMSE control. The result is conditional on this channel model, allocation and estimator family.
The energy frontier tells the more useful story. Increasing OFDM pilot density supplies more explicit channel samples, but the tested family stays near the same BER region while reference energy rises. The single-reference point occupies a different part of that cost-performance plane. Block training remains the strongest control, but it spends a complete training block and should not be confused with a sparse-pilot design.
Low-error points need care. Nine observed errors at 27.5 dB make that point statistically delicate, and the zero-error point at 30 dB is an upper bound. The evidence supports a receiver operating region, not a universal waveform ranking.
05The guard is a design variable
The guard is not a fixed tax. It is a design variable coupled to the effective delay support. Define the normalized support margin as
A negative margin means that part of the channel response lies outside the protected observation band. Payload energy then contaminates the bins used to estimate the channel. A modest positive margin isolates the reference response. An excessively large margin can also hurt the current estimator because it admits more noise-dominated bins while removing additional payload resources.
The 100 ns case makes the threshold visible. Moving from \(\eta_G=-0.013\) to \(0.140\) reduces BER from roughly \(3.4\times10^{-3}\) to \(9.8\times10^{-5}\). Extending the margin toward 2.2 raises BER again to roughly \(4.2\times10^{-3}\). The useful region is a basin, not a monotonic rule that more guard is always better.
06Scope and next steps
The exact beat-bin statement assumes integer circular delay, a CP-valid observation window and a channel that is approximately coherent over one useful block. Fractional delay is handled empirically through guard margin. Doppler shifts the beat response away from its integer bin; the present one-dimensional readout is intended for \(|f_DT_u|\ll1\).
For the 4096-sample, 122.88 MHz example at 3.5 GHz, a radial speed of 60 km/h produces a normalized Doppler of about \(6.5\times10^{-3}\). The phase rotation during one useful block is small. Narrower bandwidths, longer blocks, higher carrier frequencies or faster motion will require a two-dimensional range-Doppler observation rather than the present delay-only readout.
Several questions remain open: off-grid estimation could reduce the guard needed for fractional delays; window selection must become robust to synchronization error; uplink operation needs separation among multiple reference sources; and hardware validation must include RF impairments, pulse shaping and nonlinear amplification. The current result establishes the observation mechanism and its operating law. It does not close the receiver-design problem.
One chirp is enough to expose the full processed channel only when the receiver protects the physical support that the channel actually occupies.
The reported comparisons are specific to the audited NR-scale processed grid, TDL-B channel family, allocation and receiver controls. They do not claim universal superiority over OFDM, standardized reference-signal designs or other OCDM estimators.