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Choosing the Right Waveform for the Right Task

We now have the tools to compare four major sensing waveforms: FMCW (the workhorse of automotive radar), OFDM (the workhorse of communications), OTFS (the delay-Doppler newcomer), and PMCW (phase-modulated continuous wave, used in advanced digital radars).

The central insight of this chapter is that the waveform choice determines the structure of $\mathbf{A}$ , and the structure of $\mathbf{A}$ determines resolution, sidelobe behaviour, and computational complexity. There is no universally "best" waveform --- the choice depends on the application: standalone sensing, ISAC, high-Doppler environments, or hardware constraints.

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Definition:
FMCW Sensing Matrix Structure

An FMCW radar transmits a chirp with bandwidth $W$ and sweep time $T_{\mathrm{sweep}}$ . After dechirping (mixing with the transmit signal), the beat frequency of target $k$ at range $R_k$ is

$f_k = \frac{2 R_k W}{c \, T_{\mathrm{sweep}}}$

For a burst of $M$ chirps, the sensing matrix has the same Fourier structure as OFDM:

$[\mathbf{A}_{\mathrm{FMCW}}]_{(n,m),k} = e^{-j2\pi n f_k / f_s} \, e^{j2\pi m T_{\mathrm{PRI}} \nu_k}$

where $f_s$ is the sampling rate and $T_{\mathrm{PRI}}$ is the pulse repetition interval.

The key difference from OFDM: FMCW is a dedicated sensing waveform with no data modulation, so there are no null subcarriers and no data compensation step. The entire bandwidth is available for sensing.

The FMCW sensing matrix is structurally identical to the OFDM sensing matrix --- both are partial 2D Fourier matrices. The practical differences lie in the CP constraint (OFDM) vs PRI constraint (FMCW), and in the ability to carry data (OFDM) vs dedicated sensing (FMCW).

Definition:
PMCW Sensing

Phase-Modulated Continuous Wave (PMCW) radar transmits a binary phase-coded sequence (e.g., m-sequence, Gold code, ZCZ sequence) repeated over $M$ repetitions. The code of length $L$ with chip duration $T_c$ provides:

Range resolution: $\Delta R = c T_c / 2 = c / (2 L \cdot B_{\mathrm{chip}})$
Doppler resolution: same as FMCW/OFDM: $\Delta v = \lambda / (2 M T_{\mathrm{PRI}})$

The sensing matrix has entries

$[\mathbf{A}_{\mathrm{PMCW}}]_{(n,m),k} = s[(n - \tau_k / T_c)_L] \, e^{j2\pi m T_{\mathrm{PRI}} \nu_k}$

where $s[n]$ is the phase code sequence. Unlike FMCW and OFDM, the range columns are determined by cyclic shifts of the code rather than complex exponentials.

PMCW has several advantages: (1) excellent range sidelobes with well-chosen codes (approaching $-20 \log_{10}(L)$ dB), (2) fully digital --- no analog chirp generator needed, (3) natural CDMA-style multi-user separation. The disadvantage is higher PAPR sensitivity and code-length constraints.

FMCW vs OFDM vs OTFS vs PMCW for Sensing

Property	FMCW	OFDM	OTFS	PMCW
Range resolution	$c/(2W)$	$c/(2 N_c \Delta f)$	$c/(2 N_c \Delta f)$	$c T_c / 2$
Doppler resolution	$\lambda/(2 M T_{\mathrm{PRI}})$	$\lambda/(2 M T_{\mathrm{sym}})$	$\lambda/(2 M T_{\mathrm{sym}})$	$\lambda/(2 M T_{\mathrm{PRI}})$
Max unamb. range	$c T_{\mathrm{PRI}} / 2$	$c T_{\mathrm{cp}} / 2$	$c T_{\mathrm{cp}} / 2$	$c L T_c / 2$
Ambiguity shape	Ridge (range-Doppler coupled)	Thumbtack (separable)	Thumbtack (separable)	Thumbtack (code-dependent)
Sidelobe level	Low ( $-30$ to $-60$ dB with window)	$-13$ dB (Dirichlet, $-43$ dB with Hamming)	$-13$ dB (Dirichlet)	Code-dependent ( $-20\log_{10} L$ dB)
PAPR	Low (constant envelope)	High ( $\sim 10$ -- $12$ dB)	High (same as OFDM)	Low (constant envelope)
High-Doppler robustness	Good (range-Doppler coupling)	Poor (ICI for $\nu > 0.1\Delta f$ )	Excellent (sparse in DD domain)	Good (code correlation robust)
Data carrying (ISAC)	No (dedicated sensing)	Yes (native)	Yes (native)	Limited (code modulation)
$\mathbf{A}$ structure	2D Fourier (Kronecker)	2D Fourier (Kronecker)	Block-circulant	Cyclic shift + Fourier
Spectral efficiency	Low (dedicated spectrum)	High (shared spectrum)	High (shared spectrum)	Moderate

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Theorem: Waveform Choice Determines $\mathbf{A}$ Structure

For a scene discretised into $Q$ voxels on a delay-Doppler grid, each waveform produces a sensing matrix $\mathbf{A} \in \mathbb{C}^{P \times Q}$ with the following structure:

FMCW: $\mathbf{A} = \mathbf{F}_M \otimes \mathbf{F}_{N_s}$ (Kronecker product of two DFT matrices). No data-dependence.
OFDM: $\mathbf{A} = \mathbf{F}_M \otimes \mathbf{F}_{N_c}$ after data compensation. Rows corresponding to null subcarriers are missing (partial Fourier).
OTFS: $\mathbf{A} = \mathrm{diag}(\mathbf{x}) \cdot \mathbf{C}_{\mathrm{2D}}$ where $\mathbf{C}_{\mathrm{2D}}$ is a 2D circulant matrix. Block-circulant structure enables FFT-based operations.
PMCW: $\mathbf{A} = \mathbf{F}_M \otimes \mathbf{S}_L$ where $\mathbf{S}_L$ is a circulant code matrix with rows being cyclic shifts of the code. The Kronecker factor $\mathbf{S}_L$ is not a DFT matrix.

All four sensing matrices support efficient matrix-vector products in $O(PQ / \min(P,Q) \cdot \log)$ time, but their conditioning and coherence properties differ.

The golden thread: every waveform produces a structured $\mathbf{A}$ that admits fast multiplication. The differences lie in (a) how much of the Fourier structure is preserved (affects conditioning), (b) whether data modulation creates missing entries (affects RIP-like properties), and (c) whether the structure is Kronecker (separable range-Doppler) or block-circulant (coupled via circular convolution).

Proof

FMCW: dechirp converts to Fourier sensing

After dechirping, each range cell maps to a frequency bin via $f_k = 2R_k B / (c T_{\mathrm{sweep}})$ . Sampling at rate $f_s$ gives $N_s$ range samples per chirp. The slow-time Doppler processing adds the second DFT factor $\mathbf{F}_M$ .

OFDM: data compensation restores Fourier structure

Dividing by $d[n,m]$ removes the data modulation, yielding $Z[n,m] = \sum_k \alpha_k e^{-j2\pi n \Delta f \tau_k} e^{j2\pi m T_{\mathrm{sym}} \nu_k}$ . This is a sum of 2D complex exponentials --- the sensing matrix is $\mathbf{F}_M \otimes \mathbf{F}_{N_c}$ evaluated at target locations. Null subcarriers create missing rows.

OTFS: SFFT converts to block-circulant

The OTFS input-output relation is a 2D circular convolution with kernel $h_{\mathrm{DD}}$ . Vectorising gives a block-circulant matrix modulated by the pilot symbols.

PMCW: code correlation replaces DFT in range

Range processing correlates the received signal with cyclic shifts of the transmit code $s[n]$ . This replaces the DFT matrix with the circulant code matrix $\mathbf{S}_L$ . Doppler processing remains a DFT across repetitions. $\blacksquare$

Waveform Comparison: Resolution and Sidelobes

Compare the key metrics of FMCW, OFDM, OTFS, and PMCW waveforms. Adjust the system parameters (bandwidth, number of symbols, code length) and observe how range resolution, Doppler resolution, sidelobe levels, and PAPR change across waveforms.

Parameters

Bandwidth (MHz)100

Number of symbols

M

64

f_0

(GHz)28

Metric to compare

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Example: Waveform Selection for an ISAC Base Station

A 5G NR base station at $f_0 = 28$ GHz with $W = 400$ MHz must simultaneously serve communication users and sense a $500$ -m radius area with targets moving up to $100$ km/h. Which waveform(s) are suitable? What are the limitations?

Solution

Range requirements

Range resolution: $\Delta R = c/(2W) = 0.375$ m (excellent). Maximum range $R_{\max} = 500$ m requires $T_{\mathrm{cp}} \geq 2R_{\max}/c = 3.33\;\mu\text{s}$ .

With $\Delta f = 30$ kHz, $T_{\mathrm{cp}} \approx 2.34\;\mu\text{s}$ (insufficient). With $\Delta f = 15$ kHz, $T_{\mathrm{cp}} \approx 4.69\;\mu\text{s}$ (sufficient).

Doppler requirements

Maximum Doppler: $\nu_{\max} = 2v_{\max}f_0/c = 2 \times 27.8 \times 28 \times 10^9 / (3 \times 10^8) \approx 5.2$ kHz. Normalised: $\nu_{\max} / \Delta f = 5.2/15 \approx 0.35$ for $\Delta f = 15$ kHz.

OFDM assessment

With $\nu/\Delta f \approx 0.35$ , the ICI is severe: $\mathrm{SIR}_{\mathrm{ICI}} \approx 3/(\pi \times 0.35)^2 \approx 2.5 \approx 4$ dB. OFDM sensing will suffer significant degradation at maximum velocity.

OTFS assessment

OTFS with the same parameters avoids ICI entirely. The delay-Doppler channel has $K$ sparse entries regardless of velocity. OTFS is the preferred waveform for this scenario.

Recommendation

Use OTFS (pre/post-processing on top of existing OFDM hardware) for the sensing function, while the communication function uses standard OFDM. The ISFFT/SFFT adds minimal computational overhead. FMCW is ruled out because it cannot carry data.

Range Resolution vs Signal Bandwidth

Explore how range resolution improves with signal bandwidth for each waveform. All waveforms achieve $\Delta R = c/(2W)$ , but the achievable bandwidth depends on the waveform and system constraints (subcarrier spacing, code rate, chirp slope).

Parameters

Max bandwidth (MHz)500

f_0

(GHz)28

⚠️Engineering Note

PAPR Constraints in Practice

OFDM and OTFS suffer from high peak-to-average power ratio (PAPR), typically 10--12 dB for $N_c \geq 256$ subcarriers. This forces the power amplifier to operate with significant back-off, reducing the effective radiated power and hence the sensing range.

FMCW and PMCW have near-constant envelope (PAPR $\approx 0$ -- $3$ dB), allowing the PA to operate near saturation. For automotive radar at 77 GHz where PA efficiency is critical, this is a significant advantage of FMCW/PMCW.

PAPR reduction techniques for OFDM (clipping, tone reservation, selected mapping) can help but add complexity and may distort the sensing waveform. OTFS inherits the PAPR of its underlying OFDM implementation.

🎓CommIT Contribution(2020)

OTFS for Joint Radar and Communication

L. Gaudio, M. Kobayashi, G. Caire, G. Colavolpe — IEEE Trans. Wireless Commun., vol. 19, no. 9

Gaudio, Kobayashi, Caire, and Colavolpe provided one of the first rigorous analyses of OTFS for simultaneous radar parameter estimation and data communication. The key contributions are:

Unified ISAC model: The same OTFS frame serves as both communication signal and radar waveform, with the delay-Doppler channel providing target parameters.
Performance analysis: Derived the Cramer-Rao bound for range and velocity estimation under OTFS, showing that OTFS achieves the same estimation accuracy as a dedicated radar waveform while simultaneously communicating data.
Comparison with OFDM: Demonstrated that OTFS outperforms OFDM for radar estimation at high Doppler, with the gap growing as target velocity increases.

This work is foundational for the ISAC framework developed in Chapter 34.

OTFSISACradarCommITView Paper →

Common Mistake: FMCW Cannot Carry Data

Mistake:

Proposing FMCW for ISAC applications, assuming it can simultaneously carry communication data.

Correction:

FMCW is a dedicated sensing waveform with no native data modulation capability. While some researchers have proposed embedding data in chirp parameters (slope, phase offset), the achievable data rate is extremely low compared to OFDM or OTFS. For true ISAC, communication- native waveforms (OFDM, OTFS) are required.

Quick Check

Which waveform is best suited for an ISAC system that must sense targets at velocities up to 300 km/h while serving communication users?

FMCW

OFDM

OTFS

PMCW

Correction:

OTFS

OTFS handles high Doppler natively (sparse delay-Doppler channel) and carries data like OFDM. It is the natural choice for high-mobility ISAC.

Key Takeaway

The waveform choice determines the structure of $\mathbf{A}$ : FMCW and OFDM produce Kronecker-Fourier matrices, OTFS produces block-circulant matrices, and PMCW produces code-circulant matrices. For ISAC, OFDM and OTFS are preferred because they natively carry data. OTFS is superior to OFDM at high Doppler. FMCW remains the best choice for dedicated sensing with maximum PA efficiency. There is no universal best waveform --- the choice depends on the application requirements.

Looking Ahead: Beyond Single-Waveform Sensing

This chapter has treated each waveform in isolation. In practice, modern sensing systems may combine waveforms --- e.g., FMCW for long-range detection followed by OFDM for fine imaging at shorter range. Moreover, when multiple base stations or vehicles cooperate (Chapter 11), the joint sensing matrix $\mathbf{A}$ aggregates measurements from multiple waveforms and geometries, and the analysis becomes a multi-view, multi-frequency problem (Chapter 08).

Comparison of Sensing Waveforms

Choosing the Right Waveform for the Right Task

Definition: FMCW Sensing Matrix Structure

Definition: PMCW Sensing

FMCW vs OFDM vs OTFS vs PMCW for Sensing

Theorem: Waveform Choice Determines A\mathbf{A}A Structure

FMCW: dechirp converts to Fourier sensing

OFDM: data compensation restores Fourier structure

OTFS: SFFT converts to block-circulant

PMCW: code correlation replaces DFT in range

Waveform Comparison: Resolution and Sidelobes

Parameters

Example: Waveform Selection for an ISAC Base Station

Range requirements

Doppler requirements

OFDM assessment

OTFS assessment

Recommendation

Range Resolution vs Signal Bandwidth

Parameters

PAPR Constraints in Practice

OTFS for Joint Radar and Communication

Common Mistake: FMCW Cannot Carry Data

Quick Check

Key Takeaway

Looking Ahead: Beyond Single-Waveform Sensing

Definition:
FMCW Sensing Matrix Structure

Definition:
PMCW Sensing

Theorem: Waveform Choice Determines $\mathbf{A}$ Structure