OFDM Ambiguity Function and Performance

Why the Ambiguity Function Matters for OFDM Sensing

The ambiguity function ∣χ(τ,ν)∣2|\chi(\tau, \nu)|^2 tells us everything about a waveform's joint range-Doppler resolution. In Chapter 09 we studied the ambiguity function for classical radar waveforms --- LFM chirps, phase-coded pulses. Now we ask: what does the ambiguity function look like for an OFDM waveform with random data symbols?

The answer is remarkably clean: random data symbols create a near-ideal thumbtack ambiguity function, where range and Doppler are decoupled. But this clean picture breaks down when the Doppler shift becomes comparable to the subcarrier spacing, introducing inter-carrier interference (ICI). Understanding these limits is essential for system design.

Theorem: OFDM Ambiguity Function

For an OFDM frame with NcN_c subcarriers, subcarrier spacing Δf\Delta f, and MM symbols with i.i.d. unit-modulus data symbols, the expected ambiguity function is

E[∣χ(τ,ν)∣2]=∣sin⁡(πNcΔfτ)sin⁡(πΔfτ)∣2⋅∣sin⁡(πMTsymν)sin⁡(πTsymν)∣2\mathbb{E}\bigl[|\chi(\tau, \nu)|^2\bigr] = \left|\frac{\sin(\pi N_c \Delta f \tau)}{\sin(\pi \Delta f \tau)}\right|^2 \cdot \left|\frac{\sin(\pi M T_{\mathrm{sym}} \nu)}{\sin(\pi T_{\mathrm{sym}} \nu)}\right|^2

This is a product of two Dirichlet kernels. The main-lobe widths are:

  • Range (delay): Δτ=1/(NcΔf)=1/W\Delta \tau = 1/(N_c \Delta f) = 1/W
  • Doppler: Δν=1/(MTsym)\Delta \nu = 1/(M T_{\mathrm{sym}})

The peak sidelobe level of each Dirichlet kernel is −13.2-13.2 dB.

Random data symbols decorrelate the sidelobes between different delay-Doppler cuts, producing a near-ideal thumbtack shape. This is in stark contrast to deterministic waveforms like LFM whose ambiguity function has a ridge coupling range and Doppler. The OFDM ambiguity function is separable: range resolution and Doppler resolution are independent.

Proof
Single-target cross-correlation

For a single target at (τ0,ν0)(\tau_0, \nu_0), the compensated signal is Z[n,m]=α0e−j2πnΔfτ0ej2πmTsymν0+W~[n,m]Z[n,m] = \alpha_0 e^{-j2\pi n \Delta f \tau_0} e^{j2\pi m T_{\mathrm{sym}} \nu_0} + \tilde{W}[n,m]. The matched filter output at test point (τ,ν)(\tau, \nu) is

χ(τ,ν)=∑n=0Nc−1∑m=0M−1Z[n,m] ej2πnΔfτe−j2πmTsymν\chi(\tau, \nu) = \sum_{n=0}^{N_c-1} \sum_{m=0}^{M-1} Z[n,m] \, e^{j2\pi n \Delta f \tau} e^{-j2\pi m T_{\mathrm{sym}} \nu}

Separate the sums

Because the delay phase depends only on nn and the Doppler phase only on mm, we factor

χ(τ,ν)=(∑n=0Nc−1ej2πnΔf(τ−τ0))⏟range kernel⋅(∑m=0M−1e−j2πmTsym(ν−ν0))⏟Doppler kernel\chi(\tau, \nu) = \underbrace{\left(\sum_{n=0}^{N_c-1} e^{j2\pi n \Delta f (\tau - \tau_0)}\right)}_{\text{range kernel}} \cdot \underbrace{\left(\sum_{m=0}^{M-1} e^{-j2\pi m T_{\mathrm{sym}} (\nu - \nu_0)}\right)}_{\text{Doppler kernel}}

Evaluate the geometric sums

Each factor is a Dirichlet kernel:

∑n=0Nc−1ej2πnΔfδτ=sin⁡(πNcΔfδτ)sin⁡(πΔfδτ) ejπ(Nc−1)Δfδτ\sum_{n=0}^{N_c-1} e^{j2\pi n \Delta f \delta\tau} = \frac{\sin(\pi N_c \Delta f \delta\tau)}{\sin(\pi \Delta f \delta\tau)} \, e^{j\pi(N_c-1)\Delta f \delta\tau}

Taking the squared magnitude yields the stated result. ■\blacksquare

,

OFDM Ambiguity Function Surface

Explore the 2D ambiguity function ∣χ(τ,ν)∣2|\chi(\tau, \nu)|^2 of an OFDM waveform. Adjust the number of subcarriers and OFDM symbols to see how they independently control range and Doppler resolution. The −13.2-13.2 dB Dirichlet sidelobes are visible along both axes.

Parameters
64
32
60

Definition:

Cyclic Prefix and Maximum Unambiguous Range

The cyclic prefix of length TcpT_{\mathrm{cp}} ensures orthogonality between subcarriers as long as the maximum round-trip delay satisfies

τmax⁡≤Tcp\tau_{\max} \leq T_{\mathrm{cp}}

This imposes a hard limit on the maximum unambiguous range:

Rmax⁡=c Tcp2R_{\max} = \frac{c \, T_{\mathrm{cp}}}{2}

For 5G NR with Δf=30\Delta f = 30 kHz (normal CP), Tcp=2.34  μsT_{\mathrm{cp}} = 2.34\;\mu\text{s}, giving Rmax⁡≈351R_{\max} \approx 351 m. For Δf=120\Delta f = 120 kHz, Tcp≈0.59  μsT_{\mathrm{cp}} \approx 0.59\;\mu\text{s} and Rmax⁡≈88R_{\max} \approx 88 m.

Targets beyond Rmax⁡R_{\max} cause inter-symbol interference (ISI) that corrupts the range-Doppler map.

This is a fundamental limitation of OFDM sensing that does not exist in FMCW or pulsed radar. The CP was designed for multipath delay spread in communications (typically < 5 μ\mus), which inadvertently limits the sensing range. System designers must choose Δf\Delta f carefully to balance range coverage against spectral efficiency.

Theorem: Inter-Carrier Interference at High Doppler

When the Doppler shift ν\nu of a target satisfies ν≥Δf/10\nu \geq \Delta f / 10, the inter-carrier interference (ICI) power on adjacent subcarriers becomes non-negligible. Specifically, for a single target at Doppler ν\nu, the received signal on subcarrier nn after demodulation contains interference from all other subcarriers:

Y[n,m]=α d[n,m] ej2πmTsymν sin⁡(πν/Δf)Ncsin⁡(πν/(NcΔf))⏟desired, attenuated+ICI terms+W[n,m]Y[n, m] = \alpha \, d[n, m] \, e^{j2\pi m T_{\mathrm{sym}} \nu} \, \underbrace{\frac{\sin(\pi \nu / \Delta f)}{N_c \sin(\pi \nu / (N_c \Delta f))}}_{\text{desired, attenuated}} + \text{ICI terms} + W[n,m]

The signal-to-interference ratio due to ICI is approximately

SIRICI≈1(πνTsym)2/3\mathrm{SIR}_{\mathrm{ICI}} \approx \frac{1}{(\pi \nu T_{\mathrm{sym}})^2 / 3}

For ν/Δf=0.1\nu / \Delta f = 0.1, SIRICI≈24\mathrm{SIR}_{\mathrm{ICI}} \approx 24 dB. For ν/Δf=0.5\nu / \Delta f = 0.5, SIRICI≈6\mathrm{SIR}_{\mathrm{ICI}} \approx 6 dB --- rendering OFDM sensing unreliable.

OFDM relies on subcarrier orthogonality, which holds only when the channel is approximately constant over one symbol period. High Doppler destroys this orthogonality by introducing rapid phase rotation within each symbol, spreading each target's energy across adjacent subcarriers. This is the fundamental weakness of OFDM for high-mobility sensing, and the primary motivation for OTFS (Section 10.3).

Proof
Doppler-induced phase within one symbol

A target at Doppler ν\nu introduces a time-varying phase ej2πνte^{j2\pi \nu t} within the OFDM symbol duration TsymT_{\mathrm{sym}}. The DFT at the receiver evaluates the windowed signal over [0,1/Δf][0, 1/\Delta f], which is no longer a pure complex exponential.

DFT leakage calculation

The DFT of ej2πνte^{j2\pi \nu t} over [0,1/Δf][0, 1/\Delta f] at subcarrier nn yields

H[n]=1−ej2π(ν/Δf−n)/Nc⋅Nc1−ej2π(ν/Δf−n)/Nc=sin⁡(π(ν/Δf−n))sin⁡(π(ν/Δf−n)/Nc)H[n] = \frac{1 - e^{j2\pi(\nu/\Delta f - n)/N_c \cdot N_c}}{1 - e^{j2\pi(\nu/\Delta f - n)/N_c}} = \frac{\sin(\pi(\nu/\Delta f - n))}{\sin(\pi(\nu/\Delta f - n)/N_c)}

For nn not equal to the nearest integer to ν/Δf\nu/\Delta f, this is non-zero --- this is the ICI.

SIR approximation

Summing the ICI power from all interfering subcarriers and comparing to the desired subcarrier power gives

SIRICI≈3(πν/Δf)2⋅π2/3≈1(πνTsym)2/3\mathrm{SIR}_{\mathrm{ICI}} \approx \frac{3}{(\pi \nu / \Delta f)^2 \cdot \pi^2 / 3} \approx \frac{1}{(\pi \nu T_{\mathrm{sym}})^2 / 3}

which degrades quadratically with the normalized Doppler ν/Δf\nu / \Delta f. ■\blacksquare

Example: ICI in Automotive OFDM Radar

An automotive OFDM radar operates at f0=77f_0 = 77 GHz with Δf=120\Delta f = 120 kHz. A target vehicle approaches at a relative velocity of v=200v = 200 km/h. Compute the Doppler shift, the normalized Doppler ν/Δf\nu / \Delta f, and the SIR due to ICI. Is OFDM adequate for this scenario?

Solution
Doppler shift

ν=2vf0c=2×55.6×77×1093×108≈28.5  kHz\nu = \frac{2v f_0}{c} = \frac{2 \times 55.6 \times 77 \times 10^9}{3 \times 10^8} \approx 28.5\;\text{kHz}$

Normalized Doppler

νΔf=28.5120≈0.24\frac{\nu}{\Delta f} = \frac{28.5}{120} \approx 0.24$

SIR calculation

SIRICI≈3(π×0.24)2≈30.568≈5.3≈7.2  dB\mathrm{SIR}_{\mathrm{ICI}} \approx \frac{3}{(\pi \times 0.24)^2} \approx \frac{3}{0.568} \approx 5.3 \approx 7.2\;\text{dB}$

At 7 dB SIR, the ICI is significant --- the range-Doppler map will have severe smearing. OFDM is inadequate for this high-Doppler scenario. This motivates OTFS modulation, which handles high Doppler natively (Section 10.3).

Common Mistake: −13-13 dB Sidelobes Mask Weak Targets

Mistake:

Assuming the 2D-FFT range-Doppler map can detect all targets, regardless of their relative amplitudes.

Correction:

The Dirichlet kernel sidelobes are at −13.2-13.2 dB. A target that is more than 13 dB weaker than its neighbour will be masked by the sidelobes. Windowing (Hamming, Hann, Blackman) reduces sidelobes at the cost of wider main lobe (worse resolution). Alternatively, compressed sensing methods (Chapter 13) bypass the sidelobe issue entirely by exploiting sparsity.

⚠️Engineering Note

Windowing Trade-off in OFDM Sensing

In practice, a Hamming or Hann window is applied before the 2D-FFT to suppress sidelobes. The trade-off:

  • No window (rectangular): −13-13 dB sidelobes, narrowest main lobe
  • Hamming: −43-43 dB sidelobes, main lobe 1.5×1.5\times wider
  • Blackman: −58-58 dB sidelobes, main lobe 1.7×1.7\times wider

For automotive radar where dynamic range exceeds 40 dB (strong vehicle

  • weak pedestrian), Hamming or better is essential. For imaging of similarly-strong targets, the rectangular window preserves resolution.

Quick Check

A 5G NR system with Δf=60\Delta f = 60 kHz and normal cyclic prefix (Tcp≈1.17  μsT_{\mathrm{cp}} \approx 1.17\;\mu\text{s}) is used for sensing. What is the maximum unambiguous range?

≈88\approx 88 m

≈175\approx 175 m

≈351\approx 351 m

≈5000\approx 5000 m

Correction:
≈175\approx 175 m

Rmax⁡=cTcp/2=3×108×1.17×10−6/2≈175R_{\max} = c T_{\mathrm{cp}}/2 = 3 \times 10^8 \times 1.17 \times 10^{-6} / 2 \approx 175 m.

Inter-Carrier Interference (ICI)

Leakage of energy between OFDM subcarriers caused by a Doppler shift that destroys subcarrier orthogonality. ICI power scales as (ν/Δf)2(\nu/\Delta f)^2 and becomes a dominant impairment when ν/Δf≳0.1\nu / \Delta f \gtrsim 0.1.

Related: Doppler Resolution

Key Takeaway

The OFDM ambiguity function is a product of two Dirichlet kernels --- a thumbtack with separable range and Doppler resolution. The cyclic prefix imposes a hard limit on maximum sensing range. At high Doppler (ν/Δf≳0.1\nu / \Delta f \gtrsim 0.1), inter-carrier interference degrades performance quadratically, motivating the move to OTFS.

OFDM-Based SensingOTFS Modulation for Sensing