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Why Waveform Design Matters for Imaging

The transmitted waveform determines what the radar can resolve. A waveform with duration $T$ and bandwidth $W$ cannot simultaneously achieve arbitrarily fine range and Doppler resolution — there is a fundamental trade-off encoded in the ambiguity function.

For imaging, the ambiguity function is the point-spread function (PSF) of the matched-filter estimator. Its shape directly determines the sidelobe structure we will encounter in Chapter 13 when we study backpropagation imaging.

Definition:
The Ambiguity Function

For a transmitted waveform $s(t)$ with unit energy ( $\int |s(t)|^2\,dt = 1$ ), the ambiguity function is:

$\chi_A(\tau, \nu) = \int_{-\infty}^{\infty} s(t)\,s^*(t - \tau)\,e^{j2\pi\nu t}\,dt,$

where $\tau$ is the delay (range) variable and $\nu$ is the Doppler (velocity) variable.

The squared magnitude $|\chi_A(\tau, \nu)|^2$ is the ambiguity diagram — it measures how well the radar can distinguish a target at delay $\tau$ and Doppler $\nu$ from a target at the origin.

At the origin, $|\chi_A(0, 0)|^2 = 1$ (perfect self-correlation). A target at $(\tau, \nu)$ produces a response proportional to $|\chi_A(\tau, \nu)|^2$ in the matched-filter output — this is the range-Doppler sidelobe level.

Theorem: The Radar Uncertainty Principle

For any unit-energy waveform $s(t)$ , the total volume of the ambiguity function is constant:

$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} |\chi_A(\tau, \nu)|^2\,d\tau\,d\nu = 1.$

Consequently, it is impossible to design a waveform with $|\chi_A(\tau, \nu)|^2 \approx 0$ for all $(\tau, \nu) \neq (0,0)$ . Compressing the ambiguity function in one dimension necessarily spreads it in another.

This is the radar analog of the Heisenberg uncertainty principle in quantum mechanics: energy is conserved and must appear somewhere in the delay-Doppler plane. A narrow mainlobe requires sidelobes elsewhere.

Proof

Apply Parseval's theorem

Write $\chi_A(\tau, \nu)$ as the Fourier transform (in $t$ ) of $s(t) s^*(t-\tau)$ . Then:

$\int |\chi_A(\tau, \nu)|^2\,d\nu = \int |s(t)|^2 |s(t-\tau)|^2\,dt.$

Integrate over delay

Definition:
Linear Frequency Modulation (LFM) Chirp

The LFM chirp (or linear FM, "chirp") waveform is:

$s(t) = \frac{1}{\sqrt{T}}\,\text{rect}\!\left(\frac{t}{T}\right) e^{j\pi \mu t^2},$

where $T$ is the pulse duration, $\mu = W/T$ is the chirp rate, and $W$ is the swept bandwidth.

The instantaneous frequency is $f_i(t) = \mu t$ , sweeping linearly from $-W/2$ to $+W/2$ over $[{-T/2}, {T/2}]$ .

The time-bandwidth product $W T$ determines the pulse compression ratio: a chirp with $W T = 1000$ achieves 30 dB processing gain.

Theorem: Ambiguity Function of the LFM Chirp

The ambiguity function of an LFM chirp with bandwidth $W$ and duration $T$ is:

$|\chi_A(\tau, \nu)| \approx \left|\text{sinc}\!\left[(W\tau - \nu T)\left(1 - \frac{|\tau|}{T}\right)\right]\right|\left(1 - \frac{|\tau|}{T}\right),$

for $|\tau| < T$ . The zero-delay cut gives $|\chi_A(0, \nu)| = |\text{sinc}(\nu T)|$ (Doppler resolution $1/T$ ). The zero-Doppler cut gives $|\chi_A(\tau, 0)| \approx |\text{sinc}(W\tau)|$ (range resolution $1/W$ , i.e., $\Delta R = c/(2W)$ ).

The ambiguity diagram has a sheared ridge along the line $\nu = \mu\tau = (W/T)\tau$ , reflecting range-Doppler coupling.

The LFM trades a clean thumbtack ambiguity for a ridge-shaped one. The ridge means that a Doppler shift appears as a range shift in the matched filter output — this coupling must be corrected in radar processing.

Proof

Substitute LFM into ambiguity definition

$\chi_A(\tau, \nu) = \frac{1}{T}\int_{-T/2}^{T/2} e^{j\pi\mu t^2} e^{-j\pi\mu(t-\tau)^2} e^{j2\pi\nu t}\,dt$ .

Simplify the exponent

The quadratic terms cancel, leaving $e^{j2\pi(\nu + \mu\tau)t} e^{-j\pi\mu\tau^2}$ . The integral becomes $\text{sinc}[(\nu + \mu\tau)T_{\text{eff}}]$ where $T_{\text{eff}} = T - |\tau|$ .

Example: Designing an LFM Chirp for Automotive Radar

An automotive radar at $f_0 = 77$ GHz requires range resolution $\Delta R = 0.15$ m and unambiguous range $R_{\text{ua}} = 200$ m. Design the LFM waveform parameters.

Solution

Compute required bandwidth

$\Delta R = c/(2W)$ , so $W = c/(2 \cdot 0.15) = 1$ GHz.

Compute maximum PRI

$R_{\text{ua}} = c T_{\text{PRI}}/2$ , so $T_{\text{PRI}} = 2 \cdot 200 / (3 \times 10^8) = 1.33$ $\mu$ s.

Choose pulse duration

For FMCW operation (continuous chirp), $T \approx T_{\text{PRI}} = 1.33$ $\mu$ s. The time-bandwidth product is $W T = 10^9 \cdot 1.33 \times 10^{-6} = 1330$ . Chirp rate: $\mu = 10^9 / 1.33 \times 10^{-6} = 7.5 \times 10^{14}$ Hz/s.

Check Doppler coupling

At $v = 100$ km/h $\approx 27.8$ m/s, $f_d = 2v/\lambda = 2 \cdot 27.8 / (3.9 \times 10^{-3}) = 14.3$ kHz. Range error from coupling: $\Delta R = v T / (W T) \cdot c/2 = f_d \cdot c / (2\mu) = 2.9$ mm. This is negligible compared to $\Delta R = 0.15$ m.

Definition:
Phase-Coded Waveforms

A phase-coded waveform consists of $N$ sub-pulses of duration $T_c$ , each modulated by a phase $\phi_n$ :

$s(t) = \frac{1}{\sqrt{N}} \sum_{n=0}^{N-1} e^{j\phi_n}\,\text{rect}\!\left(\frac{t - nT_c}{T_c}\right).$

The total duration is $T = NT_c$ and the bandwidth is $W \approx 1/T_c$ , giving time-bandwidth product $W T = N$ .

Barker codes achieve the minimum possible peak sidelobe level $1/N$ among binary ( $\phi_n \in \{0, \pi\}$ ) codes. They exist only for $N \leq 13$ .

Polyphase codes (Frank, Zadoff-Chu) allow larger $N$ with near-ideal ambiguity properties.

Ambiguity Function Explorer

Compare the ambiguity functions of LFM, Barker-13, and rectangular pulse waveforms. The LFM shows the characteristic sheared ridge; the Barker code approaches a thumbtack with low sidelobes; the rectangular pulse has a diamond shape with no range resolution.

Parameters

Waveform type

BT

product100

Ambiguity Function as the Point-Spread Function

The connection to imaging (Ch 07.2, Ch 13) is direct: the matched-filter image of a point scatterer is precisely $|\chi_A(\tau, \nu)|$ centered at the scatterer's delay and Doppler. The PSF of the sensing operator $\mathbf{A}^{H}\mathbf{A}$ in the range-Doppler plane is a product of the waveform's ambiguity function and the array's spatial beampattern.

This means that waveform design is not just a radar engineering concern — it directly determines the sidelobe structure of the imaging operator, which in turn determines whether sparse recovery algorithms (Ch 14) can separate closely spaced targets.

Comparison of Radar Waveform Types

Property	Rectangular Pulse	LFM Chirp	Barker Code	Zadoff-Chu
$W T$ product	1	$\gg 1$	$N \leq 13$	$N$ (arbitrary)
Range resolution	$cT/2$ (poor)	$c/(2W)$ (fine)	$cT_c/2$	$cT_c/2$
Doppler resolution	$1/T$ (good)	$1/T$ (good)	$1/(NT_c)$	$1/(NT_c)$
Range sidelobes	N/A	$-13.3$ dB (unweighted)	$-22.3$ dB ( $N=13$ )	$\approx -13$ dB
Ambiguity shape	Diamond	Sheared ridge	Near thumbtack	Near thumbtack
Range-Doppler coupling	None	Yes ( $\nu = \mu\tau$ )	Minimal	Minimal
Doppler tolerance	Good	Good (with correction)	Poor for large $N$	Moderate
Hardware complexity	Low	Low (simple chirp)	Moderate (phase switching)	Moderate

Common Mistake: Attempting to Design a Thumbtack Ambiguity Function

Mistake:

Trying to design a waveform with simultaneously narrow mainlobe and no sidelobes in both range and Doppler.

Correction:

The volume constraint $\iint |\chi_A|^2 = 1$ makes this impossible. A thumbtack-like ambiguity (e.g., Barker code) suppresses close-in sidelobes but necessarily has sidelobes at larger delay/Doppler offsets. Waveform design is about managing where the sidelobe energy goes, not eliminating it.

Historical Note: The Invention of Pulse Compression

1950s

Pulse compression via LFM was invented independently by several groups in the early 1950s. The key insight — that a long waveform with internal modulation can achieve the energy of a long pulse and the resolution of a short one — was first described in a classified 1951 patent by Sidney Darlington of Bell Labs. The name "chirp" comes from the similarity of the LFM waveform to a bird's call with rising pitch.

Barker codes were introduced by R.H. Barker in 1953 as binary sequences with optimal autocorrelation properties. Despite 70+ years of searching, no Barker code longer than 13 has been found, and it is widely conjectured that none exists.

Ambiguity Function

The cross-correlation of a waveform with a delayed-and-Doppler-shifted version of itself: $\chi_A(\tau, \nu) = \int s(t) s^*(t-\tau) e^{j2\pi\nu t}\,dt$ . Characterizes the joint range-Doppler resolution of the waveform.

Key Takeaway

The ambiguity function $|\chi_A(\tau, \nu)|^2$ is the fundamental tool for waveform design. Its volume is constant (the radar uncertainty principle), so the designer chooses where to place sidelobe energy. For imaging, the ambiguity function is the range-Doppler component of the PSF of $\mathbf{A}^{H}\mathbf{A}$ .

LFM Chirp Pulse Compression

An LFM (linear frequency modulated) chirp waveform is transmitted and passes through the matched filter. The animation shows the instantaneous frequency sweeping across the pulse bandwidth

W

, followed by the matched filter output contracting into a narrow sinc-like peak with range resolution

\Delta R = c/(2W)

. Sidelobe structure and the effect of windowing are highlighted.

Waveforms and the Ambiguity Function