Long-Range Stretch Processing#

FMCW and stretch radars recover range by mixing the received echo against a reference chirp, producing a beat tone proportional to the delay difference. By default RadarSimPy deramps against a zero-delay reference: the beat frequency is \(k \tau\), where \(k\) is the chirp slope and \(\tau = 2R/c\) the full round-trip delay.

That is fine at short range, but it breaks down long before the ranges real long-range radars operate at.

Why a Range Gate Is Needed#

Consider an X-band stretch radar imaging ships at 60 nautical miles:

Parameter	Value
Carrier \(f_c\)	9 GHz
Bandwidth \(B\)	300 MHz
Chirp length \(T\)	50 µs
Chirp slope \(k = B/T\)	6 × 10¹² Hz/s
IF/ADC rate \(f_s\) (complex)	40 MHz
Max un-gated range \(f_s c / 2k\)	999 m
Target range \(R\)	111.12 km
Round-trip delay \(\tau\)	741.4 µs
Beat frequency \(k \tau\)	4.45 GHz

The beat tone is over 200 times the Nyquist limit of ±20 MHz. The target aliases into an essentially arbitrary range bin and is unrecoverable. Note also that \(\tau\) is nearly 15 chirp lengths, so the echo arrives long after the chirp that produced it has ended.

Real long-range stretch radars avoid this by mixing against a reference chirp delayed to a range gate, so the beat depends only on the target’s offset from the gate rather than on its absolute range.

Using `gate_delay`#

Set Receiver(gate_delay=...) to the two-way delay of the range you want to gate on:

import scipy.constants as const
from radarsimpy import Radar, Transmitter, Receiver

gate_range = 111.12e3  # m

tx = Transmitter(
    f=[8.85e9, 9.15e9],   # 300 MHz sweep about 9 GHz
    t=50e-6,              # 50 us chirp
    tx_power=40,
    prp=200e-6,
    pulses=256,
)

rx = Receiver(
    fs=40e6,
    noise_figure=8,
    rf_gain=20,
    baseband_gain=30,
    gate_delay=2 * gate_range / const.c,   # ~741.4 us
)

radar = Radar(transmitter=tx, receiver=rx)

The receive window now opens at the gate and the deramp reference is the transmit chirp delayed by the same amount. A target at exactly gate_range beats at DC; a target offset by \(\Delta R\) beats at \(2 k \Delta R / c\).

Note

gate_delay defaults to 0, which reproduces the zero-delay behavior exactly. Existing simulations are unaffected.

Swath and Resolution#

The gate does not change resolution, only where the usable window sits:

\[\text{swath about the gate} = \pm \frac{f_s c}{4 k} \qquad \text{range resolution} = \frac{c}{2B}\]

For the configuration above that is a ±500 m swath at 0.5 m resolution, i.e. 2000 range bins — exactly pulse_length * fs, the number of samples per pulse. A 200–300 m ship fits comfortably inside one gate.

Both are available from the radar object:

radar.chirp_slope                # 6e12 Hz/s
radar.unambiguous_range_span     # ~999.3 m
radar.unambiguous_range_window   # (110620.4, 111619.7)
radar.receiver.gate_range        # 111120.0 m

Note

Where the window sits depends on whether a gate is configured. Un-gated, every target has a positive round-trip delay, so all beat tones are positive and the whole [0, fs) band is usable — the window is [0, span]. Once a gate makes the residual delay signed, the usable band becomes (-fs/2, +fs/2) and the window straddles the gate at +/- span/2.

Converting Beat Bins to Range#

Range is measured relative to the gate:

\[R = R_\text{gate} + \frac{f_\text{beat}\, c}{2 k}\]

import numpy as np
import scipy.constants as const

profile = np.fft.fft(baseband[0, 0, :])
freqs = np.fft.fftfreq(profile.size, d=1 / rx.bb_prop["fs"])
ranges = radar.receiver.gate_range + freqs * const.c / (2 * radar.chirp_slope)

Negative beat frequencies correspond to targets inside the gate, so with complex baseband the swath is two-sided about the gate.

Covering More Than One Swath#

A single simulation has a single gate. To cover a wider area, run the simulation once per gate and stitch the results:

for gate_range in np.arange(100e3, 120e3, 1e3):
    rx = Receiver(..., gate_delay=2 * gate_range / const.c)
    ...

Targets outside the swath alias exactly as they would without a gate. Sizing the gate is up to you; the simulator does not check target ranges against the window, since the beat-frequency relationship only applies to deramp processing of a linear FM waveform and not to the pulsed, CW, or arbitrary-waveform configurations RadarSimPy also supports.

Effect on Phase Noise#

Transmitter phase noise partially cancels in a coherent radar because the reference and the echo are drawn from the same oscillator a short time apart — the range correlation effect. With a gate, the two are separated by the residual delay \(\tau - \text{gate\_delay}\) rather than by the full round trip, which is the correct model for a digitally generated delayed reference (the usual long-range stretch architecture).

Important

Phase-noise results at long range are only meaningful with a gate configured. Without one, the reference and echo are separated by the full round-trip delay and the phase-noise lookup wraps, so close-in phase noise does not cancel as it would in real hardware.

Limitations#

Note

The echo of a gated return physically belongs to the chirp transmitted \(\text{round}(\text{gate\_delay} / \text{prp})\) pulses earlier. Per-pulse f_offset and pulse_amp / pulse_phs are applied using the current pulse index, so frequency-hopped or phase-coded pulse trains are not modeled exactly at long gate delays. Identical chirps — the common FMCW case — are unaffected.

Near the edges of the window the reference and echo chirps only partially overlap. The simulator extrapolates the chirp there, modeling an ideal infinite chirp, which keeps the full window usable. Real hardware would see reduced correlation gain over roughly \(|\Delta \tau| f_s\) samples.

Long-Range Stretch Processing#

Why a Range Gate Is Needed#

Using gate_delay#

Swath and Resolution#

Converting Beat Bins to Range#

Covering More Than One Swath#

Effect on Phase Noise#

Limitations#

See Also#

Using `gate_delay`#