In [11]:

import radarsimpy

print("`RadarSimPy` used in this example is version: " + str(radarsimpy.__version__))

`RadarSimPy` used in this example is version: 15.2.0

Receiver Operating Characteristic (ROC) Analysis¶

ROC analysis quantifies the trade-off between detection probability $P_d$ and false alarm probability $P_{fa}$ as a function of SNR. RadarSimPy provides two tools:

roc_pd(pfa, snr, N, model) — given SNR, compute $P_d$
roc_snr(pfa, pd, N, model) — given $P_d$, compute required SNR

Swerling target models supported:

Model	RCS distribution	Fluctuation rate
Swerling 1	Exponential (Rayleigh amp.)	Scan-to-scan
Swerling 2	Exponential	Pulse-to-pulse
Swerling 3	Chi-squared (4 DOF)	Scan-to-scan
Swerling 4	Chi-squared (4 DOF)	Pulse-to-pulse
Swerling 5	Constant RCS	None
Coherent	Ideal coherent integration	—
Real	Based on actual distributions	—

N is the number of non-coherent integration pulses — more pulses improve detection but increase processing time and reduce Doppler ambiguity range.

Example 1: $P_d$ vs SNR — Swerling 3, N = 64¶

$P_{fa} \in \{10^{-4},\, 10^{-5}\}$, SNR from −10 to 10 dB.

In [12]:

from radarsimpy.tools import roc_pd, roc_snr
import numpy as np
import plotly.graph_objs as go
from IPython.display import Image, display

# Set to True for interactive plots; False renders a static JPEG (e.g. for HTML export)
INTERACTIVE = False


def show(fig):
    if INTERACTIVE:
        fig.show()
    else:
        display(Image(fig.to_image(format="jpg", scale=2)))


snr = np.arange(-10, 10, 0.1)
pfa = np.array([1e-4, 1e-5])
pd = roc_pd(pfa, snr, 64, "Swerling 3")

In [13]:

fig = go.Figure()

for i, p in enumerate(pfa):
    fig.add_trace(go.Scatter(x=snr, y=pd[i, :], name=f"Pfa = {p}"))

fig.update_layout(
    title="Swerling 3, N = 64",
    yaxis=dict(title="Probability of detection Pd"),
    xaxis=dict(title="SNR (dB)"),
    height=600,
)

show(fig)

No description has been provided for this image

Example 2: Effect of Integration — Swerling 3, N = 20¶

Reducing from N = 64 to N = 20 lowers integration gain, shifting detection curves to the right (higher SNR required for the same $P_d$).

In [14]:

pd = roc_pd(pfa, snr, 20, "Swerling 3")

In [15]:

fig = go.Figure()

for i, p in enumerate(pfa):
    fig.add_trace(go.Scatter(x=snr, y=pd[i, :], name=f"Pfa = {p}"))

fig.update_layout(
    title="Swerling 3, N = 20",
    yaxis=dict(title="Probability of detection Pd"),
    xaxis=dict(title="SNR (dB)"),
)

show(fig)

Example 3: All Swerling Models — $P_{fa} = 10^{-6}$, N = 5¶

Coherent integration is the best-case bound. Swerling 1/2 are hardest to detect due to deep RCS fades; Swerling 5 (constant RCS) performs closest to the coherent bound.

SNR range: −10 to 30 dB.

In [16]:

snr = np.arange(-10, 30, 0.1)
pfa = 1e-6
N = 5
pd_sw1 = roc_pd(pfa, snr, N, "Swerling 1")
pd_sw2 = roc_pd(pfa, snr, N, "Swerling 2")
pd_sw3 = roc_pd(pfa, snr, N, "Swerling 3")
pd_sw4 = roc_pd(pfa, snr, N, "Swerling 4")
pd_sw5 = roc_pd(pfa, snr, N, "Swerling 5")
pd_coherent = roc_pd(pfa, snr, N, "Coherent")
pd_real = roc_pd(pfa, snr, N, "Real")

In [17]:

fig = go.Figure()

for name, pd_vals in [
    ("Swerling 1", pd_sw1),
    ("Swerling 2", pd_sw2),
    ("Swerling 3", pd_sw3),
    ("Swerling 4", pd_sw4),
    ("Swerling 5", pd_sw5),
    ("Coherent", pd_coherent),
    ("Real", pd_real),
]:
    fig.add_trace(go.Scatter(x=snr, y=pd_vals, name=name))

fig.update_layout(
    title=f"Pfa = {pfa}, N = {N}",
    yaxis=dict(title="Probability of detection Pd"),
    xaxis=dict(title="SNR (dB)"),
)

show(fig)

Example 4: Required SNR vs $P_d$ — $P_{fa} = 10^{-5}$, N = 10¶

Inverse calculation: given a detection requirement, how much SNR is needed? Useful for radar link budget design.

In [18]:

pfa = 1e-5
pd = np.arange(0.1, 0.9, 0.01)
N = 10
snr_real = roc_snr(1e-4, pd, N, "Real")

snr_sw1 = roc_snr(pfa, pd, N, "Swerling 1")
snr_sw2 = roc_snr(pfa, pd, N, "Swerling 2")
snr_sw3 = roc_snr(pfa, pd, N, "Swerling 3")
snr_sw4 = roc_snr(pfa, pd, N, "Swerling 4")
snr_sw5 = roc_snr(pfa, pd, N, "Swerling 5")
snr_coherent = roc_snr(pfa, pd, N, "Coherent")
snr_real = roc_snr(pfa, pd, N, "Real")

In [19]:

fig = go.Figure()

for name, snr_vals in [
    ("Swerling 1", snr_sw1),
    ("Swerling 2", snr_sw2),
    ("Swerling 3", snr_sw3),
    ("Swerling 4", snr_sw4),
    ("Swerling 5", snr_sw5),
    ("Coherent", snr_coherent),
    ("Real", snr_real),
]:
    fig.add_trace(go.Scatter(x=pd, y=snr_vals, name=name))

fig.update_layout(
    title=f"Pfa = {pfa}, N = {N}",
    xaxis=dict(title="Probability of detection Pd"),
    yaxis=dict(title="Minimal SNR (dB)"),
)

show(fig)

Generate SNR Lookup Table (Optional)¶

Uncomment cell below to pre-compute required SNR for N = 1…256 at $P_d = 0.5$, $P_{fa} = 10^{-4}$ and save to snr.csv. Useful for embedding in real-time systems that need fast lookups instead of calling roc_snr() at runtime.

In [20]:

# snr_sw1 = np.zeros(256)
# snr_sw2 = np.zeros(256)
# snr_sw3 = np.zeros(256)
# snr_sw4 = np.zeros(256)
# snr_sw5 = np.zeros(256)
# snr_ci = np.zeros(256)
# pd = 0.5
# pfa = 1e-4

# for N in range(1, 257):
#     snr_sw1[N-1] = roc_snr(pfa, pd, N, 'Swerling 1')
#     snr_sw2[N-1] = roc_snr(pfa, pd, N, 'Swerling 2')
#     snr_sw3[N-1] = roc_snr(pfa, pd, N, 'Swerling 3')
#     snr_sw4[N-1] = roc_snr(pfa, pd, N, 'Swerling 4')
#     snr_sw5[N-1] = roc_snr(pfa, pd, N, 'Swerling 5')
#     snr_ci[N-1] = roc_snr(pfa, pd, N, 'Coherent')


# mat = np.zeros((256, 6))
# mat[:, 0] = snr_sw1
# mat[:, 1] = snr_sw2
# mat[:, 2] = snr_sw3
# mat[:, 3] = snr_sw4
# mat[:, 4] = snr_sw5
# mat[:, 5] = snr_ci

# np.savetxt("../data/snr.csv", mat, delimiter=",")

Summary¶

roc_pd() and roc_snr() let you work the detection problem in both directions.
Swerling 1/2 (Rayleigh-distributed RCS) require the most SNR; Swerling 5 and Coherent require the least.
Increasing N improves detection at the cost of update rate and Doppler ambiguity.

Things to Try¶

Experiment	Change	Observable effect
Integration sweep	N = 1, 10, 50, 100	Observe diminishing returns
Stricter false alarm	$P_{fa} = 10^{-9}$	SNR requirement rises significantly
Fix $P_d = 0.95$	Use `roc_snr` for all models	Quantify worst- vs best-case SNR gap
Generate design curve	Sweep N, fixed $P_d$/$P_{fa}$	Integration gain vs N plot