Pfa is the probability that the radar receiver declares a target present when, in reality, only noise (or clutter) is present.
Pfa = P(Detect | Noise Only)

A false alarm occurs when the noise-only voltage at the detector output exceeds the set detection threshold (VT). The Pfa is determined by:
1. The statistical distribution of noise: For random noise, it's often modeled as Gaussian (or Rayleigh after envelope detection). The shape of this distribution dictates how likely it is for noise to exceed a certain threshold.
2. The detection threshold (VT): A higher threshold reduces Pfa (fewer false alarms), but also reduces Pd. A lower threshold increases Pfa (more false alarms), but also increases Pd.
3. Receiver Bandwidth and Integration Time: These affect the noise power. For a fixed threshold, Pfa can be thought of as the fraction of time that noise alone would exceed the threshold. This directly relates to the clutter and noise "spikes" that an operator might see on a display in the absence of targets. Typical values for radar systems are very small, e.g., 10−6 (one false alarm per million decision opportunities) to 10−8 or even lower, depending on the system's operational requirements.

Pd is the probability that the radar receiver correctly declares a target present when a target signal is actually present, along with noise.
Pd = P(Detect | Signal + Noise)

Pd is determined by:
1. Signal-to-Noise Ratio (SNR): This is the most dominant factor. A higher SNR means the target signal is stronger relative to the noise, making it easier to distinguish from noise, and thus leading to a higher Pd.
2. The detection threshold (VT): As discussed, a lower threshold increases Pd.
3. Target Fluctuation Characteristics (Swerling Models): Real targets do not reflect radar energy with a constant amplitude. They "fluctuate" due to changes in aspect angle, polarization, and multipath effects. These fluctuations significantly impact Pd.
4. Number of Integrated Pulses (N): By coherently or non-coherently integrating (summing) multiple echoes from the same target, the effective SNR improves, leading to a higher Pd.

Pd and Pfa are intimately related through the detection threshold and the SNR. For a given SNR, increasing Pd (e.g., by lowering the threshold) will inevitably increase Pfa. Conversely, decreasing Pfa (e.g., by raising the threshold) will inevitably decrease Pd. This fundamental trade-off is precisely what ROC curves illustrate.

Factors Influencing Pfa and Pd:

• Signal-to-Noise Ratio (SNR): As mentioned, SNR is the primary determinant of Pd for a given Pfa. The higher the SNR, the better the detection performance. SNR depends on:
• Transmitted Power (Pt)
• Antenna Gain (G)
• Target Radar Cross Section (σ)
• Range (R)
• System Noise Temperature (Ts)
• Receiver Bandwidth (B)
• Pulse Integration (N)
• Noise Statistics: Typically assumed to be additive white Gaussian noise (AWGN). If noise is non-Gaussian or colored, more complex processing is needed.
• Detection Threshold: The judicious selection of the detection threshold is crucial. It is often set to achieve a desired constant Pfa over time, which might require adaptive thresholding (e.g., Constant False Alarm Rate - CFAR - processing) to account for varying noise or clutter levels.
• Target Fluctuation (Swerling Models): This is a very significant factor. If a target's radar cross-section (RCS) fluctuates, it makes detection harder. Strong pulses might occur, but also very weak ones that fall below the threshold. The average RCS might be high, but the instantaneous RCS can be low.
• Number of Integrated Pulses (N): When multiple pulses are integrated, the SNR increases by a factor related to N (e.g., N for coherent integration, or √N for non-coherent integration). This improves Pd for a given Pfa.

Let's assume a specific fixed Pfa =10−6. For a simple "square-law" detector (typical in non-coherent processing), the relationship between Pd, Pfa, and SNR is often depicted in universal detection curves or through more complex numerical evaluations (e.g., using Marcum's Q-function for non-fluctuating targets).
Without going into the complex mathematical functions (as they require external reference or complex derivation), let's illustrate the concept:
- If SNR = 10 dB, for Pfa =10−6, Pd might be around 0.5 (50%).
- If SNR = 13 dB (an increase of 3 dB, or a doubling of signal power), for the same Pfa =10−6, Pd might increase significantly to around 0.9 (90%).
This clearly demonstrates that even a small increase in SNR can lead to a substantial improvement in the probability of detection. This highlights the importance of maximizing SNR through good radar design and matched filtering.