Questions: Signal Detection and Statistical Hypothesis Testing

Question 1

A radar designer raises the detection threshold to reduce false alarms. What is the unavoidable consequence?

• A. The probability of detection also increases — fewer false alarms means the system is more reliable overall
• B. The probability of missed detections increases — real targets are more likely to go unrecognized
• C. SNR improves because the threshold filters out more noise
• D. The ROC curve shifts toward the upper-left corner, improving overall performance

Answer: (B)

Raising the threshold makes the detector more conservative: it requires stronger evidence before declaring a signal present. This reduces false alarms (Pfa), but it simultaneously increases missed detections — cases where a real signal falls below the threshold and goes undetected. The ROC curve itself does not shift; the designer is simply moving to a different operating point on the same curve. Threshold adjustment is a tradeoff within a fixed performance ceiling set by SNR, not a way to escape that ceiling.

Question 2

What does increasing SNR do to the receiver operating characteristic (ROC) curve?

• A. It shifts the operating point along the existing ROC curve toward lower false-alarm rates
• B. It shifts the entire ROC curve toward the upper-left, enabling simultaneously higher detection and lower false-alarm rates
• C. It has no effect on the ROC curve — SNR only matters for the matched filter, not the detector
• D. It makes the ROC curve steeper near Pfa = 0 but flatter elsewhere

Answer: (B)

SNR determines how well separated the signal-plus-noise and noise-only distributions are in observation space. Higher SNR means greater separation, so there exists a threshold that simultaneously achieves high probability of detection and low probability of false alarm — the ROC curve bows further toward the ideal upper-left corner. Threshold adjustment moves you along the ROC curve; SNR improvement lifts the curve itself. This is why reducing noise or increasing signal power fundamentally improves detection in a way no threshold tuning can achieve.

Question 3

In the likelihood ratio test, raising the detection threshold γ reduces the probability of false alarm.

• T. True
• F. False

Answer: True

A higher threshold γ requires the likelihood ratio to be larger before declaring H₁, meaning you demand stronger evidence that a signal is present. This makes it less likely that pure noise — which generates smaller likelihood ratios on average — exceeds the threshold. So Pfa decreases. The cost is that weaker real signals also fail to exceed the higher threshold, increasing the probability of missed detections. The threshold γ directly controls the tradeoff point on the ROC curve.

Question 4

Raising the detection threshold improves detection performance because it reduces errors.

• T. True
• F. False

Answer: False

This is the core misconception in detection theory. Raising the threshold reduces one type of error (false alarms) while increasing another (missed detections). There is no threshold setting that eliminates both simultaneously — the ROC curve defines the fundamental tradeoff. What 'improves' detection in the sense of lifting overall performance is increasing SNR, not adjusting the threshold. The threshold allocates the available performance between error types according to application priorities; it cannot create performance that SNR does not permit.

Question 5

Why can't adjusting the detection threshold substitute for improving SNR when the goal is to simultaneously reduce both false alarms and missed detections?

Model answer: The ROC curve traces all achievable (Pfa, Pd) pairs for a fixed SNR. Every point on the curve corresponds to a different threshold. Moving along the curve by adjusting the threshold reduces one error type only by increasing the other — you cannot simultaneously improve both. Only increasing SNR separates the signal and noise distributions further, lifting the ROC curve toward the ideal corner and making it possible to achieve lower Pfa and higher Pd at the same time.

The key insight is that SNR sets the ceiling on performance — it determines the shape of the ROC curve — while the threshold determines where on that curve the system operates. A system with poor SNR is stuck on a ROC curve close to the diagonal, no matter how carefully the threshold is set. Improving SNR (by boosting signal power, reducing noise bandwidth, or integrating over longer intervals) stretches the curve toward the upper-left, expanding the region of achievable performance. Threshold tuning is local; SNR improvement is global.