The concept of roll-off refers to the rate at which the gain of a system decreases with increasing frequency. In the case of a single-pole model, the roll-off of the gain is -20 dB/decade.

Explanation:

1. Single-pole Model:
A single-pole model is a simplified representation of a system or circuit that exhibits a first-order response. It is commonly used to analyze the frequency response of systems and circuits. The transfer function of a single-pole model is given by:

H(s) = 1 / (s + a)

where H(s) is the transfer function, s is the Laplace variable, and a is the pole location.

2. Gain Roll-off:
The gain of a system is defined as the ratio of the output amplitude to the input amplitude, expressed in decibels (dB). The roll-off of the gain refers to the rate at which the gain decreases with increasing frequency.

In the case of a single-pole model, the gain roll-off is determined by the pole location. The pole location, a, is a measure of the rate at which the gain decreases with increasing frequency. It is given by the reciprocal of the time constant, τ, of the system.

The time constant, τ, is the time it takes for the output of the system to reach 63.2% of its steady-state value in response to a step input. It is calculated as:

τ = 1 / a

3. -20 dB/decade Roll-off:
The roll-off of the gain of a single-pole model is determined by the pole location, a. The rate at which the gain decreases with increasing frequency is given by -20 dB/decade.

This means that for every decade increase in frequency, the gain of the system decreases by 20 dB. Similarly, for every factor of 10 increase in frequency, the gain decreases by a factor of 10^(20/20) = 0.1.

This roll-off rate is characteristic of a first-order system and is commonly observed in various electrical and electronic circuits. It is important to consider the roll-off of the gain when designing systems or circuits to ensure the desired frequency response is achieved.