Introduction:
In radar systems, the minimum range is the shortest distance at which the radar can detect a target. The minimum range is determined by the pulse width of the transmitted radar signal. To increase the minimum range, the pulse width needs to be increased. However, increasing the pulse width also affects the peak power of the radar signal. In order to maintain a constant signal-to-noise ratio, the peak power must be increased when the pulse width is increased.

Explanation:
When the minimum range is doubled, it means that the radar needs to be able to detect targets at twice the distance compared to its current capability. To achieve this, the pulse width of the radar signal needs to be doubled.

Relationship between pulse width and range:
The pulse width of a radar signal is directly related to the range resolution. Range resolution refers to the ability of the radar to distinguish between two targets that are closely spaced in range. The shorter the pulse width, the better the range resolution. Conversely, a longer pulse width results in a poorer range resolution.

Effect of pulse width on peak power:
When the pulse width is increased, the energy of the radar signal is spread over a longer time duration. This means that the peak power of the radar signal decreases. In order to maintain a constant signal-to-noise ratio, the peak power needs to be increased when the pulse width is increased.

Relationship between pulse width and peak power:
The relationship between pulse width and peak power is inverse. As the pulse width increases, the peak power decreases, and vice versa. Mathematically, this relationship can be expressed as:

Peak Power ∝ 1 / Pulse Width

Doubling the minimum range:
When the minimum range is doubled, the pulse width needs to be doubled. Since the relationship between pulse width and peak power is inverse, the peak power needs to be increased by a factor of 2^2 = 4 in order to maintain a constant signal-to-noise ratio.

Therefore, the correct answer is option 'C' - The peak power has to be increased by a factor of 16.