You plan to acquire a sinusoidal signal that repeats 250 microseconds. What should be the minimum sampling rate of the digital signal processor you plan to use in order to acquire this sinusoidal signal?

F = 25.1 kHz

Step-by-step explanation:

Time (period) = 250 × 10^-6 s

Let's first calculate the folding frequency. Given by 

NF (rad s-1) = π/Δt

where NF = Nyquist frequency

 t  = time increment between observations.

The original signal must be filtered by analog or physical methods to remove all frequencies higher than the Nyquist frequency before the signal is sampled or digitized.

F = 3.143/0.00025

F = 12566.37Hz

minimum sampling rate = 2F

2 × 12566.37 = 25132.74Hz

minimum sampling rate  = 25.1 KHz

t = 3

step-by-step explanation:

put t = 3 in this expression :

(3²-1)(3²-2)(3² - 3)(3² - 4) = 8×7×6×5 = 56×30 = 1680