# DFT, Band Pass Sampling.

Cuthbert Nyack
In order to be able to recover a signal correctly after sampling, then the sampling frequency must be greater than twice the highest frequency in the signal.
For the case of a band pass signal a lower sampling frequency can be used if it is permissible to reproduce the band pass signal with a lower center frequency. In this case the sampling frequency must be greater than twice the bandwidth.

eg parameters:-
(6.0, 8.0, 0.01, 24.78, 70, NA, 70, 343.0, 34.5, 0.25) the band pass signal is between 6 and 8rad/s. The sampling frequency is 17.5rad/s so the signal occurs in the spectrum between 6 and 8 rad/s. ( wL between 246 and 343)

(6.0, 8.0, 0.01, 24.49, 40, NA, 40, 301.0, 34.5, 0.25) the band pass signal is between 6 and 8rad/s. The sampling frequency is 10.0rad/s so the signal occurs in the spectrum between 6 and 8 rad/s. above half the sampling frequency ( wL between 450 and 600). However ws/2 is now 5 rad/s (wL = 375) and the original spectrum is reproduced but inverted between 2 and 4rad/s (wL between 150 and 300)

(6.0, 8.0, 0.01, 24.05, 23, NA, 23, 375.0, 34.5, 0.65) ws/2 is now 2.875 rad/s less than the lowest frequency in the signal but greater than twice the bandwidth (wL = 375) and the original spectrum is reproduced correctly between 0.245 and 2.246rad/s (wL between 32 and 293).
Setting Nf2 = 50 shows that the original spectrum occurs between the sampling frequency and 1.5 times the sampling frequency. (wL between 360 and 480)

(14.0, 16.0, 0.01, 24.0, 22, NA, 69, 375.0, 34.5, 0.7) ws/2 is now 2.75 rad/s less than the lowest frequency in the signal but greater than twice the bandwidth and the original spectrum is reproduced but inverted between 0.483 and 2.484rad/s (wL between 21 and 108).
The original spectrum occurs between the 2.5 times the sampling frequency and 3 times the sampling frequency. (wL between 608 and 695)

(14.0, 16.0, 0.01, 24.21, 27, NA, 69, 294.0, 34.5, 0.7) ws/2 is now 3.275 rad/s less than the lowest frequency in the signal but greater than twice the bandwidth and the original spectrum is reproduced correctly between 0.483 and 2.484rad/s (wL between 21 and 108).
The original spectrum occurs between the 2.0 times the sampling frequency and 2.5 times the sampling frequency. (wL between 608 and 695)

The original spectrum is reproduced correctly between 0 and ws/2 provided the original signal is between nws and (n + ½)ws and the sampling frequency is greater than twice the bandwidth.