Sampling Theorem
•Effect of sampling on the frequencies of the signal
•Consider a sinusoidal signal of frequency f, x(t) = A sin (2 pi f t), and we sample it at a sampling frequency fs or a sampling period T (fs T = 1), the discrete signal is
x(n) = A sin (2 pi nf T), n = 0, +-1, +-2, +-3, ….
•Consider another sinusoidal signal of frequency (f + fs), x1(t) = A sin (2 pi (f+fs) t), and we sample it at the same sampling frequency fs, the discrete signal is
x1(n) = A sin (2 pi (f+fs) n T) = A sin (2 pi nf T + 2 pi n) = A sin (2 pi nf T) = x(n)
•Similarly, all signals with frequencies f +- mfs , m = 0, , +-1, +-2, +-3, …. will produce the same discrete signals.
•The result means that when we sample an analog signal of frequency f into a discrete signal, the discrete signal will carry all frequencies f +- mfs , m = 0, , +-1, +-2, +-3, ….
•Effect of sampling on the frequencies of the signal
•Consider a sinusoidal signal of frequency f, x(t) = A sin (2 pi f t), and we sample it at a sampling frequency fs or a sampling period T (fs T = 1), the discrete signal is
x(n) = A sin (2 pi nf T), n = 0, +-1, +-2, +-3, ….
•Consider another sinusoidal signal of frequency (f + fs), x1(t) = A sin (2 pi (f+fs) t), and we sample it at the same sampling frequency fs, the discrete signal is
x1(n) = A sin (2 pi (f+fs) n T) = A sin (2 pi nf T + 2 pi n) = A sin (2 pi nf T) = x(n)
•Similarly, all signals with frequencies f +- mfs , m = 0, , +-1, +-2, +-3, …. will produce the same discrete signals.
•The result means that when we sample an analog signal of frequency f into a discrete signal, the discrete signal will carry all frequencies f +- mfs , m = 0, , +-1, +-2, +-3, ….
