This definition generalizes in a straightforward manner to a discrete signal with a countably infinite number of values x n such as a signal sampled at discrete times t n = t 0 + ( n Δ Δ t ) : S ¯ ¯ x x ( f ) = lim N → → ∞ ∞ ( Δ Δ t ) 2 | ∑ ∑ n = − − N N x n e − − i 2 π π f n Δ Δ t | 2 ⏟ ⏟ | x ^ ^ d ( f ) | 2 , where x ^ ^ d ( f ) is the discrete-time Fourier transform of x n . The sampling interval Δ Δ t is needed to keep the correct physical units and to ensure that we recover the continuous case in the limit Δ Δ t → → 0. But in the mathematical sciences the interval is often set to 1, which simplifies the results at the expense of generality. (also see normalized frequency)