Oscillating systems. Description and analysis - Building Acoustics - page 15

x ( t )

Time, t

Δ t

Figure 1.10 Sampling of a continuous signal by time intervals Δ t.

x 1 x 2

x ( t )

x 0

x n

x 3

x N-1

Time, t

Figure 1.11 Allocating amplitude values to the samples. The marked bars indicate the values x 0 and x N - 1
allocated to the first and the last sample.

The continuous time signal x ( t ) is thereby substituted by a set of numbers x n =
x ( n Δ t ) with n = 0, 1, 2, 3,..., N - 1. The corresponding continuous transform X ( f ) (see
Equation (1.17)) will turn into a discrete sequence X k = X ( k Δ f ) with k = 0, 1, 2, 3,…, N -
1. Per definition, the discrete Fourier transform (DFT) is given by