Building Acoustics

Oscillating systems 15

x(t)

Time, t

Δt

Figure 1.10 Sampling of a continuous signal by time intervals Δt.

x(t)

Time, t

x 0

x 1

x 2

x 3

xN-1

xn

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

The continuous time signal x(t) is thereby substituted by a set of numbers xn =
x(nΔt) with n = 0, 1, 2, 3,..., N – 1. The corresponding continuous transform X(f) (see
Equation (1.17)) will turn into a discrete sequence Xk = X(kΔf) with k = 0, 1, 2, 3,..., N –

1. Per definition, the discrete Fourier transform (DFT) is given by