Oscillating systems. Description and analysis - Building Acoustics - page 8

1.4.1.2 Frequency analysis of a periodic function (periodic signal)
An example of a periodic function is shown in Figure 1.3 . Performing Fourier analysis
we get the frequency amplitudes | X k | as shown in Figure 1.4 . The calculations for this
example are indeed performed by the so-called discrete Fourier transform (DFT) (see
below), but this has no importance in this example. We have derived the time function by
just summing three sinusoidal signals having amplitudes of 1.0, 0.5 and 0.3, respectively
but also adjusting their relative phases. Performing the Fourier calculation, Figure 1.4
shows that these components will appear with the RMS-values reasonably correct.

1.4.2 Transient signals. Fourier integral

The above mathematical description may be adapted for non-periodic functions, e.g.
transient time functions such as the sound pressure pulse from a gunshot or an explosion,
the acceleration of a plate when hit by a hammer etc. Formally, we may say that the
function is still periodic but the period T now goes to infinity. This makes the distance
between frequency components infinitesimal, in the limiting case it goes to zero. We then
get a continuous frequency spectrum. The Fourier series transforms into an integral and
the Fourier coefficients will be a continuous function of the frequency, the so-called
Fourier transform. Working from Equation (1.7), we may express the transform X ( f ) as
follows

+∞

∫

− j2 π ft

Xf () = xte () d t − ∞ < < +∞ f ,

(1.13)

−∞

where X ( f ) will exist if

+∞

∫

xt () d t < ∞ .

−∞

X ( f ) is called the direct Fourier transform or spectrum. If this is a known function we
may use the inverse transform to find the corresponding time function x ( t ). Using a
similar modification of Equation (1.6) we may write

+∞

= ∫

j2 π ft

xt () X f e ( ) d f −∞ < < +∞ t .

(1.14)

−∞

Equations (1.13) and (1.14) are a Fourier transform pair. It should be noted that X ( f )
again is a complex function with both positive and negative frequencies which applies
even if the time function is real. It is also usual to express X ( f ) using a polar notation as
in Equation (1.7) , i.e. we write

Xf e θ

j( ) f

Xf () = () ,

(1.15)

where | X ( f ) | and θ ( f ) are denoted amplitude spectrum and phase spectrum, respectively.