Four examples of the relationship between the waveform of a signal in the
time domain compared to its spectrum in the frequency domain.

The last figure shows a random signal (complex and not periodical).
Most natural sound signals are complex in shape. The primary result of a
frequency analysis is to show that the signal is composed of a number of
discrete frequencies at individual levels present simultaneously.

The number of discrete frequencies displayed is a function of the accuracy of
the frequency analysis which normally can be defined by the user.

Filters

To analyze a sound signal, frequency filters or a bank of filters are used. If
the bandwidths of these filters are small a highly accuracy analysis is
achieved.
The signal flow chart shown illustrates the elements in a simple sound level
meter.

On top is a microphone for signal pick up. Then a gain amplification stage
followed by a single frequency filter - here shown as an ideal filter. In the
following we will look at real filters. After filtering follows a rectifier with the
standardised time constants Fast, Slow and Impulse and the signal level is
finally converted to dB and shown on the display.

Bandpass Filters and Bandwidth

Practical and ideal filter - I: Ideal filter - P: Practical filter - R: Ripple

Ideal filters are only a mathematical abstraction. In real life, filters do not
have a flat top and and vertical sides. The departure from the idealised flat
top is described as an amount of ripple. The bandwidth of the filter is
described as the difference between the frequencies where the level has
dropped 3 dB in level corresponding to 0.707 in absolute measures.

It is useful to define a Noise Bandwidth for a filter. This corresponds to an
ideal filter of the same level as the real filter, but with its bandwidth (Noise
Bandwidth) set to leave the two filters with the same 'area'.

Treatment and Fourier transform

The figure below shows the influence of the sampling and its spectrum (calculated by Fourier tranform).

Integral Fourier trandform.

Fourier series.

Sampled functions - Discrete in time and periodical in frequency

Discrete Fourier transform - Discrete and periodical in time and frequency

The transformation of a continuous wave in a succession of discrete points introduces a periodicity in the spectrum. If the signal is periodic in time, the spectrum is then also discrete.