Frequency domain

As an example, we discuss the case of the digital transmission or the storage on a CD-laser of the voice signal.

Let's suppose an interference (represented as a 6 kHz-frequency sinusoide, called fi) is superposed to the human voice spectrum (from 300 Hz to 3 kHz).

 

Let's assume that the transmission is achieved with a 8 kHz-sampling frequency. The sampling frequency is so greater than the greater frequency contained in the voice spectrum (3 kHz). There is so no aliasing for the voice spectrum. On the other hand, the interference is aliased because , the voice signal is so inaudible.

 

This phenomenon may be considered as a "frequency periodisation" of the analogue spectrum displayed below:

If the sampling frequency were equal to 16 kHz, the problem would have been solved.

Sampling Theorem

TEMPORAL DOMAIN

An analogue signal s(t), multiplied by a Dirac comb |_|_|(t), becomes the digital signal s*(t).

 

FREQUENCY DOMAIN

S(f) is the spectrum of the signal s(t), the convolution between S(f) and the spectrum of the Dirac comb gives S*(f), obtained as a periodisation of S(f).

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=

 

*

 

=

 

If the period of the Dirac comb is short enough, the space between two impulses of the comb spectrum is large enough and there is no aliasing. If the period of the comb is too long, it is no possible to extract from the spectrum of the sampled signal the spectrum of the original one.

The condition linking the sampling frequency and the greater frequency contained in the signal spectrum is called the Nyquist condition. It means that fe > 2fmax (theoretically fe 2fmax).

Application : FFT You can display some signals and observe their spectra calculated by FFT.

Fast Fourier Transform

Flash animation - Frequency domain