# Aliasing

March 29, 2018

Back to: Random Testing

Digital sampling of sine waves has the peculiarity that different sine waves can give identical sample values for a fixed sample rate. Therefore, the frequency value of an FFT result is not unique, in general. This is illustrated in Fig. 11 with SR = 2000 Hz. The digital samples of the two sine waves shown with different frequencies are identical. For the FFT computation, the higher frequency “looks” like the lower frequency, thus the origin of the name “aliasing.”

In order to avoid this ambiguity in the FFT, the lowest frequency values are assumed. For a given Sample Rate these frequencies lie between 0 and SR/2 = *f _{ q}*. Thus the need to remove from the signal all frequencies above the Nyquist frequency. This is usually done by passing the signal through an analog low-pass filter before it is converted to digital samples by an A/D convertor. A typical low-pass filter begins attenuating the signal at about 80% of the nominal cut-off frequency, so most systems provide good data for frequencies up to

*f*= 0.4 SR.

For completeness it should be noted that the aliasing occurs as a mirror image about the Nyquist frequency (and odd number multiples). The frequency *f *_{a} = *f _{ q}* +

*df*looks the same as the frequency

*f*

_{b}=

*f*–

_{ q}*df*(with 0 <

*df*<

*f*) for a sample rate SR = 2

_{ q}*f*. And so does

_{ q}*f*

_{c}= 3

*f*–

_{ q}*df*, and

*f*

_{d}= 3

*f*+

_{ q}*df*, and so on for all odd number multiples of

*f*.

_{ q}