(April 25 1997)

## SamplingThe voltage across the loudspeaker output of a receiver is a function of time. Let us call it x( t ). Of course time is continuos, but since x( t ) can not vary very fast because of the limited bandwidth of the receiver, the function x( t ) is completely known if it is known at a limited number of points. We may use a computer to sample at 8kHz. This is standard .WAV format with a Sound Blaster and a PC, and the real wave form will be identical to a smooth wave form that is fitted to these points if the receiver does not produce any signal energy at all above 4kHz.Aliasing is a phenomenon that limits the useful spectrum width when using fourier transforms. It has nothing to do with this particular use of the sampled data, it comes from the sampling process itself. I think it is best described by an example. Assume data is sampled at 8kHz. The highest frequency that can be represented with such data points is 4kHz. With all odd points = +1 and all even points = -1, the data corresponds to a sine wave of frequency 4 kHz and amplitude 1, and this is one possible result when sampling a 4kHz signal at 8kHz. Another possible result is that all points are = 0. That would happen if the sampling points accidentally happened to coincide with the zero crossings of the sine wave. If the frequency is slightly below 4kHz, say (4-x)kHz then the phase will vary slowly, so the data points will be +/- 1 for a while, and then gradually go through 0 towards +/- 1 again. If we look at the odd points only, they will be a perfect sine wave of frequency x. The even points will just have reversed sign. When this signal is fourier transformed, the transform will give a large amplitude for the frequency (4-x) kHz and nothing else. If we feed (4+x) kHz into the sampling input, still sampling at 8 kHz, the data points will be identical to the ones we got at (4-x) kHz. When we take the fourier transform, the result will be a large amplitude at (4-x) kHz and this is incorrect. Aliasing means seeing frequencies within the passband that really belong outside.
## FilteringAliasing has to be avoided. It corresponds to mirror frequencies while the sampling process corresponds to frequency mixing with half the sampling frequency. When using fourier transforms for weak signal monitoring a reasonable maximum level for alias frequencies could be -60dB. This means that local stations have to be 60 dB stronger than the signals we look for before they start to cause confusion through aliasing.Usually it is assumed that one needs a filter that removes all signals above the Nyquist frequency, half the sampling frequency. This is however not necessary. If we have a low pass filter that falls from 0 to -60 dB in half an octave, we may place -60dB at 4.0kHz, but then the flat response goes only up to 2.82kHz, square root of two lower. By placing the -60dB point at 4*1.189 kHz = 4.76kHz, the flat response will go up to 3.36 kHz, still square root of two below. By doing that, the digital data may be incorrect for frequencies between 3.36 and 4.0kHz, but by removing that frequency range in the digital signal we have a larger bandwidth with flat frequency response, without any aliasing. This is particularly convenient if the data is used for calculating FFT spectra. Removing frequencies above some limit just means discard the points above some upper limit in the spectrum. There are more aliasing frequencies corresponding to mixing with overtones of the Nyquist frequency. It is trivial to make sure they do not impair the digital data because they are far off in frequency. |