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That paper includes an example of frequency aliasing dating back to 1922. The first published use of the term "aliasing" in this context is due to Blackman and Tukey in 1958. [ 5 ] In their preface to the Dover reprint [ 6 ] of this paper, they point out that the idea of aliasing had been illustrated graphically by Stumpf [ 7 ] ten years prior.
In this example, f s is the sampling rate, and 0.5 cycle/sample × f s is the corresponding Nyquist frequency. The black dot plotted at 0.6 f s represents the amplitude and frequency of a sinusoidal function whose frequency is 60% of the sample rate. The other three dots indicate the frequencies and amplitudes of three other sinusoids that ...
Effects of aliasing, blurring, and sharpening may be adjusted with digital filtering implemented in software, which necessarily follows the theoretical principles. A family of sinusoids at the critical frequency, all having the same sample sequences of alternating +1 and –1.
Fig 1: Typical example of Nyquist frequency and rate. They are rarely equal, because that would require over-sampling by a factor of 2 (i.e. 4 times the bandwidth). In signal processing , the Nyquist rate , named after Harry Nyquist , is a value equal to twice the highest frequency ( bandwidth ) of a given function or signal.
The bandwidth, B, in this example is just small enough that the slower sampling does not cause overlap (aliasing). Sometimes, a sampled function is resampled at a lower rate by keeping only every M th sample and discarding the others, commonly called "decimation". Potential aliasing is prevented by lowpass-filtering the samples before decimation.
In the case of optical image sampling, as by image sensors in digital cameras, the anti-aliasing filter is also known as an optical low-pass filter (OLPF), blur filter, or AA filter. The mathematics of sampling in two spatial dimensions is similar to the mathematics of time-domain sampling, but the filter implementation technologies are different.
That fidelity is reduced when () contains frequency components whose cycle length (period) is less than 2 sample intervals (see Aliasing). The corresponding frequency limit, in cycles per second ( hertz ), is 0.5 {\displaystyle 0.5} cycle/sample × f s {\displaystyle f_{s}} samples/second = f s / 2 {\displaystyle f_{s}/2} , known as the Nyquist ...
The sinc function as audio, at 2000 Hz (±1.5 seconds around zero) In mathematics, the historical unnormalized sinc function is defined for x ≠ 0 by = .. Alternatively, the unnormalized sinc function is often called the sampling function, indicated as Sa(x).