infinite set of samples. Bandpass signals Sometimes aliasing is used intentionally on signals with no low-frequency content, called bandpass signals. Undersampling, which creates low-frequency aliases, can produce the same result, with less effort, as frequency-shifting the signal to lower frequencies before sampling at the lower rate. Some digital channelizers exploit aliasing in this way for computational efficiency. See Sampling (signal processing), Nyquist rate (relative to sampling), and Filter bank. Sampling sinusoidal functions Sinusoids are an important type of periodic function, because realistic signals are often modeled as the summation of many sinusoids of different frequencies and different amplitudes (for example, with a Fourier series or transform). Understanding what aliasing does to the individual sinusoids is useful in understanding what happens to their sum. When sampling a function at frequency (intervals ), the following functions yield identical sets of samples: }. A frequency spectrum of the samples produces equally strong responses at all those frequencies. Without collateral information, the frequency of the original function is ambiguous. So the functions and their frequencies are said to be aliases of each other. Noting the trigonometric identity: we can write all the alias frequencies as positive values: . For example, here a plot depicts a set of samples with parameter , and two different sinusoids that could have produced the samples. Nine cycles of the red sinusoid and one cycle of the blue sinusoid span an interval of 10 samples. The corresponding number of cycles per sample are and . So the alias of is (and vice versa). Aliasing matters when one attempts to reconstruct the original waveform from its samples. The most common reconstruction technique produces the smallest of the frequencies. So it is usually important that be the unique minimum. A necessary and sufficient condition for that is where is commonly called the Nyquist frequency of a system that samples at rate In our example, the Nyquist condition is satisfied if the original signal is the blue sinusoid (). But if the usual reconstruction method will produce the blue sinusoid instead of the red one. Folding In the example above, and are symmetrical around the frequency And in general, as increases from 0 to decreases from to Similarly, as increases from to continues decreasing from to 0. A graph of amplitude vs frequency for a single sinusoid at frequency and some of its aliases at and would look like the 4 black dots in the first figure below. The red lines depict the paths (loci) of the 4 dots if we were to adjust the frequency and amplitude of the sinusoid along the solid red segment (between and ). No matter what function we choose to change the amplitude vs frequency, the graph will exhibit symmetry between 0 and This symmetry is commonly referred to as folding, and another name for (the Nyquist frequency) is folding frequency. Folding is often observed in practice when viewing the frequency spectrum of real-valued samples, such as the second figure below. Complex sinusoids Complex sinusoids are waveforms whose samples are complex numbers, and the concept of negative frequency is necessary to distinguish them. In that case, the frequencies of the aliases are given by just:' Therefore, as increases from to goes from up to 0. Consequently, complex sinusoids do not exhibit folding. Complex samples of real-valued sinusoids have zero-valued imaginary parts and do exhibit folding. Sample frequency When the condition is met for the highest frequency component of the original signal, then it is met for all the frequency components, a condition called the Nyquist criterion. That is typically approximated by filtering the original signal to