Aliasing - Sampling Sinusoidal Functions

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 (with a Fourier series or transform). Understanding what aliasing does to the individual sinusoids is useful in understanding what happens to their sum.

Here a plot depicts a set of samples whose sample-interval is 1, and two (of many) different sinusoids that could have produced the samples. The sample-rate in this case is = 1. For instance, if the interval is 1 second, the rate is 1 sample per second. Nine cycles of the red sinusoid and 1 cycle of the blue sinusoid span an interval of 10. The respective sinusoid frequencies are = 0.9 and = 0.1. In general, when a sinusoid of frequency is sampled with frequency the resulting samples are indistinguishable from those of another sinusoid of frequency for any integer N. The values corresponding to N ≠ 0 are called images or aliases of frequency In our example, the N=±1 aliases of are and A negative frequency is equivalent to its absolute value, because sin(‑wt+θ)=sin(wt‑θ+π), and cos(‑wt+θ)=cos(wt‑θ). Therefore we can express all the image frequencies as for any integer N (with being the actual signal frequency). Then the N=1 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.