The relationship between two sinusoidal signals and is shown for and The signals are assumed to be observed at times and the observed points are indicated. When is an integer the points on the curve coincide and the signals are said to be aliased. Considering all frequencies in the range , the largest value of so that all signals may be identified from the observed points is called the Nyquist frequency. It is well known that .
The spectral density function for a stationary time series , with autocovariance function , can also be written as
The sdf can be interpreted as the amount of variability accounted for by a sinusoid with frequency .
Let . Then we can write where is an integer and . Then we see that . Consequently all such frequencies are said to be aliased with . The highest frequency that can be represented in a discrete time series sampled at times is 0.5 and is known as the Nyquist frequency. More generally, if the time interval between observations is Δ then the Nyquist frequency is Δ/2.