Aliasing in Time Series Analysis

frequency

phase

m

The relationship between two sinusoidal signals

a

t

=cos(2πf+ϕ)

and

b

t

=cos(2π(f+m)+ϕ)

is shown for

|f|≤0.5,

|ϕ|≤π/2,

and

|m|≤3.

The signals are assumed to be observed at times

t=0,1,…,6

and the observed points are indicated. When

m

is an integer the points on the curve coincide and the signals are said to be aliased. Considering all frequencies in the range

|f|≤

f

m

, the largest value of

f

m

so that all signals may be identified from the observed points is called the Nyquist frequency. It is well known that

f

m

=0.5

.