Dirichlet energy of a signal. Suppose the T-vector x represents a time series or signal. The quantity

D(X) = (x ₁ - X ₂ ) ² + - - - + (x _T-1 - x _T ) ² ,

the sum of the differences of adjacent values of the signal, is called the Dirichlet energy of the signal, named after the mathematician Peter Gustav Lejeune Dirichlet. The Dirichlet energy is a measure of the roughness or wiggliness of the time series. It is sometimes divided by T — 1, to give the mean square difference of adjacent values.

(a) Express D(x) in vector notation. (You can use vector slicing, vector addition or subtraction, inner product, norm, and angle.)

(b) How small can D(x) be? What signals x have this minimum value of the Dirichlet energy?

(c) Find a signal x with entries no more than one in absolute value that has the largest possible value of D(x). Give the value of the Dirichlet energy achieved.