phase function, the Hilbert transform will have ’i for ω > 0

spikes at the beginning and end of the pulse. H (ω) = (16.10)

i for ω < 0.

This is because the decomposition of the pulse

function into trigonometric components requires Thus

contributions from many components, and each of

|H (ω)| = 1

the components is shifted by its own quarter of

x H x H (ω) = x x (ω) (16.11)

a period. This example indicates that the Hilbert

transform can only be interpreted as a ˜time rate Var XH = Var(X) (16.12)

of change™ when most of the variability of X t is

and

con¬ned to a relatively narrow frequency band.

The Hilbert transform is used to augment the

’ x x (ω) for ω ≥ 0

x H x (ω) =

information contained in a vector time series by

x x (ω) for ω < 0

adding information about its future behaviour. This

x H x (ω) = 0.

is accomplished by combining the original vector (16.13)

16.2: Hilbert Transform 355

Note also that

±=0.9

AR(1),

κx H x (ω) = 1 for all ω = 0.

5

That is, there is perfect coherence between Xt and

0

its Hilbert transform at all nonzero frequencies.

This is as it should be since XH is just a phase-

-10 -5

t

shifted version of Xt at each frequency.

So far we have de¬ned the Hilbert transform in Input

Hilbert Transform

the frequency domain. To obtain the ¬lter in the

time domain we use the following theorem from

0 20 40 60 80 100

Brillinger [66, pp. 31,395]:

If Xt is a stationary multivariate process with

AR(1),=0.3

±

4

absolutely summable auto-covariance function

γx x , then the process

2

Yt = lim YtT (16.14)

0

T ’∞

-4 -2

where

T Input

Hilbert Transform

=

YtT h δ Xt’δ (16.15)

δ=’T

0 20 40 60 80 100

and

1

2

hδ = H (ω)e2πiδω dω (16.16) Figure 16.2: Realizations of AR(1) processes

(solid) with ±1 = 0.9 (top) and ±2 = 0.3

’1

2

(bottom) and their Hilbert transforms XH (dashed)

t

exists and has ¬nite variance.

The application of (16.16) to (16.10) yields computed with (16.15) and T = 20.

(cf. Rasmusson et al. [329])

preferred frequency, the connection between the

if δ is odd

2

hδ = δπ (16.17) input and its Hilbert transform is rather loose. The

if δ is even.

0

Hilbert transforms lead the input series. Visually,

Note that h δ ¤ 0 for negative δ and δ h δ = 0 so the lead seems to be longer when ±1 = 0.9 than

that the time mean of XH is zero. when ±1 = 0.3. This impression is substantiated

t

H in the time

Thus, the Hilbert transform Xt by the cross-covariance between the input time

domain of a stationary process Xt is series and its Hilbert transform (Figure 16.3).

Maximum cross-correlations for the short memory

∞

2

XH = Xt+2δ+1 ’ Xt’(2δ+1) . process are obtained for lag-1, while the long

t

(2δ + 1)π memory process exhibits almost uniform lag

δ=0

(16.18) correlations for a wide range of lags.

There is a more rigid link between the input and

Note that the series in (16.18) does not converge its Hilbert transform when the input is the AR(2)

for sine time functions and other non-stationary process with ±1 = 0.9 and ±2 = ’0.8, which

time series because their auto-covariance functions is shown in Figure 16.4. This process is quasi-

are not absolutely summable. oscillatory with a period of about 6 time steps

(cf. [10.3.4“6], [11.1.7] and [11.2.6]). Since this

16.2.3 Examples: The Hilbert Transform of AR process has a preferred frequency, the phase shift

Processes. We now apply the Hilbert transform between the Hilbert transform and the input series

to the AR(1) and AR(2) processes discussed in is about 1.5 time steps. Large Hilbert transform

Chapter 11. values regularly precede large changes of the input

Figure 16.2 displays realizations of AR(1) series, con¬rming the interpretation of the Hilbert

processes with ±1 = 0.9 and ±1 = 0.3 and their transform as the ˜momentum™ of the input process.

Hilbert transforms (using (16.15) and T = 20). This impression is further substantiated by the

Since AR(1) processes have a red spectrum and no lagged cross-covariance function (Figure 16.3),

16: Complex Eigentechniques

356

end-effects can not be seen because the middle of

AR(1) ± = 0.3 a longer time series is shown.

The ¬lter length T is determined by iteratively

0.3

increasing T until there is little change in the

AR(1) ± = 0.9 estimated transform.

An alternative approach is to re-express the