ensure that the ¬lter does not attenuate variability

applying ¬lters A and B in sequence:

at its point of peak response (Figure 17.13a). This

[{A · B}(X)]t = [A(B(X))]t . ¬lter has weights

±1

The ¬lter weights pk of the product P = A · B are

2 for k = 0

given by the convolution 0 for |k| = 1

(17.48) bk = 1

pk = al bk’l

’ 4 for |k| = 2

l

0 otherwise.

and the response function is the product of the

response functions of A and B:

cAB (ω) = cA (ω)cB (ω). (17.49)

17.5.5 Some Filters That Discriminate Between

Time Scales. The simple ¬lters discussed up

Thus convolution in the time domain is equal to

to this point do not have particularly desirable

multiplication in the frequency domain (and vice

properties. The (1-2-1)-¬lter and its relatives ˜cut

versa).

off™ slowly by gradually changing the attenuation

Since a ¬lter is uniquely determined by its

of variance with frequency. The running-mean

response function (17.49) proves that the sequence

¬lter also cuts off slowly, but it also has large

of the application of the two ¬lters is irrelevant.

sidelobes that allow variance leakage from high

That is,

frequencies.

A · B = B · A. (17.50) In contrast, the ideal low-pass ¬lter has a box-

car shaped frequency response function that cuts

off sharply at a prescribed cut-off frequency (see

17.5.4 Further Simple Filters. The results Figure 17.14). Unfortunately, the ideal digital ¬lter

of the preceding subsection may be used to can not actually be used because it has in¬nitely

construct other simple ¬lters from the (1-2-1)-¬lter many nonzero weights. The ideal low-pass ¬lter

has frequency response function c(ω) = 1 for

([17.5.2]).

|ω| ¤ ω0 and c(ω) = 0 elsewhere. It has

The plain (1-2-1)-¬lter may be applied re-

weights a0 = 2ω0 and ak = π|k| sin(2π |k|ω0 )

1

peatedly to suppress the high-frequency variabil-

for |k| > 0. Simply truncating the weights at

ity more ef¬ciently. For example, the response

17: Speci¬c Statistical Concepts

388

Blackmon™s ¬lters Filters for daily data Wallace et al.™s ¬lters

¤ 10 days ¤ 5 days

low-pass band-pass high-pass low-pass band-pass

0 0.09747 0.27769 0.47626 0.21196 0.45221 0.82119 0.66850

’0.31860 ’0.07287 ’0.16871 ’0.27390

1 0.09547 0.14335 0.19744

’0.10201 ’0.28851 ’0.13432

2 0.08963 0.01975 0.15769 -0.14062

’0.19477 ’0.10059

3 0.08049 0.10098 0.10288 0.09733 0.00000

’0.09233 ’0.01860 ’0.05682

4 0.06883 0.04625 0.03951 0.06204

’0.05468 ’0.01752

5 0.05564 0.02830 0.0 0.02833 0.04669

’0.02820

6 0.04196 0.04193 0.01678 0.03316 0.01118 0.00000

’0.03684 ’0.07089 ’0.02810

7 0.02882 0.00335 0.03331 0.02646

’0.01445 ’0.03003 ’0.00227 ’0.02194

8 0.01707 0.00411 0.02906

’0.02073 ’0.01518

9 0.00734 0.03281 0.00302 0.02250 0.00000

’0.0

10 0.0 0.03043 0.01179 0.00708 0.01163 0.02193

’0.00488 ’0.00200

11 0.01257 ” ” 0.00101 0.00965

’0.00748 ’0.01917 ’0.00900 ’0.00624

12 ” ” 0.00000

’0.00818 ’0.00967 ’0.00715 ’0.00903 ’0.00461

13 ” ”

’0.00749 ’0.00013 ’0.00801 ’0.00274

14 0.00627 ” ”

’0.00596 ’0.00491

15 -0.00304 0.00362 ” ” 0.00000

Table 17.1: Digital ¬lters designed to extract variability on speci¬c time scales. The ¬lters are symmetric

(i.e., ak = a’k ).

Blackmon™s [47] ¬lters for separating low-frequency, baroclinic, and high-frequency time scales from

12-hourly data are listed in columns 2 to 4. The response functions are shown in Figure 17.15.

Two additional ¬lters are listed in columns 5 and 6.

Wallace et al.™s [410] high-pass ¬lters are listed in columns 7 and 8. The response functions are shown

in Figure 17.17.

a ¬xed lag K does not yield a particularly ideal

¬lter. As illustrated in Figure 17.14, this results

in a ¬lter with frequency response function that

0.8

Response

has large Gibbsian overshoots and side lobes.18

The solution to this problem is to strive for

0.4

a frequency response function shape that does

not cut off as abruptly as the ˜ideal™ ¬lter.

In this way, excellent digital ¬lters can be

0.0

constructed by carefully selecting a ¬nite number

of weights.

-0.4 -0.2 0.0 0.2 0.4

A set of three such ¬lters designed by Blackmon

Frequency

[47] for the twice-daily data that are often used

in atmospheric science. The low-pass ¬lter in this

set suppresses variability related to day-to-day Figure 17.14: The frequency response function of

weather events and keeps variability on time scales the ideal low-pass ¬lter (dashed curve) and the

of weeks (such as ˜blocking™ events). The band- ¬lter obtained by truncating the weights of the

pass ¬lter extracts variability in the baroclinic ideal ¬lter at K = 7.

time scale (approximately 2.25 to 5 days), and

the high-pass ¬lter retains only variability on the

one- to two-day time scale (see below for the exact

Blackmon™s ¬lters have 2K + 1 = 31 weights,

de¬nition).19

which are listed in Table 17.1. The response

functions are plotted in Figure 17.15.

18 This problem is similar to the one that motivates the use of

Another pair of ¬lters designed to retain low-

data tapers. (cf. [12.3.8]).

19 These ¬lters are applied in [3.1.6]. frequency and baroclinic variability in daily data

17.5: Time Filters 389