1. Chapter 2 – Linear Filters
Setting:
Filter
(Input)
(Output)
NOTE: Some slides have blank sections. They are based on a teaching style in which the corresponding blank (derivations, theorem proofs, examples, …) are worked out in class on the board or overhead projector.

2. Linear Filter
Most time series we will consider can be viewed as output from a linear filter

3. Notes: 1.
- is called the impulse response function 2.
- frequency response function
3. Most filters we study are causal (realizable), i.e. the present output depends on present and past inputs
(not the future)

4. Question: What do we mean by
Mean Square Convergence
Random Variables
Linear Filter Notation

5. Note: We denote the process X t by
Theorem 2.1: Suppose
Note: We denote the process X t by
Proof: Appendix 2.A

6. Proof: Appendix 2.A

7. General Linear Process (GLP)
- a causal linear filter with white noise input

8. GLP in backward shift (operator) notation
Algebraic counterpart

9. Theorem 2.3
For sum to make sense

11. Theorem 2.2 for the case of a GLP
spectrum of the output
squared modulus of frequency response function
spectrum of the input
Spectral Density of GLP

12. Wold Decomposition Theorem
- emphasizes the central role that GLP’s play in the study of (weakly) stationary processes
Let { X t ; t = 0,  1, 2, … } be a (weakly) stationary time series with zero mean. Then X t can be written as the sum of two processes: X t = U t + V t where
(i) U t is a GLP with y 0 = 1
(nondeterministic component)
(ii) V t is completely determined by a linear function of its past values
(deterministic component)

13. Filter Recall: We have looked at linear filters Filter (Output)
(Input)
We have looked at linear filters

14. Filtering Applications
Will use Theorem 2.2 to design filters:
where 
of the output and input, respectively 

15. Question: What will be the effect of differencing the data?
Realization
Question: What will be the effect of differencing the data?
Differenced Data

16. Types of Filters Low-pass filters: High-pass filters:
Filters are sometimes used to “filter out” certain frequencies from a set of data:
Types of Filters
Low-pass filters:
- “pass” low frequency behavior and “filter out” higher frequency behavior
High-pass filters:
- “pass” high frequency behavior and “filter out” lower frequency behavior

17. Types of Filters - continued
Band-pass filters:
- “pass” frequencies in a certain “frequency band”
Band-stop (notch) filters:
- “pass” frequencies except those in a certain “frequency band”

18. Example:
- Suppose the goal of the filtering is to keep frequencies greater than .3 and remove frequencies less than .3. (high pass filter)
- Theorem 2.2 says that ideally we would design our filter so that
1 -
.5 -

19. Squared Frequency Response Functions
Ideal Low-Pass Filter
Ideal Band-Pass Filter

20. Filter Examples:
1. Difference

21. Notes: ● A first difference is a “high pass” filter
● It allows some low frequency behavior to leak through

22. Realization
Differenced Data

23. 2. Sum

24. Recall - Figure 1.21
Apply sum filter

25. 2. Sum data(fig1.21) x=fig1.21 n=length(x) y=rep(0,n) for (i in 2:n) {
y[i-1]=x[i]+x[i-1] }
plotts.wge(y)

26. 3. Moving Average Filters
I will consider three:

27. Frequency Response Functions
2-point moving average
3-point moving average
7-point moving average

28. Notes: ● these are low pass filters (smoothers)
● as window-length increases, the “cutoff” frequency becomes smaller

29. Realization
2-point moving average
3-point moving average
7-point moving average

30. Butterworth Filters
A low-pass Butterworth filter has frequency response function with the property
where
fc is the cut-off frequency
N is the order of the Butterworth filter

32. Notes on the Butterworth Filter:
● for very large values of N,
● although the impulse response function associated with H( f ) is of infinite order, the impulse response function of an Nth order Butterworth filter can be well approximated by a ratio of two Nth order polynomials resulting in the recursive filter

34. N=1
N=4

35. tswge demo Recall - Figure 1.21 To apply a Butterworth filter use
butterworth.wge=function(x,order=k,type,cutoff,plot=TRUE)
data(fig1.21a)
butterworth.wge(fig1.21a,order=4,type='low',cutoff=.2)
butterworth.wge(fig1.21a,order=4,type='low',cutoff=.32)
butterworth.wge(fig1.21a,order=4,type='high',cutoff=.2)
butterworth.wge(fig1.21a,order=4,type='high',cutoff=.05)

36. Filtered KK Eats (band stop)
King Kong Eats Grass
Filtered KK Eats (band stop)
Low-Pass Filtered Data
High-Pass Filtered Data