1. Definitions Periodic Function: f(t +T) = f(t)t, (Period T)(1) Ex: f(t) = A sin(2Πωt + )(2) has period T = (1/ω) and ω is said to be the frequency (angular), A is amplitude, is the phase (2) can also be written as f(t) = C sin(2Πωt) + D cos (2Πωt)(3) with A = C² + D², = tan -1 (D/C)

2. Fourier Series Theorem:If y(t) is periodic function of period 2Π then its behavior is captured by studying it on (-Π, Π) satisfying 1. 2.y(t) has a finite number of discontinuities of the first kind on (-Π, Π), then (4) Where, for j=0,1,2… (5) for j=1,2,… (6) The R.H.S. of (4) is called the Fourier series of y(t)

3. Fourier Transforms (FT) Let y(t) be a continuous function of t (not necessarily periodic ), then the FT of y(t) is defined as (7) The inverse FT of Y( ) is simply y(t) (8) Parsevals Identity: (9) (conservation of energy principle)

4. Discrete Fourier Transforms (DFT) In case y t is a discrete series defined only at discrete time points …-2,-1,0,1,2,… then the DFT of y t is (10) and (11) In applications, the series is usually observed only for N periods say 0,1,…,(N-1) and the finite Fourier transforms is then defined as (12)

5. Spectrum Let X t be a discrete covariance stationary process, with =cor{X t X t+ } the autocorrelation at lag, then spectrum h( ) of X t may be defined as (13) h( ) is periodic (of period 2 ) and hence is usually studied on (-, ) We also have (14) In case the series X t is real we may simplify (13) to (15)

6. Examples 1. If X t is a WNP,(flat spectrum) 2. If X t = t - 1 t-1 (MA(1)) then 3. X t =a X t-1 + t with |a| < 1 then

7. Spectrum Estimation Data: X(1), X(2),…,X(N)in mean deviation form (N is even) Periodogram of X(t) defined as Є (-, )(16) Fourier frequencies p=0,1,…,(N/2) Define I p =I N ( p ) Plot of I p against p (or p ) is called Periodogram

8. Theorem: Є (-, ) Where Modified Periodogram: is an asymptotically unbiased estimator of the spectrum but is inconsistent (variance does not vanish asymptotically)

9. Source of the problem All the autocorrelation possible from s=-(N-1), to (N-1) are included in but for high values of s, is an unreliable estimate of the true K(s). Solution: Sacrifice some of the information available in the later, thereby deliberately introducing bias but reducing unreliability of the estimates.

10. Two Choices Truncation: Ignore all for |s|>S 0 (some fixed no. (N-1)) Tail / Tapering: Give lower weight to for increasing s, through a properly chosen weighting scheme. This leads to the window selection problem, and a window estimate of the spectrum. with, lag window generator

11. Continued.. We get different spectral estimates by choosing different forms of 1. Truncated Periodogram Window: 2. Bartlett Window:

12. Continued.. 3. Daniell Window: 4. Parzen Window:

13. Theorem: Spectral (window ) estimates with lag windowcan be viewed as a weighted average of the modified periodogram in the frequency range with (continuous) weighting function Note: is called as the spectral window generator

14. Comparison of Windows: Bartlett window can lead to negative spectral estimates, a problem not present with the Daniell and Parzen windows. The Parzen window has lower variance but higher bias than the Daniell window. Bandwidth Concepts: Resolvability: Suppose the true spectrum has twin peaks at 1 and 2, then we should like the estimated spectrum also to have peaks at 1 and 2 Good Resolution implies that the two peaks are not merged together and shown as a single intermediate peak.This requires that the width of the spectral window generator should not exceed | 1- 2 |, the distance between the 2 peaks.Thus as an informal rule,we should try to choose as narrow a window as possible.

15. Theorem (17) The window estimates are asymptotically unbiased, consistent and normal provided that 1. 2.

16. Bandwidth Half-power Bandwidth: Parzen Bandwidth: Jenkins Bandwidth: M truncation parameter of window,,

17. Greander Uncertainty Principle: Choice of M: High values of Mlow bias, small bandwidth, high variance Low value of Mlow variance, wide bandwidth, high bias

18. Filter Suppose X(t) and Y(t) are two series s.t. is a sequence of constants Y(t) is filtered series of X(t) is parameter function of the filter

19. Slutskis Theorem: Pre-whitening: Choose or g(u) such that Y(t) is white noise. Fit a sufficiently high order AR model to X(t) and take Y(t) as the residual. Estimate the spectrum of Y(t) by choosing a low value of M, since good resolution is automatic for WNP.