Solution Manual for Spectral Analysis
for Univariate Time Series 2nd
edition Donald B. Percival, Andrew T.
Walden
Solution Manual for Spectral Analysis
for Univariate Time Series 2nd
edition Donald B. Percival, Andrew T.
Walden
,Solution Manul for Spectral Analysis for Univariate Time Series 2nd edition Donald B. Percival, Andrew T.
Walden
Spectral Analysis
for
Univariate Time Series:
Solutions Guide (Version 1.1)
Donald B. Percival
University of Washington
Andrew T. Walden
Imperial College London
Solution Manul for Spectral Analysis for Univariate Time Series 2nd edition Donald B. Percival, Andrew T.
Walden
,Solution Manual for Spectral Analysis for Univariate Time Series 2nd edition Donald B. Percival, Andrew T.
Walden
Preface to Solutions Guide
This document gives full solutions to the 244 exercises posed at the ends of Chapters
1 to 11 of the book Spectral Analysis for Univariate Time Series. The reader should note
the following conventions.
• Exercise [2.22] refers to an exercise stated at the end of Chapter 2, while Exer-
cise [37] refers to an exercise on page 37 of the book (the solution to the latter is
in an Appendix that is available on the Web site for the book – see page xx in
the Preface to the book for details about accessing this site).
• Figure 3.3 refers to a figure that is associated with the solution to Exercise [3.3],
whereas Figure 239 refers to a figure in the book on page 239. Figures 1.6a,
1.6b and so forth are multiple figures associated with the solution to Exercise [1.6].
A similar scheme is used for labeling tables: Table 6.21 refers to a table associated
with the solution to Exercise [6.21], while Table 279 refers to a table on page 279
of the book.
• References to Equation (∗) within a particular solution refer to a displayed equa- tion
defined previously within that same solution (likewise for Equations (∗∗), (∗ ∗ ∗),
(†) and (††)).
We would certainly appreciate hearing from instructors who find errors of any sort in
these solutions (or know of a better solution to a particular exercise).
Don Percival Andrew Walden
University of Washington Imperial College London
October 11, 2019
ii
Solution Manual for Spectral Analysis for Univariate Time Series 2nd edition Donald B. Percival, Andrew T.
Walden
, Answer to Exercise [1.1] 1
Answer to Exercise [1.1] Since |a — ib|2 = a2 + b2 for real-valued a and b, we have
!
2 ! 2
k Σ
N −1
k Σ
N−1
xt cos (2πfjt) Σ
N −1
x te −i2πfjt
= — i x t sin (2πfjt)
N2 N2
t=0 t=0 t=0
!2 !2
k Σ
N−1 NΣ −1
= 2 xt cos (2πfjt) + xt sin (2πfjt) .
N t=0 t=0
We obtain Equation (13a) by letting k = 2 in the above. For even N , Equation (13a)
follows by letting k = 1 because in that case fj = fN/2 = 1/2 and hence
N −1 N −1
Σ Σ
xt sin (2πfjt) = xt sin (πt) = 0.
t=0 t=0
Solution Manul for Spectral Analysis for Univariate Time Series 2nd edition Donald B. Percival, Andrew T.
Walden
for Univariate Time Series 2nd
edition Donald B. Percival, Andrew T.
Walden
Solution Manual for Spectral Analysis
for Univariate Time Series 2nd
edition Donald B. Percival, Andrew T.
Walden
,Solution Manul for Spectral Analysis for Univariate Time Series 2nd edition Donald B. Percival, Andrew T.
Walden
Spectral Analysis
for
Univariate Time Series:
Solutions Guide (Version 1.1)
Donald B. Percival
University of Washington
Andrew T. Walden
Imperial College London
Solution Manul for Spectral Analysis for Univariate Time Series 2nd edition Donald B. Percival, Andrew T.
Walden
,Solution Manual for Spectral Analysis for Univariate Time Series 2nd edition Donald B. Percival, Andrew T.
Walden
Preface to Solutions Guide
This document gives full solutions to the 244 exercises posed at the ends of Chapters
1 to 11 of the book Spectral Analysis for Univariate Time Series. The reader should note
the following conventions.
• Exercise [2.22] refers to an exercise stated at the end of Chapter 2, while Exer-
cise [37] refers to an exercise on page 37 of the book (the solution to the latter is
in an Appendix that is available on the Web site for the book – see page xx in
the Preface to the book for details about accessing this site).
• Figure 3.3 refers to a figure that is associated with the solution to Exercise [3.3],
whereas Figure 239 refers to a figure in the book on page 239. Figures 1.6a,
1.6b and so forth are multiple figures associated with the solution to Exercise [1.6].
A similar scheme is used for labeling tables: Table 6.21 refers to a table associated
with the solution to Exercise [6.21], while Table 279 refers to a table on page 279
of the book.
• References to Equation (∗) within a particular solution refer to a displayed equa- tion
defined previously within that same solution (likewise for Equations (∗∗), (∗ ∗ ∗),
(†) and (††)).
We would certainly appreciate hearing from instructors who find errors of any sort in
these solutions (or know of a better solution to a particular exercise).
Don Percival Andrew Walden
University of Washington Imperial College London
October 11, 2019
ii
Solution Manual for Spectral Analysis for Univariate Time Series 2nd edition Donald B. Percival, Andrew T.
Walden
, Answer to Exercise [1.1] 1
Answer to Exercise [1.1] Since |a — ib|2 = a2 + b2 for real-valued a and b, we have
!
2 ! 2
k Σ
N −1
k Σ
N−1
xt cos (2πfjt) Σ
N −1
x te −i2πfjt
= — i x t sin (2πfjt)
N2 N2
t=0 t=0 t=0
!2 !2
k Σ
N−1 NΣ −1
= 2 xt cos (2πfjt) + xt sin (2πfjt) .
N t=0 t=0
We obtain Equation (13a) by letting k = 2 in the above. For even N , Equation (13a)
follows by letting k = 1 because in that case fj = fN/2 = 1/2 and hence
N −1 N −1
Σ Σ
xt sin (2πfjt) = xt sin (πt) = 0.
t=0 t=0
Solution Manul for Spectral Analysis for Univariate Time Series 2nd edition Donald B. Percival, Andrew T.
Walden
