Appendix A
According to equation (8), a closed-form for the L1-norm can be derived
by solving equation (7), which reads
\(\sigma_{L1}\left([x_{i},x_{i+1}]\right)=\frac{\int_{x_{i}}^{x_{i+1}}{\left|Asin(2\pi\omega\ x+\phi)-\frac{\left(x_{i+1}-x\right)Asin(2\pi\omega\ x_{i}+\phi)+\left(x-x_{i}\right)Asin(2\pi\omega\ x_{i+1}+\phi)}{T}\right|\text{dx}}}{T}\)(A1).
Concentrating on the integration in the numerator, there are two
integrations should be completed, since
\(L1\left(\left|\varepsilon\right|\right)=L1\left(\varepsilon\right)\ \text{or}\ L1\left(\left|\varepsilon\right|\right)=L1\left(-\varepsilon\right)\ \)(A2).
In the\(L1\left(\left|\varepsilon\right|\right)=L1\left(\varepsilon\right)\)case the derivation starts with splitting the integrand into separated
terms,
\(\int_{x_{i}}^{x_{i+1}}{\left(Asin(2\pi\omega\ x+\phi)-\frac{\left(x_{i+1}-x\right)Asin(2\pi\omega\ x_{i}+\phi)+\left(x-x_{i}\right)Asin(2\pi\omega\ x_{i+1}+\phi)}{T}\right)\text{dx}}=\int_{x_{i}}^{x_{i+1}}{A\sin\left(2\pi\omega\ x+\phi\right)\text{dx}}-\int_{x_{i}}^{x_{i+1}}{\frac{x_{i+1}\operatorname{Asin}\left(2\pi\omega\ x_{i}+\phi\right)}{T}\text{dx}}+\int_{x_{i}}^{x_{i+1}}{\frac{\operatorname{Asin}\left(2\pi\omega\ x_{i}+\phi\right)}{T}\text{xdx}}-\int_{x_{i}}^{x_{i+1}}{\frac{\operatorname{Asin}\left(2\pi\omega\ x_{i+1}+\phi\right)}{T}\text{xdx}}+\int_{x_{i}}^{x_{i+1}}{\frac{x_{i}\operatorname{Asin}\left(2\pi\omega\ x_{i+1}+\phi\right)}{T}\text{dx}}\)(A3).
By performing the definite integration, and making use of
\(x_{i+1}=x_{i}+\text{T\ }\) (A4),
equation (A3) becomes
\(\left[-\frac{A}{2\pi\omega}\cos\left(2\pi\omega\ x+\phi\right)\right]_{x_{i}}^{x_{i+1}}-\left[\frac{x_{i+1}\operatorname{Asin}\left(2\pi\omega\ x_{i}+\phi\right)}{T}x\right]_{x_{i}}^{x_{i+1}}+\left[\frac{\operatorname{Asin}\left(2\pi\omega\ x_{i}+\phi\right)}{T}\frac{x^{2}}{2}\right]_{x_{i}}^{x_{i+1}}-\left[\frac{\operatorname{Asin}\left(2\pi\omega\ x_{i+1}+\phi\right)}{T}\frac{x^{2}}{2}\right]_{x_{i}}^{x_{i+1}}+\left[\frac{x_{i}\operatorname{Asin}\left(2\pi\omega\ x_{i+1}+\phi\right)}{T}x\right]_{x_{i}}^{x_{i+1}}=-\left(\frac{A}{2\pi\omega}\cos\left(2\pi\omega\ x_{i}+2\pi\omega\ T+\phi\right)-\frac{A}{2\pi\omega}\cos\left(2\pi\omega\ x_{i}+\phi\right)\right)-\left(x_{i}+T\right)\operatorname{Asin}\left(2\pi\omega\ x_{i}+\phi\right)+\operatorname{Asin}\left(2\pi\omega\ x_{i}+\phi\right)\left(x_{i}+\frac{T}{2}\right)-\operatorname{Asin}\left(2\pi\omega\ x_{i}+2\pi\omega\ T+\phi\right)\left(x_{i}+\frac{T}{2}\right)+x_{i}\operatorname{Asin}\left(2\pi\omega\ x_{i}+2\pi\omega\ T+\phi\right)=-\frac{A}{2\pi\omega}\cos\left(2\pi\omega\ x_{i}+2\pi\omega\ T+\phi\right)+\frac{A}{2\pi\omega}\cos\left(2\pi\omega\ x_{i}+\phi\right)-\frac{T}{2}\operatorname{Asin}\left(2\pi\omega\ x_{i}+\phi\right)-\frac{T}{2}\operatorname{Asin}\left(2\pi\omega\ x_{i}+2\pi\omega\ T+\phi\right)\)(A5).
Applying trigonometric identities
\(\cos\left(\alpha+\beta\right)=\cos\alpha\cos\beta-\sin\alpha\sin\text{β\ }\)(A6)
and
\(\sin\left(\alpha+\beta\right)=\sin\alpha\cos\beta+\cos\alpha\sin\text{β\ }\)(A7),
equation (A5) is reformulated as
\(-\frac{A}{2\pi\omega}\left(\cos\left(2\pi\omega\ x_{i}+\phi\right)\cos\left(2\pi\omega\ T\right)-\sin\left(2\pi\omega\ x_{i}+\phi\right)\sin\left(2\pi\omega\ T\right)\right)+\frac{A}{2\pi\omega}\cos\left(2\pi\omega\ x_{i}+\phi\right)-\frac{T}{2}\operatorname{Asin}\left(2\pi\omega\ x_{i}+\phi\right)-\frac{T}{2}A\left(\sin\left(2\pi\omega\ x_{i}+\phi\right)\cos\left(2\pi\omega\ T\right)+\cos\left(2\pi\omega\ x_{i}+\phi\right)\sin\left(2\pi\omega\ T\right)\right)=-\frac{A}{2\pi\omega}\cos\left(2\pi\omega\ T\right)\cos\left(2\pi\omega\ x_{i}+\phi\right)+\frac{A}{2\pi\omega}\sin{\left(2\pi\omega\ T\right)\sin\left(2\pi\omega\ x_{i}+\phi\right)}+\frac{A}{2\pi\omega}\cos\left(2\pi\omega\ x_{i}+\phi\right)-A\frac{T}{2}\sin\left(2\pi\omega\ x_{i}+\phi\right)-A\frac{T}{2}\cos\left(2\pi\omega\ T\right)\sin\left(2\pi\omega\ x_{i}+\phi\right)-A\frac{T}{2}\sin\left(2\pi\omega\ T\right)\cos\left(2\pi\omega\ x_{i}+\phi\right)=\left(-\frac{A}{2\pi\omega}\cos\left(2\pi\omega\ T\right)+\frac{A}{2\pi\omega}-A\frac{T}{2}\sin\left(2\pi\omega\ T\right)\right)\cos\left(2\pi\omega\ x_{i}+\phi\right)+\left(\frac{A}{2\pi\omega}\sin\left(2\pi\omega\ T\right)-A\frac{T}{2}+A\frac{T}{2}\cos\left(2\pi\omega\ T\right)\right)\sin\left(2\pi\omega\ x_{i}+\phi\right)\)(A8).
Inserting (A8) to the numerator of (A1), it becomes
\(\sigma_{L1}\left([x_{i},x_{i+1}]\right)=\left(-\frac{A}{2\pi\omega T}\cos\left(2\pi\omega\ T\right)+\frac{A}{2\pi\omega T}-\frac{A}{2}\sin\left(2\pi\omega\ T\right)\right)\cos\left(2\pi\omega\ x_{i}+\phi\right)+\left(\frac{A}{2\pi\omega T}\sin\left(2\pi\omega\ T\right)-\frac{A}{2}+\frac{A}{2}\cos\left(2\pi\omega\ T\right)\right)\sin\left(2\pi\omega\ x_{i}+\phi\right)\)(A9),
Now let’s see the\(L1\left(\left|\varepsilon\right|\right)=L1\left(-\varepsilon\right)\)case. In this integrand, similarly to (A3) becomes
\(\int_{x_{i}}^{x_{i+1}}{\left(\frac{\left(x_{i+1}-x\right)Asin(2\pi\omega\ x_{i}+\phi)+\left(x-x_{i}\right)Asin(2\pi\omega\ x_{i+1}+\phi)}{T}-Asin(2\pi\omega\ x+\phi)\right)\text{dx}}\)(A10),
so only the sign of the two main terms is exchanged. The derivation
using the steps through (A3) to (A9) are identical apart from the
consequent change of the sign, resulting in
\(\sigma_{L1}\left([x_{i},x_{i+1}]\right)=\left(\frac{A}{2\pi\omega T}\cos\left(2\pi\omega\ T\right)-\frac{A}{2\pi\omega T}+\frac{A}{2}\sin\left(2\pi\omega\ T\right)\right)\cos\left(2\pi\omega\ x_{i}+\phi\right)+\left(-\frac{A}{2\pi\omega T}\sin\left(2\pi\omega\ T\right)+\frac{A}{2}-\frac{A}{2}\cos\left(2\pi\omega\ T\right)\right)\sin\left(2\pi\omega\ x_{i}+\phi\right)\)(A11),
which is the same as (A9) apart from the opposite sign of the
coefficients of the sine and cosine functions of\(2\pi\omega\ x_{i}+\phi\).
Equations (A9) and (A11) provides a solution in a form of
\(\sigma_{L1}\left([x_{i},x_{i+1}]\right)=C\bullet\cos\left(2\pi\omega x_{i}+\phi\right)+S\bullet sin(2\pi\omega x_{i}+\phi)\)(A12).
By generalizing the solutions (A9) and (A11) in the form of (8) instead
of (A12), they are equivalent to equation (8) to (10) of the article.