# what are and why are sine and cosine modulated integrals used?

I have found the definition of the following formulas in a paper regarding active vibration control, where they are called sine and cosine modulated integrals.

$$y$$ is measurement signal with a strong periodic component of frequency $$N\Omega$$

$$y^{(i)}_{Nc}=\frac{2}{T}\int^{T}_{0}y^{(i)}(ϕ)\cos(Nϕ)dϕ$$

$$y^{(i)}_{Ns}=\frac{2}{T}\int^{T}_{0}y^{(i)}(ϕ)\sin(Nϕ)dϕ$$

where $$\phi=\Omega t$$.

From these the vector $$y_N^{(i)}$$ is defined as $$y_N^{(i)} = \begin{bmatrix} y_{Nc}^{(1)}\\ y_{Ns}^{(1)}\\\vdots\end{bmatrix}$$

The same is done for the control input(s) $$u$$. Then a quadratic cost function to be minimised at each step is defined using these newly introduced signals in this way:

$$J(k) = y^T_NQy_N+u^T_NRu_N$$

where $$Q$$ and $$R$$ are just two weighing matrices.

They should extract the harmonic component considered but is anybody able to explain them a little bit further?

Here the link for the paper

Another question: suppose I have the value $$y_N$$: how can I invert the relationship to get $$y^{(i)}$$

Here the link for the paper.

• Welcome to MSE! We'll do our best to address your question on a mathematical level; for a more applied perspective, the Signal Processing Stack Exchange may be helpful. – Semiclassical Jul 31 '14 at 14:36
• A few things: 1) The paper you cite is behind a paywall, and so will not be accessible to all readers. So it behooves you to make sure you supply as much context as possible. 2) Your definition of $y_N^{(i)}$ is quite unclear: are the next two elements supposed to be $y^{(2)}_{N,c}$ and $y^{(2)}_{N,s}$? If so, what is the $i$ in $y^{(i)}_N$ supposed to signal? – Semiclassical Aug 3 '14 at 14:44