Auto correlation of a cosine signal

Question

I have to find the autocorrelation of $ x(t) = A \cos ( 2\pi f_0 t) $ , and I know from theory that I should calculate $$ \lim_{x \rightarrow \infty } \frac{A^{2}}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} \cos (2 \pi f_0 t ) \cos(2\pi f_0 t + 2\pi f_0 \tau ) dt $$ this because Cosine is completely real. Now I don’t know how to solve this integral, I tried with trigonometric formulas but I never arrived at the correct result that should be $ \frac{A^{2}}{2} \cos(2 \pi f_0 \tau ) $ I found some sites that do the same thing but my book used this formula, with the limit

Answer 1

There a few formulas for calculating autocorrelation depending on signal feature (deterministic or probabilistic, periodic, discrete etc.). As you can find here correct formula for "periodic" signal is following: $$R_{xx}\left(\tau \right)=\frac{1}{T}\int _{t_0}^{t_0+T}f\left(t\right)f\left(t+\tau \right)dt\:$$ From there you can basically calculate autocorrolation as you said by using $$\cos \left(\alpha +\beta \right)=\cos \left(\alpha \right)\cos \left(\beta \right)-\sin \left(\alpha \right)\sin \left(\beta \right)$$ And don't forget that $T=\frac{1}{f_0}$