My textbook says:
Taking the complex conjugate of a periodical signal x(t) has the effect of complex conjugation and time reversal on the corresponding Fourier series coefficients.
I understood the complex conjugation effect however I don't get why this operation also has the effect of time reversal.
Thanks in advance.
Given the Fourier series coefficients
$$c_n = \frac{1}{2\pi}\int_{-\pi}^{\pi}x(t) e^{-itn}dt$$
we see that the coefficients of the conjugated signal are
$$c_n' = \frac{1}{2\pi}\int_{-\pi}^{\pi}x^*(t) e^{-itn}dt = \left\{\frac{1}{2\pi} \int_{-\pi}^{\pi}x(t) e^{itn}dt\right\}^* = c_{-n}^*$$
I think this means that $\widehat{\overline{x}}(m) = \overline{\widehat{x}(-m)}$. How to see this... Write $x(t) = y(t) + i z(t)$, where $y$ and $z$ are real-valued functions. The Fourier coefficients of $\overline{x}$ are $$ \begin{align*} \widehat{\overline{x}}(m) &= \int_0^1 \overline{x}(t) e^{-2\pi i t \cdot m} dt = \int_0^1 y(t) e^{-2\pi i t \cdot m} dt - i\int_0^1 z(t) e^{-2\pi i t \cdot m} dt\\ &= -\int_0^1 y(-t) e^{2\pi i t \cdot m} dt + i \int_0^1 z(-t) e^{2\pi i t \cdot m} dt\\ \end{align*} $$ by the change of variables $t \to -t$. Also, notice that the Fourier coefficients $\widehat{x}(-m)$ are $$ \begin{align*} \widehat{x}(-m) &= \int_0^1 x(t) e^{- 2\pi i m \cdot t} dt = \int_0^1 y(t) e^{-2\pi i t \cdot m} dt + i\int_0^1 z(t) e^{-2\pi i t \cdot m} dt\\ &= -\int_0^1 y(-t) e^{2\pi i t \cdot m} dt - i \int_0^1 z(-t) e^{2\pi i t \cdot m} dt \end{align*} $$ which is the conjugate of the expression obtained above.
