Consider a continuous time periodic signal x(t) with fundamental period T and Fourier series coefficient X[k]. What is the Fourier series coefficient of the signal y(t) = x(t – t₀) + x(t + t₀) ?

x(t – t₀) will also have time period T.

Let Fourier series coefficient of x(t – t₀) be x₁(k)

\( {X_1}\left[ k \right] = \frac{1}{T}\mathop \smallint \limits_T x\left( {t - {t_0}} \right){e^{ - jk{\omega _o}t}}dt\)

\(= \frac{{{e^{ - je\omega {t_0}}}}}{T}\mathop \smallint \nolimits^ x\left( τ \right){e^{ - jk{\omega _0}τ }}dτ \)

Taking t - t₀ = τ, we get:

\( = {e^{-{\rm{jk\omega }}{{\rm{t}}_0}}}X\left[ k \right] \)

Now, if the Fourier series coefficient of x(t + t₀) be X2 [k], we can write:

\({X_2}\left[ k \right] = {e^{jk\omega {t_0}}}X\left[ k \right]\)

Now X1 [k] + X2 [k]

\(= \left( {{e^{ - jk\omega {t_0}}} + {e^{ - jk\omega {t_0}}}} \right)X\left[ k \right]\)

\(= 2\cos k\omega {t_0}X\left[ k \right]\)

\(= 2\cos \left( {\frac{{2\pi }}{T}k{t_0}} \right)X\left[ k \right]\)