Multiplication or Modulation Property of Continuous-Time Fourier Series

Fourier Series

If $x(t)$ is a periodic function with period $T$, then the continuous-time exponential Fourier series of the function is defined as,

$$\mathrm{x(t)\:=\:\sum_{n= -\infty}^{\infty}C_{n}\:e^{jn\omega_{0} t}\:\:\dotso\: (1)}$$

Where, $C_{n}$ is the exponential Fourier series coefficient, which is given by,

$$\mathrm{C_{n}\:=\:\frac{1}{T}\int_{t_{0}}^{t_{0} \:+\: T}x(t)e^{-jn\omega_{0} t}dt\:\:\dotso\: (2)}$$

Modulation or Multiplication Property

Let $x_{1}(t)$ and $x_{2}(t)$ two periodic signals with time period T and with Fourier series coefficient $C_{n}$ and $D_{n}$. If

$$\mathrm{x_{1}(t)\overset{FS}{\leftrightarrow}C_{n}}$$

$$\mathrm{x_{2}(t)\overset{FS}{\leftrightarrow}D_{n}}$$

Then, the modulation or multiplication property of continuous time Fourier series states, that

$$\mathrm{x_{1}(t)\:\cdot\: x_{2}(t)\overset{FS}{\leftrightarrow}\sum_{k \:=\: -\infty}^{\infty}C_{k}\:D_{n-k}}$$

Proof

From the definition of continuous time Fourier series, we get,

$$\mathrm{FS[x_{1}(t)\:\cdot \:x_{2}(t)] \:=\: \frac{1}{T}\int_{t_{0}}^{t_{0} \:+\: T}[x_{1}(t)\:\cdot\: x_{2}(t)] e^{-jn\omega_{0} t}dt}$$

$$\mathrm{\Rightarrow\:FS[x_{1}(t)\:\cdot\: x_{2}(t)] \:=\: \frac{1}{T}\int_{t_{0}}^{t_{0} \:+\: T}x_{1}(t)\left (\sum_{k= -\infty}^{\infty} C_{k} e^{jk\omega_{0} t}\right )e^{-jn\omega_{0} t}dt}$$

$$\mathrm{\Rightarrow\:FS[x_{1}(t)\:\cdot\: x_{2}(t)] \:=\: \frac{1}{T}\int_{t_{0}}^{t_{0} \:+\: T}x_{1}(t)\left (\sum_{k=-\infty}^{\infty} C_{k} e^{-j(n\:-\:k)\omega_{0} t}\right )e^{-jn\omega_{0} t}dt\:\:\dotso\: (3)}$$

By rearranging the order of integration and summation in equation (3), we obtain,

$$\mathrm{FS[x_{1}(t)\:\cdot\: x_{2}(t)]\:=\:\sum_{k= -\infty}^{\infty} C_{k}\left(\frac{1}{T} \int_{t_{0}}^{t_{0}\:+\:T} x_{1}(t)e^{-j(n-k)\omega_{0} t}\:dt\right ) \:=\: \sum_{k= -\infty}^{\infty}C_{k}D_{n-k}}$$

Where,

$$\mathrm{D_{n-k} \:=\: \frac{1}{T}\int_{t_{0}}^{t_{0} \:+\: T}x_{1}(t)e^{-j(n-k)\omega_{0} t}\:dt}$$

Therefore,

$$\mathrm{x_{1}(t)\:\cdot\: x_{2}(t)\overset{FS}{\leftrightarrow}\sum_{k= -\infty}^{\infty} C_{k}D_{n-k} \:\:\: (Hence, \: Proved)}$$