Time Shifting, Time Reversal, and Time Scaling Properties 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,

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

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

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

Time Shifting Property of Fourier Series

Let $x(t)$ is a periodic function with time period $T$ and with Fourier series coefficient $C_{n}$. Then, if

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

Then, the time shifting property of continuous-time Fourier series states that

$$\mathrm{x(t-t_{0})\overset{FS}{\leftrightarrow}e^{-jn\omega_{0} t_{0}}C_{n}}$$

Proof

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

$$\mathrm{x(t)=\sum_{n=−\infty}^{\infty}C_{n}e^{jn\omega_{0} t}…(3)}$$

Replacing $t$ by $(t− t_{0})$ in equation (3), we have,

$$\mathrm{x(t− t_{0})=\sum_{n=−\infty}^{\infty}C_{n}e^{jn\omega_{0}(t− t_{0})}}$$

$$\mathrm{\Rightarrow\:x(t− t_{0})=\sum_{n=−\infty}^{\infty}(C_{n}e^{-jn\omega_{0}t_{0}})e^{jn\omega_{0}t}… (4)}$$

$$\mathrm{∵\:\sum_{n=−\infty}^{\infty}(C_{n}e^{-jn\omega_{0}t_{0}})e^{jn\omega_{0}t}=FS^{-1}[C_{n}e^{-jn\omega_{0}t_{0}}]… (5)}$$

From equations (4) & (5), we get,

$$\mathrm{x(t− t_{0})\overset{FT}{\leftrightarrow}e^{-jn\omega_{0}t_{0}}C_{n}\:\:(Hence,\:\:Proved)}$$

Time Reversal Property of Fourier Series

Let $x(t)$ is a periodic function with time period $T$ and with Fourier series coefficient $C_{n}$. Then, if

$$\mathrm{x(t)\overset{FT}{\leftrightarrow}C_{n}}$$

Then, the time reversal property of continuous-time Fourier series states that

$$\mathrm{x(-t)\overset{FT}{\leftrightarrow}C_{-n}}$$

Proof

From the definition of continuous-time Fourier series, we have,

$$\mathrm{x(t)=\sum_{n=−\infty}^{\infty}C_{n}e^{jn\omega_{0} t}… (6)}$$

Replacing $t$ by $(−t)$ in equation (6), we get,

$$\mathrm{x(-t)=\sum_{n=−\infty}^{\infty}C_{n}e^{jn\omega_{0} (-t)}… (7)}$$

Substituting $(n = −k)$ in the RHS of equation (7), we have,

$$\mathrm{x(-t)=\sum_{k=−\infty}^{-\infty}C_{-k}e^{j(-k)\omega_{0}(-t)}}$$

$$\mathrm{\Rightarrow\:x(-t)=\sum_{k=−\infty}^{\infty}C_{-k}e^{jk\omega_{0}t}… (8)}$$

Now, by substituting $(k= n)$ in equation (8), we get,

$$\mathrm{x(-t)=\sum_{n=−\infty}^{\infty}C_{-k}e^{jn\omega_{0}t}=FS^{-1}[C_{-n}]}$$

$$\mathrm{\therefore\:x(-t)\overset{FT}{\leftrightarrow}C_{-n}\:\:(Hence\:proved)}$$

Time Scaling Property of Fourier Series

Let $x(t)$ is a periodic function with time period $T$ and with Fourier series coefficient $C_{n}$. Then, if

$$\mathrm{x(t)\overset{FT}{\leftrightarrow}C_{n}}$$

Then, the time scaling property of continuous-time Fourier series states that

$$\mathrm{x(at)\overset{FT}{\leftrightarrow}C_{n}\:\:with\:\omega_{0}\rightarrow\:a\omega_{0}}$$

Proof

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

$$\mathrm{x(t)=\sum_{n=−\infty}^{\infty}C_{n}\:e^{jn\omega_{0} t}… (9)}$$

Replacing $t$ by $(at)$ in equation (9), we get,

$$\mathrm{x(at)=\sum_{n=−\infty}^{\infty}C_{n}\:e^{jn\omega_{0} at}}$$

$$\mathrm{\Rightarrow\:x(at)=\sum_{n=−\infty}^{\infty}C_{n}\:e^{jn(a\omega_{0}) t}=FS^{-1}[C_{n}]… (10)}$$

Therefore,

$$\mathrm{x(at)\overset{FT}{\leftrightarrow}C_{n}\:\:with\:\omega \rightarrow a\omega_{0}\:\:(Hence,\:\:proved)}$$