Time Shifting, Time Reversal, and Time Scaling Properties of Continuous-Time Fourier Series

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

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

Time Shifting Property of Fourier Series

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

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

Replacing t by $\mathrm{(t \:-\: t_{0})}$ in equation (3), we have,

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

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

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

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,

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

Replacing t by (-t) in equation (6), we get,

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

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

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

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

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

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

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

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,

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

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

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

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

Therefore,

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