Time Convolution and Frequency Convolution Properties of Discrete-Time Fourier Transform

Signals and Systems Electronics & Electrical Digital Electronics

Discrete-Time Fourier Transform

The Fourier transform of a discrete-time sequence is known as the discrete-time Fourier transform (DTFT). Mathematically, the discrete-time Fourier transform of a discrete-time sequence $\mathrm{\mathit{x\left ( n \right )}}$ is defined as −

$$\mathrm{\mathit{F\left [ x\left ( n \right ) \right ]=X\left ( \omega \right )=\sum_{n=-\infty }^{\infty }x\left ( n \right )e^{-j\, \omega n}}}$$

Time Convolution Property of DTFT

Statement – The time convolution property of DTFT states that the discretetime Fourier transform of convolution of two sequences in time domain is equivalent to multiplication of their discrete-time Fourier transforms. Therefore, if

$$\mathrm{\mathit{x_{\mathrm{1}}\left ( n \right )\overset{FT}{\leftrightarrow}X_{\mathrm{1}}\left ( \omega \right )\: \: \mathrm{and}\: \: x_{\mathrm{2}}\left ( n \right )\overset{FT}{\leftrightarrow}X_{\mathrm{2}}\left ( \omega \right )}}$$

Then,

$$\mathrm{\mathit{F\left [ x_{\mathrm{1}}\left ( n \right )\ast x_{\mathrm{2}}\left ( n \right ) \right ]\mathrm{\, =\, }X_{\mathrm{1}}\left ( \omega \right )\ast X_{\mathrm{2}}\left ( \omega \right )}}$$

Proof

From the definition of DTFT, we have,

$$\mathrm{\mathit{F\left [ x\left ( n \right ) \right ]\mathrm{\, =\, }\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x\left ( n \right )e^{-j\, \omega n}}}$$

$$\mathrm{\mathit{\therefore F\left [ x_{\mathrm{1}}\left ( n \right )\ast x_{\mathrm{2}}\left ( n \right ) \right ]\mathrm{\, =\, }\sum_{n\mathrm{\, =\, }-\infty }^{\infty }\left [ x_{\mathrm{1}}\left ( n \right )\ast x_{\mathrm{2}}\left ( n \right ) \right ]e^{-j\, \omega n}}}$$

But, the convolution of two sequences is defined as,

$$\mathrm{\mathit{x_{\mathrm{1}}\left ( n \right )\ast x_{\mathrm{2}}\left ( n \right )\mathrm{\, =\, }\sum_{k\mathrm{\, =\, }-\infty }^{\infty }x_{\mathrm{1}}\left ( k \right )\, x_{\mathrm{2}}\left ( n-k \right )}}$$

$$\mathrm{\mathit{\therefore F\left [ x_{\mathrm{1}}\left ( n \right )\ast x_{\mathrm{2}}\left ( n \right ) \right ]\mathrm{\, =\, }\sum_{n\mathrm{\, =\, }-\infty }^{\infty }\left [ \sum_{k\mathrm{\, =\, }-\infty }^{\infty }x_{\mathrm{1}}\left ( k \right )\, x_{\mathrm{2}}\left ( n-k \right ) \right ]e^{-j\, \omega n}}}$$

By interchanging the order of summations, we get,

$$\mathrm{\mathit{F\left [ x_{\mathrm{1}}\left ( n \right )\ast x_{\mathrm{2}}\left ( n \right ) \right ]\mathrm{\, =\, }\sum_{k\mathrm{\, =\, }-\infty }^{\infty }x_{\mathrm{1}}\left ( k \right )\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x_{\mathrm{2}}\left ( n-k \right )e^{-j\, \omega n}}}$$

Substituting $\mathrm{\mathit{\left ( n-k \right )\mathrm{\, =\, }m}} $ and $\mathrm{\mathit{n\mathrm{\, =\, }\left ( m\mathrm{\, +\, }k \right )}}$ in the second summation, we have,

$$\mathrm{\mathit{F\left [ x_{\mathrm{1}}\left ( n \right )\ast x_{\mathrm{2}}\left ( n \right ) \right ]\mathrm{\, =\, }\sum_{k\mathrm{\, =\, }-\infty }^{\infty }x_{\mathrm{1}}\left ( k \right )\sum_{m\mathrm{\, =\, }-\infty }^{\infty }x_{\mathrm{2}}\left ( m \right )e^{-j\, \omega \left ( m\mathrm{\, +\, }k \right )}}}$$

$$\mathrm{\mathit{\Rightarrow F\left [ x_{\mathrm{1}}\left ( n \right )\ast x_{\mathrm{2}}\left ( n \right ) \right ]\mathrm{\, =\, }\sum_{k\mathrm{\, =\, }-\infty }^{\infty }x_{\mathrm{1}}\left ( k \right )e^{-j\, \omega k}\sum_{m\mathrm{\, =\, }-\infty }^{\infty }x_{\mathrm{2}}\left ( m \right )e^{-j\, \omega m}}}$$

$$\mathrm{\mathit{\therefore F\left [ x_{\mathrm{1}}\left ( n \right )\ast x_{\mathrm{2}}\left ( n \right ) \right ]\mathrm{\, =\, }X_{\mathrm{1}}\left ( \omega \right )X_{\mathrm{2}}\left ( \omega \right )}}$$

Therefore, the convolution of sequences in time domain is equal to the product of their spectra in the frequency domain.

Frequency Convolution Property of DTFT

Statement – The frequency convolution property of DTFT states that the discrete-time Fourier transform of multiplication of two sequences in time domain is equivalent to convolution of their spectra in frequency domain. Therefore, if

Then,

$$\mathrm{\mathit{F\left [ x_{\mathrm{1}}\left ( n \right ) x_{\mathrm{2}}\left ( n \right ) \right ]\mathrm{\, =\, }X_{\mathrm{1}}\left ( \omega \right )\ast X_{\mathrm{2}}\left ( \omega \right )}}$$

Proof

From the definition of DTFT, we have,

$$\mathrm{\mathit{F\left [ x\left ( n \right ) \right ]\mathrm{\, =\, }X\left ( \omega \right )\mathrm{=}\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x\left ( n \right )e^{-j\, \omega n}}}$$

$$\mathrm{\mathit{\therefore F\left [ x_{\mathrm{1}}\left ( n \right ) x_{\mathrm{2}}\left ( n \right ) \right ]\mathrm{\, =\, }\sum_{n\mathrm{\, =\, }-\infty }^{\infty }\left [ x_{\mathrm{1}}\left ( n \right )x_{\mathrm{2}}\left ( n \right ) \right ]e^{-j\, \omega n}}}$$

But, from the definition of inverse DTFT, we have,

$$\mathrm{\mathit{x_{\mathrm{1}}\left ( n \right )\mathrm{\, =\, }\frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\pi }^{\pi }X_{\mathrm{1}}\left ( \theta \right )e^{j\, \theta n}d\theta }}$$

$$\mathrm{\mathit{\therefore F\left [ x_{\mathrm{1}}\left ( n \right ) x_{\mathrm{2}}\left ( n \right ) \right ]\mathrm{\, =\, }\sum_{n\mathrm{\, =\, }-\infty }^{\infty }\left [ \frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\pi }^{\pi }X_{\mathrm{1}}\left ( \theta \right )e^{j\, \theta n}d\theta \right ]e^{-j\, \omega n }x_{\mathrm{2}}\left ( n \right )}}$$

Now, by interchanging the order of summation and integration, we get,

$$\mathrm{\mathit{F\left [ x_{\mathrm{1}}\left ( n \right ) x_{\mathrm{2}}\left ( n \right ) \right ]\mathrm{\, =\, }\frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\pi }^{\pi }X_{\mathrm{1}}\left ( \theta \right ) \left [\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x_{\mathrm{2}}\left ( n \right ) e^{-j \left ( \omega-\theta \right ) n } \right ]d\theta }}$$

$$\mathrm{\mathit{\mathrm{\, =\, } \frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\pi }^{\pi }X_{\mathrm{1}}\left ( \theta \right )X_{\mathrm{2}} \left ( \omega-\theta \right ) d\theta }} $$

$$\mathrm{\mathit{\therefore F\left [ x_{\mathrm{1}}\left ( n \right ) x_{\mathrm{2}}\left ( n \right ) \right ]\mathrm{\, =\, }X_{\mathrm{1}}\left ( \omega \right )\ast X_{\mathrm{2}}\left ( \omega \right ) }}$$

Manish Kumar Saini

Updated on: 29-Jan-2022

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