Linearity, Periodicity and Symmetry Properties of Discrete-Time Fourier Transform

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 −

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

Linearity Property of Discrete-Time Fourier Transform

Statement – The linearity property of discrete-time Fourier transform states that, the DTFT of a weighted sum of two discrete-time sequences is equal to the weighted sum of individual discrete-time Fourier transforms. Therefore, if

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

Then,

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

Proof

From the definition of discrete-time Fourier transform, we have,

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

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

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

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

Periodicity Property of Discrete-Time Fourier Transform

The periodicity property of discrete-time Fourier transform states that the DTFT X(𝜔) is periodic in 𝜔 with period 2π, that is

$$\mathrm{\mathit{X\left ( \omega \right )\mathrm{\, =\, }X\left ( \omega \mathrm{\, +\, }\mathrm{2}n\pi \right )}}$$

Therefore, using the periodicity property of DTFT, we need only one period of X(𝜔) for the analysis and not the whole range −∞ < 𝜔 < ∞.

Symmetry Property of Discrete-Time Fourier Transform

The discrete-time Fourier transform (DTFT) X(𝜔) is a complex function of 𝜔 and hence can be expressed as −

$$\mathrm{\mathit{X\left ( \omega \right )\mathrm{\, =\, }X_{r}\left ( \omega \right )\mathrm{\, +\, }j\, X_{i}\left ( \omega \right )}}$$

Where,

• $\mathrm{\mathit{X_{r}\left ( \omega \right )}}$ is the real part of $\mathrm{\mathit{X\left ( \omega \right )}}$, and

• $\mathrm{\mathit{X_{i}\left ( \omega \right )}}$ is the imaginary part of $\mathrm{\mathit{X\left ( \omega \right )}}$.

Now, from the definition of the DTFT, we have,

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

$$\mathrm{\mathit{\Rightarrow X\left ( \omega \right )\mathrm{\, =\, }\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x\left ( n \right )\mathrm{cos}\, \omega n-j\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x\left ( n \right )\mathrm{sin}\, \omega n}}$$

$$\mathrm{\mathit{\Rightarrow X_{r}\left ( \omega \right )\mathrm{\, +\, }j\, X_{i}\left ( \omega \right )\mathrm{\, =\, }\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x\left ( n \right )\mathrm{cos}\, \omega n-j\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x\left ( n \right )\mathrm{sin}\, \omega n}}$$

On comparing LHS and RHS, we get,

$$\mathrm{\mathit{ X_{r}\left ( \omega \right ) \mathrm{\, =\, }\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x\left ( n \right )\mathrm{cos}\, \omega n}}$$

And,

$$\mathrm{\mathit{X_{i}\left ( \omega \right )\mathrm{\, =\, }-\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x\left ( n \right )\mathrm{sin}\, \omega n}} $$

$$\mathrm{\mathit{\because \mathrm{cos}\left ( -\omega \right )n\mathrm{\, =\, }\mathrm{cos}\, \omega n\: \: \mathrm{and}\: \: \mathrm{sin}\left ( -\omega \right )n\mathrm{\, =\, }-\mathrm{sin}\, \omega n }}$$

$$\mathrm{\mathit{\therefore X_{r}\left ( -\omega \right ) \mathrm{\, =\, }\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x\left ( n \right )\mathrm{cos}\, \left ( -\omega \right ) n\mathrm{\, =\, }\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x\left ( n \right )\mathrm{cos}\, \omega n}}$$

$$\mathrm{\mathit{\Rightarrow X_{r}\left ( -\omega \right ) \mathrm{\, =\, }X_{r}\left ( \omega \right )}}$$

i.e., the real part of DTFT $\mathrm{\mathit{X_{r}\left ( \omega \right )}}$ is an even function of 𝜔, i.e., it has even symmetry property.

Also,

$$\mathrm{\mathit{X_{i}\left ( -\omega \right )\mathrm{\, =\, }-\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x\left ( n \right )\mathrm{sin}\left ( -\omega \right ) n\mathrm{\, =\, }\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x\left ( n \right )\mathrm{sin}\,\omega n}}$$

$$\mathrm{\mathit{\therefore X_{i}\left ( -\omega \right )\mathrm{\, =\, }-X_{i}\left ( \omega \right )}}$$

Therefore, the imaginary part of DTFT $\mathrm{\mathit{X_{i}\left (\omega \right )}}$ is an odd function of 𝜔, i.e., it has odd symmetry property.