In this topic, you study the Properties of Discrete Fourier Transform (DFT) as Linearity, Time Shifting, Frequency Shifting, Time Reversal, Conjugation, Multiplication in Time, and Circular Convolution.
Linearity
if
\[a{x_1}[n]{\text{ }} \leftrightarrow a{X_1}[k]{\text{ }}\]
\[b{x_2}[n]{\text{ }} \leftrightarrow b{X_2}[k]{\text{ }}\]
Then
\[a{x_1}[n] + b{x_2}[n] \leftrightarrow a{X_1}[k] + b{X_2}[k]\]
Time Shifting
if
\[{x}[n]{\text{ }} \leftrightarrow {X}[k]{\text{ }}\]
Then
\[x{[n – {n_0}]_{\bmod N}} \leftrightarrow {e^{ – j\,k{n_0}(2\pi /N)}}X(k)\]
Frequency Shifting
if
\[{x}[n]{\text{ }} \leftrightarrow {X}[k]{\text{ }}\]
Then
\[{e^{j\,{k_0}n(2\pi /N)}}x[n] \leftrightarrow X{[k – {k_0}]_{\bmod N}}\]
Time Reversal
if
\[{x}[n]{\text{ }} \leftrightarrow {X}[k]{\text{ }}\]
Then
\[x{[ – {n_0}]_{\bmod N}} \leftrightarrow X{[{k_0}]_{\bmod N}}\]
Conjugation
if
\[{x}[n]{\text{ }} \leftrightarrow {X}[k]{\text{ }}\]
Then
\[{x^*}[n] \leftrightarrow {X^*}{[ – {k_0}]_{\bmod N}}\]
where * denotes the complex conjugate.
Multlplicatlon in Time
if
\[a{x_1}[n]{\text{ }} \leftrightarrow a{X_1}[k]{\text{ }}\]
\[b{x_2}[n]{\text{ }} \leftrightarrow b{X_2}[k]{\text{ }}\]
Then
\[{x_1}[n]{x_2}[n] \leftrightarrow \frac{1}{N}\left[ {{X_1}[k] \otimes {X_2}[k]} \right]\]
Circular Convolution
if
\[a{x_1}[n]{\text{ }} \leftrightarrow a{X_1}[k]{\text{ }}\]
\[b{x_2}[n]{\text{ }} \leftrightarrow b{X_2}[k]{\text{ }}\]
Then
\[{x_1}[n] \otimes {x_2}[n] \leftrightarrow {X_1}[k]{X_2}[k]\]
