The Fourier Transform Properties

By Joel B. HarleyEducation, Foundations of Digital Signal Processing, Uncategorized

The following tables illustrate the Fourier transform properties. Notice a close similarity between the each of the different transforms.

Continuous-Time Fourier Series Properties

For each property, assume:

    \[x(t) \stackrel{CTFS}{\longleftrightarrow} c_k \quad \textrm{and} \quad y(t) \stackrel{CTFS}{\longleftrightarrow} d_k\]

    \begin{align*} \hline & \textbf{Property} &\qquad & \textbf{Time domain} &\qquad & \textbf{CTFT domain} \\ \hline \\[.05em] & \textrm{Linearity} &\qquad & A x(t) + B y(t) &\qquad & A c_k + B d_k \\ \\[.05em] & \textrm{Time Shifting} &\qquad & x(t-t_0) &\qquad & c_k e^{-j k \Omega_0 t_0} \\ \\[.05em] & \textrm{Frequency Shifting} &\qquad & x(t) e^{j M \Omega_0 t} &\qquad & c_{k-M} \\ \\[.05em] & \textrm{Conjugation} &\qquad & x^{*}(t) &\qquad & c^*_{-k} \\ \\[.05em] & \textrm{Time Reversal} &\qquad & x(-t) &\qquad & c_{-k} \\ \\[.05em] & \textrm{Circular Conv.} &\qquad & x(t) \circledast y(t) &\qquad & T_0 c_k d_k \\ \\[.05em] & \textrm{Multiplication} &\qquad & x(t) y(t) &\qquad & c_k * d_k \\ \\[.05em] & \textrm{Differentiation} &\qquad & \frac{d }{dt} x(t) &\qquad & (j k \Omega_0) c_k \\ \\[.05em] & \textrm{Integration} &\qquad & \int_{-\infty}^{t} x(\tau) \, d \tau &\qquad & \left( \frac{1}{j k \Omega_0} \right) c_k \\ \\[.05em] & \textrm{Conjugate Symmetry} &\qquad & x(t) \textrm{ is real } &\qquad & c_k = c^*_{-k} \\ \\[.05em] & \textrm{Real and Even Signals} &\qquad & x(t) \textrm{ is real and even } &\qquad & c_k \textrm{ is real and even} \\ \\[.05em] & \textrm{Real and Odd Signals} &\qquad & x(t) \textrm{ is real and odd } &\qquad & c_k \textrm{ is purely imaginary and odd } \\ \\[.05em] & \textrm{Parseval's Relation} &\qquad & \frac{1}{T_0} \int_{T_0} |x(t)|^2 \, dt &\qquad & \sum_{k=-\infty}^{\infty} |c_k|^2 \\ \hline & \end{align*}

Continuous-Time Fourier Transform Properties

For each property, assume:

    \[x(t) \stackrel{CTFT}{\longleftrightarrow} X(\Omega) \quad \textrm{and} \quad y(t) \stackrel{CTFT}{\longleftrightarrow} Y(\Omega)\]

    \begin{align*} \hline & \textbf{Property} &\qquad & \textbf{Time domain} &\qquad & \textbf{CTFT domain} \\ \hline \\[.05em] & \textrm{Linearity} &\qquad & A x(t) + B y(t) &\qquad & A X(\Omega) + B Y(\Omega) \\[.05em] & \textrm{Time Shifting} &\qquad & x(t-t_0) &\qquad & X(\Omega) e^{-j \Omega t_0} \\[.05em] & \textrm{Time Scaling} &\qquad & x(\alpha t) &\qquad & \frac{1}{|\alpha|} X\left( \frac{\Omega}{\alpha} \right) \\[.05em] & \textrm{Frequency Shifting} &\qquad & x(t) e^{j \Omega_0 t} &\qquad & X(\Omega-\Omega_0) \\[.05em] & \textrm{Conjugation} &\qquad & x^{}(t) &\qquad & X^*(-\Omega) \\[.05em] & \textrm{Time Reversal} &\qquad & x(-t) &\qquad & X(-\Omega) \\[.05em] & \textrm{Convolution} &\qquad & x(t) * y(t) &\qquad & X(\Omega) Y(\Omega) \\[.05em] & \textrm{Multiplication} &\qquad & x(t) y(t) &\qquad & \frac{1}{2 \pi} X(\Omega) * Y(\Omega) \\[.05em] & \textrm{Differentiation} &\qquad & \frac{d }{dt} x(t) &\qquad & (j \Omega) X(\Omega) \\[.05em] & \textrm{Integration} &\qquad & \int_{-\infty}^{t} x(\tau) \, d \tau &\qquad & \left( \frac{1}{j \Omega} \right) X(\Omega) \\[.05em] & \textrm{Conjugate Symmetry} &\qquad & x(t) \textrm{ is real } &\qquad & X(\Omega) = X^(-\Omega) \\[.05em] & \textrm{Real and Even Signals} &\qquad & x(t) \textrm{ is real and even } &\qquad & X(\Omega) \textrm{ is real and even} \\[.05em] & \textrm{Real and Odd Signals} &\qquad & x(t) \textrm{ is real and odd } &\qquad & X(\Omega) \textrm{ is purely imaginary and odd } \\[.05em] & \textrm{Parseval's Relation} &\qquad & \int_{-\infty}^{\infty} |x(t)|^2 \, dt &\qquad & \frac{1}{2 \pi} \int_{-\infty}^{\infty} |X(\Omega)|^2 \, d\Omega \\ \hline \end{align*}

Discrete-Time Fourier Transform Properties

For each property, assume:

    \[x[n] \stackrel{DTFT}{\longleftrightarrow} X(\omega) \quad \textrm{and} \quad y[n] \stackrel{DTFT}{\longleftrightarrow} Y(\omega)\]

    \begin{align*} \hline & \textbf{Property} &\qquad & \textbf{Time domain} &\qquad & \textbf{DTFT domain} \\ \hline \\[.05em] & \textrm{Linearity} &\qquad & A x[n] + B y[n] &\qquad & A X(\omega) + B Y(\omega) \\[.05em] & \textrm{Time Shifting} &\qquad & x[n-n_0] &\qquad & X(\omega) e^{-j \omega n_0} \\[.05em] & \textrm{Frequency Shifting} &\qquad & x[n] e^{j \omega_0 n} &\qquad & X(\omega-\omega_0) \\[.05em] & \textrm{Conjugation} &\qquad & x^[n] &\qquad & X^*(-\omega) \\[.05em] & \textrm{Time Reversal} &\qquad & x[-n] &\qquad & X(-\omega) \\[.05em] & \textrm{Convolution} &\qquad & x[n] * y[n] &\qquad & X(\omega) Y(\omega) \\[.05em] & \textrm{Multiplication} &\qquad & x[n] y[n] &\qquad & \displaystyle \frac{1}{2 \pi} \int_{2 \pi} X(\theta) Y(\omega-\theta) d \theta \\[.05em] & \textrm{Differencing in Time} &\qquad & x[n] - x[n-1] &\qquad & (1-e^{-j \omega})X(\omega) \\[.05em] & \textrm{Accumulation} &\qquad & \sum_{k=-\infty}^{\infty} x[k] &\qquad & \frac{1}{1 - e^{-j \omega}} + \pi X(0) \sum_{k=-\infty}^{\infty} \delta(\omega - 2 \pi k) \\[.05em] & \textrm{Frequency Differentiation} &\qquad & n x[n] &\qquad & \displaystyle j \frac{d X(\omega)}{d \omega} \\[.05em] & \textrm{Conjugate Symmetry} &\qquad & x[n] \textrm{ is real } &\qquad & X(\omega) = X^(-\omega) \\[.05em] & \textrm{Real and Even Signals} &\qquad & x[n] \textrm{ is real and even } &\qquad & X(\omega) \textrm{ is real and even} \\[.05em] & \textrm{Real and Odd Signals} &\qquad & x[n] \textrm{ is real and odd } &\qquad & X(\omega) \textrm{ is purely imaginary and odd } \\[.05em] & \textrm{Parseval's Relation} &\qquad & \sum_{k=-\infty}^{\infty} |x[k]|^2 &\qquad & \frac{1}{2 \pi} \int_{2 \pi} |X(\omega)|^2 d \omega \\\hline \end{align*}