Class 13 (Continuous-time Fourier transform: properties)

The Properties of the Fourier Transform

For each property we two continuous-time, periodic signals $x(t)$ and $y(t)$ with fundamental frequency $\omega_0 = 2 \pi / T_0$ such that $$ x(t) \stackrel{\mathcal{F}}{\longleftrightarrow} X(\omega) \quad \textrm{and} \quad y(t) \stackrel{\mathcal{F}}{\longleftrightarrow} Y(\omega) \; .$$ In this above statement, $c_k$ are the Fourier coefficients for $x(t)$ and $d_k$ are the Fourier coefficients for $y(t)$.

Linearity

$$ A x(t) + B y(t) \stackrel{\mathcal{F}}{\longleftrightarrow} A X(\omega) + B Y(\omega) $$

Time Shifting

$$ x(t-t_0) \stackrel{\mathcal{F}}{\longleftrightarrow} X(\omega) e^{-j \omega t_0} $$

Time Scaling

$$ x(\alpha t) \stackrel{\mathcal{F}}{\longleftrightarrow} \frac{1}{|\alpha|} X\left( \frac{\omega}{\alpha} \right) $$

Frequency shifting

$$ x(t) e^{j \omega_0 t} \stackrel{\mathcal{F}}{\longleftrightarrow} X(\omega-\omega_0) $$

Conjugation

$$ x^{*}(t) \stackrel{\mathcal{F}}{\longleftrightarrow} X^*(-\omega) $$

Time Reversal

$$ x(-t) \stackrel{\mathcal{F}}{\longleftrightarrow} X(-\omega) $$

Convolution

$$ x(t) * y(t) = \int_{-\infty}^{\infty} x(\tau) y(t - \tau) \, d\tau \stackrel{\mathcal{F}}{\longleftrightarrow} X(\omega) Y(\omega) $$

Multiplication

$$ x(t) y(t) \stackrel{\mathcal{F}}{\longleftrightarrow} \frac{1}{2 \pi} X(\omega) * Y(\omega) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} X(\Omega) Y(\omega - \Omega) d \Omega $$

Differentiation

$$ \frac{d x(t)}{dt} \stackrel{\mathcal{F}}{\longleftrightarrow} (j \omega) X(\omega) $$

Integration

$$ \int_{-\infty}^{t} x(\tau) \, d \tau \stackrel{\mathcal{F}}{\longleftrightarrow} \left( \frac{1}{j \omega} \right) X(\omega) $$

Conjugate Symmetry for Real Signals

$$ x(t) \textrm{ is real } \stackrel{\mathcal{F}}{\longleftrightarrow} X(\omega) = X^*(-\omega) $$

Real and even signals

$$ x(t) \textrm{ is real and even } \stackrel{\mathcal{F}}{\longleftrightarrow} X(\omega) \textrm{ is real and even}$$

Real and odd signals

$$ x(t) \textrm{ is real and odd }\stackrel{\mathcal{F}}{\longleftrightarrow} X(\omega) \textrm{ is purely imaginary and odd }$$

Parseval's Relation for Aperiodic signals

$$ \int_{-\infty}^{\infty} |x(t)|^2 \, dt = \frac{1}{2 \pi} \int_{-\infty}^{\infty} |X(\omega)|^2 \, d\omega $$

Example

Time shifting property

Question: Compute the Fourier transform of $x(t) = \sin(\pi (t + 1))$.

Solution We know the Fourier coefficients of $2 \sin(\pi t)$ are $$ \begin{eqnarray*} 2 \sin(\pi t) \stackrel{\mathcal{F}}{\longleftrightarrow} X(\omega) &=& \frac{-\pi}{j} \delta(\omega + \pi) + \frac{\pi}{j} \delta(\omega - \pi) \\ &=& \pi j \delta(\omega + \pi) - \pi j \delta(\omega - \pi) \\ \; . \end{eqnarray*} $$ Therefore, based on the the time-shifting property, the Fourier transform of $2 \sin(\pi (t + 1))$ is $$ \begin{eqnarray*} 2 \sin(\pi (t + 1)) \stackrel{\mathcal{F}}{\longleftrightarrow} X(\omega) &=& j \delta(\omega + \pi) e^{j \omega (1)} - j \delta(\omega - \pi) e^{j \omega (1)} \\ &=& j \delta(\omega + \pi) e^{j \pi} - j \delta(\omega - \pi) e^{j \pi} \\ &=& j \delta(\omega - \pi) - j \delta(\omega + \pi) \\ \end{eqnarray*} $$

Additional Resources