Fourier Transform Properties

Properties of the Fourier Transform.

Frequency Shifting

If $x(t)⟺X(\omega )$, then $x(t)e^{j{\omega }_{0}t}⟺X(\omega -{\omega }_0)$.

Amplitude Modulation

The implication of this is amplitude modulation: multiplication of a signal $x(t)$ by a sinusoid of frequency ${\omega }_0$ will shift the spectrum of $X(\omega )$ by ${\omega }_0$.

How this works is we have:

$$x(t)e^{-j{\omega }_{0}t}⟺X(\omega +{\omega }_0)$$

Using the identity $\cos {\omega }_{0}t=\frac{e^{j{\omega }_{0}t}+e^{-j{\omega }_{0}t}}{2}$, we will have:

$$x(t)\cos {\omega }_{0}t=x(t)\frac{1}{2}[e^{j{\omega }_{0}t}+e^{-j{\omega }_{0}t}]=\frac{1}{2}[x(t)e^{j{\omega }_{0}t}+x(t)e^{-j{\omega }_{0}t}]$$

So we will have:

$$x(t)\cos {\omega }_{0}t⟺\frac{1}{2}[X(\omega -{\omega }_0)+X(\omega +{\omega }_0)]$$

Time Shifting

If $x(t)⟺X(\omega )$, we have $x(t-t_0)⟺X(\omega )e^{-j\omega t_0}$.

Phase Delay

We can write $X(\omega )$ in terms of its amplitude and phase:

X(\omega) = | X(\omega) | \; e^{j \angle X(\omega)}$

So, we can do a phase delay since we have:

$$X(\omega )\cdot e^{-j\omega t_0}=|X(\omega )|\cdot e^{j(\angle X(\omega )-\omega t_0)}$$

The magnitude remains unchanged but we have shifted the phase by $-\omega t_0$. This is a phase shift with a slope of $t_0$ as a function of $\omega$.

Time-Frequency Duality

Let’s take a look at the Inverse Fourier Transform and Fourier Transform:

$$x(t)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }X(\omega )e^{j\omega t}d\omega ,X(\omega )={\int }_{-\infty }^{\infty }x(t)e^{-j\omega t}dt$$

They’re very similar! The two differences are:

• The factor of $\frac{1}{2\pi }$
• The sign of the exponentials

Thus, if we have $x(t)⟺X(\omega )$, we have $X(t)⟺2\pi x(-\omega )$.

Linearity

Pretty simple:

$$a_1{x}_1(t)+a_2{x}_2(t)⟺a_1{X}_1(\omega )+a_2{X}_2(\omega )$$

Scaling

If we have $x(t)⟺X(\omega )$, then:

$$x(at)⟺\frac{1}{|a|}X\left(\frac{\omega }{a}\right)$$

Compression in the time domain causes expansion in the frequency domain and vice versa.

Impulse Function

Wider $x(t)$ causes narrower spectral bandwidth and vice versa, so:

$$\delta (t)⟺1$$

Reflection Property

If $a=-1$,

$$x(-t)⟺X(-\omega )$$

Convolution

If we have $x_1(t)⟺X_1(\omega )$ and $x_2(t)⟺X_2(\omega )$, we have these two.

Time convolution:

$$x_1(t)\ast x_2(t)⟺X_1(\omega )X_2(\omega )$$

Frequency convolution:

$$x_1(t)x_2(t)⟺\frac{1}{2\pi }{X}_1(\omega )\ast X_2(\omega )$$

Time Differentiation and Integration

We have:

$$\begin{array}{rl}\frac{dx(t)}{dt}& ⟺j\omega X(\omega )\\ \frac{d^{n}x(t)}{d{t}^n}& ⟺(j\omega {)}^{n}X(\omega )\end{array}$$

and

$${\int }_{-\infty }^{t}x(\tau )d\tau ⟺\frac{X(\omega )}{j\omega }+\pi X(0)\delta (\omega )$$