Introduction
In this part we’ll define and understand Fourier transforms. In the last part we covered the concept of Fourier series.
Which was a way of describing periodic functions as a linear combination (sum) of simple trigonometric functions. The Fourier transform takes this one step further, it’s a way to extend the Fourier series of non-periodic functions.
Definition 1 (Fourier transform) Given a function in the time-domain, the equivalent function in the frequency domain is:
F ( ω ) = ∫ − ∞ ∞ f ( t ) e − j ω t d t . F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} \, dt. F ( ω ) = ∫ − ∞ ∞ f ( t ) e − j ω t d t . The other way around:
f ( t ) = 1 2 π ∫ − ∞ ∞ F ( ω ) e j ω t d ω . f(t) = \dfrac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{j\omega t} \, d\omega. f ( t ) = 2 π 1 ∫ − ∞ ∞ F ( ω ) e j ω t d ω . We call these functions the Fourier pair. Meaning they are the same functions, in different domains. They represent the same signal.
f ( t ) ⟺ F ( ω ) . f(t) \iff F(\omega). f ( t ) ⟺ F ( ω ) . So the notation for the Fourier transform is:
F ( ω ) = F [ f ( t ) ] , f ( t ) = F − 1 [ F ( ω ) ] . F(\omega) = \mathcal{F}[f(t)], \quad f(t) = \mathcal{F}^{-1}[F(\omega)]. F ( ω ) = F [ f ( t )] , f ( t ) = F − 1 [ F ( ω )] . Let’s do our first Fourier transform!
Example 1 Let’s do this on a simple unit gate function, let’s do it on a general unit gate function.
f ( t ) = rect ( t τ ) , f(t) = \operatorname{rect}\left(\dfrac{t}{\tau}\right), f ( t ) = rect ( τ t ) , F ( ω ) = ∫ − ∞ ∞ rect ( t τ ) e − j ω t d t . F(\omega) = \int_{-\infty}^{\infty} \operatorname{rect}\left(\dfrac{t}{\tau}\right) e^{-j\omega t} \, dt. F ( ω ) = ∫ − ∞ ∞ rect ( τ t ) e − j ω t d t . Since the gate function is only defined at t = − τ 2 t = -\dfrac{\tau}{2} t = − 2 τ t = τ 2 t = \dfrac{\tau}{2} t = 2 τ
F ( ω ) = ∫ − τ 2 τ 2 e − j ω t d t = − 1 j ω e − j ω t ∣ − τ 2 τ 2 = − 1 j ω ( e − j ω τ 2 − e j ω τ 2 ) . \begin{align*}
F(\omega) & = \int_{-\frac{\tau}{2}}^{\frac{\tau}{2}} e^{-j\omega t} \, dt \newline
& = -\dfrac{1}{j\omega} e^{-j\omega t} \bigg\rvert_{-\frac{\tau}{2}}^{\frac{\tau}{2}} \newline
& = -\dfrac{1}{j\omega} \left(e^{-j\omega \frac{\tau}{2}} - e^{j\omega \frac{\tau}{2}}\right).
\end{align*} F ( ω ) = ∫ − 2 τ 2 τ e − j ω t d t = − j ω 1 e − j ω t − 2 τ 2 τ = − j ω 1 ( e − j ω 2 τ − e j ω 2 τ ) . Using Euler’s formula:
F ( ω ) = 2 sin ( ω τ 2 ) ω . F(\omega) = \dfrac{2 \sin\left(\dfrac{\omega \tau}{2}\right)}{\omega}. F ( ω ) = ω 2 sin ( 2 ω τ ) . Let’s rewrite it as:
F ( ω ) = τ sin ( ω τ 2 ) ω τ 2 . F(\omega) = \tau \dfrac{\sin\left(\dfrac{\omega \tau}{2}\right)}{\dfrac{\omega \tau}{2}}. F ( ω ) = τ 2 ω τ sin ( 2 ω τ ) . We can rewrite using sinc ( x ) = sin ( x ) x \operatorname{sinc}(x) = \dfrac{\sin(x)}{x} sinc ( x ) = x sin ( x )
F ( ω ) = τ sinc ( ω τ 2 ) . F(\omega) = \tau \operatorname{sinc}\left(\dfrac{\omega \tau}{2}\right). F ( ω ) = τ sinc ( 2 ω τ ) . So, this means that:
rect ( t τ ) ⟺ τ sinc ( ω τ 2 ) . \operatorname{rect}\left(\dfrac{t}{\tau}\right) \iff \tau \operatorname{sinc}\left(\dfrac{\omega \tau}{2}\right). rect ( τ t ) ⟺ τ sinc ( 2 ω τ ) . Now, we can do this manually each time, but that’s tedious, that’s why we’ll usually use a Fourier table.
Let’s cover some important Fourier transforms.
cos ( ω 0 t ) ⟺ π [ δ ( ω + ω 0 ) + δ ( ω − ω 0 ) ] , sin ( ω 0 t ) ⟺ j π [ δ ( ω + ω 0 ) − δ ( ω − ω 0 ) ] . \begin{align*}
\cos(\omega_0 t) & \iff \pi [\delta(\omega + \omega_0) + \delta(\omega - \omega_0)], \newline
\sin(\omega_0 t) & \iff j\pi [\delta(\omega + \omega_0) - \delta(\omega - \omega_0)].
\end{align*} cos ( ω 0 t ) sin ( ω 0 t ) ⟺ π [ δ ( ω + ω 0 ) + δ ( ω − ω 0 )] , ⟺ j π [ δ ( ω + ω 0 ) − δ ( ω − ω 0 )] . We’ll use these often.
Symmetry
After we have obtained one representation of the signal, it doesn’t stop us from just replacing t t t ω \omega ω
For example:
rect ( t τ ) ⟺ τ sinc ( ω τ 2 ) . \operatorname{rect}\left(\dfrac{t}{\tau}\right) \iff \tau \operatorname{sinc}\left(\dfrac{\omega \tau}{2}\right). rect ( τ t ) ⟺ τ sinc ( 2 ω τ ) . We can do:
F ( t ) = τ sinc ( t τ 2 ) . F(t) = \tau \operatorname{sinc}\left(\dfrac{t \tau}{2}\right). F ( t ) = τ sinc ( 2 t τ ) . We can utilize:
F ( t ) ⟺ 2 π f ( − ω ) . F(t) \iff 2\pi f(-\omega). F ( t ) ⟺ 2 π f ( − ω ) . Now this means:
F ( t ) ⟺ 2 π f ( − ω ) = 2 π rect ( − ω τ ) . F(t) \iff 2\pi f(-\omega) = 2\pi \operatorname{rect}\left(\dfrac{-\omega}{\tau}\right). F ( t ) ⟺ 2 π f ( − ω ) = 2 π rect ( τ − ω ) . Since the unit gate function is an even function:
F ( t ) ⟺ 2 π f ( − ω ) = 2 π rect ( ω τ ) . F(t) \iff 2\pi f(-\omega) = 2\pi \operatorname{rect}\left(\dfrac{\omega}{\tau}\right). F ( t ) ⟺ 2 π f ( − ω ) = 2 π rect ( τ ω ) . Scaling
f ( a t ) ⟺ 1 ∣ a ∣ F ( ω a ) . f(at) \iff \dfrac{1}{|a|} F\left(\dfrac{\omega}{a}\right). f ( a t ) ⟺ ∣ a ∣ 1 F ( a ω ) . As we can see, an expansion in the time domain, means a compression in the frequency domain and vice-versa.
From this we can also gain the time and frequency inversion property:
f ( − t ) ⟺ F ( − ω ) . f(-t) \iff F(-\omega). f ( − t ) ⟺ F ( − ω ) . Time and frequency shifting
f ( t − t 0 ) ⟺ F ( ω ) e − j ω t 0 . f(t - t_0) \iff F(\omega) e^{-j\omega t_0}. f ( t − t 0 ) ⟺ F ( ω ) e − j ω t 0 . This also means:
f ( t ) e j ω 0 t ⟺ F ( ω − ω 0 ) . f(t) e^{j\omega_0 t} \iff F(\omega - \omega_0). f ( t ) e j ω 0 t ⟺ F ( ω − ω 0 ) . Time differentiation and integration
d f d t ⟺ j ω F ( ω ) , ∫ − ∞ t f ( τ ) d τ ⟺ F ( ω ) j ω + π F ( 0 ) δ ( ω ) . \begin{align*}
\dfrac{df}{dt} & \iff j\omega F(\omega), \newline
\int_{-\infty}^{t} f(\tau) \, d\tau & \iff \dfrac{F(\omega)}{j\omega} + \pi F(0)\delta(\omega).
\end{align*} d t df ∫ − ∞ t f ( τ ) d τ ⟺ j ω F ( ω ) , ⟺ j ω F ( ω ) + π F ( 0 ) δ ( ω ) . Now, for the most powerful and beautiful property, which we’ll use.
Convolution
Given f 1 ( t ) ⟺ F 1 ( ω ) f_1(t) \iff F_1(\omega) f 1 ( t ) ⟺ F 1 ( ω ) f 2 ( t ) ⟺ F 2 ( ω ) f_2(t) \iff F_2(\omega) f 2 ( t ) ⟺ F 2 ( ω )
Time convolution:
f 1 ( t ) ∗ f 2 ( t ) ⟺ F 1 ( ω ) F 2 ( ω ) . f_1(t) * f_2(t) \iff F_1(\omega) F_2(\omega). f 1 ( t ) ∗ f 2 ( t ) ⟺ F 1 ( ω ) F 2 ( ω ) . Frequency convolution:
f 1 ( t ) f 2 ( t ) ⟺ 1 2 π F 1 ( ω ) ∗ F 2 ( ω ) . f_1(t) f_2(t) \iff \dfrac{1}{2\pi} F_1(\omega) * F_2(\omega). f 1 ( t ) f 2 ( t ) ⟺ 2 π 1 F 1 ( ω ) ∗ F 2 ( ω ) . The Fourier transform exists, if:
∫ − ∞ ∞ ∣ f ( t ) ∣ d t < ∞ . \int_{-\infty}^{\infty} |f(t)| \, dt < \infty. ∫ − ∞ ∞ ∣ f ( t ) ∣ d t < ∞.