11.2 Fourier Transform and Its Properties

The Fourier transform is a powerful tool that breaks down signals into their frequency components. It's like a musical ear that can pick out individual notes from a complex chord, allowing us to analyze and manipulate signals in ways we couldn't before.

This section dives into the math behind the Fourier transform and its key properties. We'll see how it relates to other transforms and learn to apply it to common functions, unlocking new ways to understand and work with signals.

Fourier transform definition and interpretation

Definition and mathematical representation

• The Fourier transform decomposes a function into its constituent frequencies, representing the function in the frequency domain
• Defined as an integral of the form $F(\omega) = \int_{-\infty}^{\infty} f(t)e^{-j\omega t} dt$, where $f(t)$ is the time-domain function, $F(\omega)$ is the frequency-domain function, and $\omega$ is the angular frequency
• The inverse Fourier transform recovers the original time-domain function from its frequency-domain representation, defined as $f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega)e^{j\omega t} d\omega$

Complex-valued function and applications

• The Fourier transform is a complex-valued function
  • Real part represents the amplitude of the frequency components
  • Imaginary part represents the phase of the frequency components
• Widely used in various fields (signal processing, communications, control systems) to analyze and manipulate signals in the frequency domain

Properties of Fourier transforms

Linearity, scaling, and shifting properties

• Linearity: The Fourier transform is a linear operation
  • Transform of a sum of functions equals the sum of their individual transforms
  • Transform of a scalar multiple of a function equals the scalar multiple of its transform
• Scaling: If $f(t)$ has a Fourier transform $F(\omega)$, then $f(at)$ has a Fourier transform $\frac{1}{|a|}F(\frac{\omega}{a})$, where $a$ is a non-zero scalar
  • Relates the scaling of a function in the time domain to the scaling of its Fourier transform in the frequency domain
• Time shifting: If $f(t)$ has a Fourier transform $F(\omega)$, then $f(t-t_0)$ has a Fourier transform $e^{-j\omega t_0}F(\omega)$, where $t_0$ is a real constant
  • A time shift in the time domain results in a phase shift in the frequency domain
• Frequency shifting: If $f(t)$ has a Fourier transform $F(\omega)$, then $e^{j\omega_0 t}f(t)$ has a Fourier transform $F(\omega-\omega_0)$, where $\omega_0$ is a real constant
  • Multiplying a function by a complex exponential in the time domain results in a frequency shift in the frequency domain

Convolution and Parseval's theorem

• Convolution: The convolution of two functions in the time domain is equivalent to the multiplication of their Fourier transforms in the frequency domain
  • If $f(t)$ and $g(t)$ have Fourier transforms $F(\omega)$ and $G(\omega)$, respectively, then the Fourier transform of their convolution, $(f*g)(t)$, is equal to $F(\omega)G(\omega)$
• Parseval's theorem: Relates the energy of a function in the time domain to the energy of its Fourier transform in the frequency domain
  • The total energy of a signal is preserved when transformed between the time and frequency domains

Fourier transform evaluation

Fourier transforms of common functions

• Rectangular pulse: The Fourier transform of a rectangular pulse with width $\tau$ and amplitude $A$ is given by $F(\omega) = A\tau \text{sinc}(\frac{\omega\tau}{2})$, where $\text{sinc}(x) = \frac{\sin(x)}{x}$
• Gaussian pulse: The Fourier transform of a Gaussian pulse, $f(t) = e^{-at^2}$, is another Gaussian function, $F(\omega) = \sqrt{\frac{\pi}{a}} e^{-\frac{\omega^2}{4a}}$, where $a > 0$
• Signum function: The signum function, $\text{sgn}(t)$, has a Fourier transform given by $F(\omega) = \frac{2}{j\omega}$, which is purely imaginary and has a singularity at $\omega = 0$

Inverse Fourier transforms of common functions

• Rectangular pulse: The inverse Fourier transform of a rectangular pulse with width $\Omega$ and amplitude $B$ is given by $f(t) = \frac{B\Omega}{2\pi} \text{sinc}(\frac{\Omega t}{2})$
• Gaussian pulse: The inverse Fourier transform of a Gaussian pulse, $F(\omega) = e^{-a\omega^2}$, is another Gaussian function, $f(t) = \sqrt{\frac{1}{4\pi a}} e^{-\frac{t^2}{4a}}$, where $a > 0$

Fourier vs Laplace transforms

Relationship between Fourier and Laplace transforms

• The Laplace transform is a generalization of the Fourier transform, extending the concept to complex frequencies ($s = \sigma + j\omega$) and causal signals (signals that are zero for $t < 0$)
• The Fourier transform can be considered a special case of the Laplace transform, where the real part of the complex frequency, $\sigma$, is set to zero

Definitions and applications

• The Laplace transform is defined as $\mathcal{L}\{f(t)\} = F(s) = \int_0^{\infty} f(t)e^{-st} dt$, while the Fourier transform is defined as $\mathcal{F}\{f(t)\} = F(\omega) = \int_{-\infty}^{\infty} f(t)e^{-j\omega t} dt$
• The Laplace transform is particularly useful for analyzing systems described by linear differential equations, as it allows for the conversion of differential equations into algebraic equations in the complex frequency domain
• The Fourier transform is more suitable for analyzing the frequency content of signals and systems, while the Laplace transform is more appropriate for studying the stability and transient behavior of systems