Fourier transform

CQT-piano-chord

unfasor

Onfreq

Rectangular function

Sinc function (normalized)

Commutative diagram illustrating problem solving via the Fourier transform

Fourier Transform is a mathematical transform used in engineering, physics, mathematics, and signal processing to convert a function of time (or space) into a function of frequency. It decomposes a function into its constituent frequencies, much like a musical chord can be expressed as the frequencies of its constituent notes. The Fourier Transform is essential in the analysis of linear time-invariant systems, modulation, and in solving partial differential equations, among other applications.

Definition

The Fourier Transform of a continuous, time-domain function f(t) is given by the integral: \[F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} dt\] where:

• \(F(\omega)\) is the Fourier Transform of f(t),
• \(e^{-j\omega t}\) is the complex exponential function,
• \(j\) is the imaginary unit, and
• \(\omega\) is the angular frequency in radians per second.

The inverse Fourier Transform allows the reconstruction of the original time-domain function from its frequency-domain representation and is given by: \[f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{j\omega t} d\omega\]

Applications

The Fourier Transform has a wide range of applications across various fields:

• In Electrical Engineering, it is used to analyze the frequency components of electrical signals.
• In Physics, it helps in studying the wave phenomena and quantum mechanics.
• In Image Processing and Signal Processing, it is crucial for filtering, image reconstruction, and compression.
• In Mathematics, it is used in solving differential equations and in number theory.
• In Chemistry and Biology, Fourier transforms are used in NMR spectroscopy and MRI imaging to analyze molecular structures and to visualize the internal structure of the body, respectively.

Properties

The Fourier Transform has several important properties that make it a powerful tool in analysis and computation:

• Linearity: The Fourier Transform of a linear combination of functions is the same linear combination of their Fourier Transforms.
• Time and Frequency Shifting: Shifting a function in time shifts its Fourier Transform in frequency, and vice versa.
• Convolution: The Fourier Transform converts convolution in the time domain to multiplication in the frequency domain, simplifying the analysis of linear systems.
• Parseval's Theorem: The total energy of a signal in the time domain is equal to the total energy in the frequency domain.

Fourier Series

For periodic functions, the Fourier Transform is related to the Fourier Series, where the function is represented as a sum of sine and cosine functions. The Fourier Series coefficients are computed using the Fourier Transform of one period of the function.

Discrete Fourier Transform and FFT

For digital signal processing, the Discrete Fourier Transform (DFT) is used, which is a discrete version of the Fourier Transform applied to a sequence of values. The Fast Fourier Transform (FFT) is an efficient algorithm for computing the DFT, significantly reducing the computation time, especially for long sequences.

Limitations

While the Fourier Transform is a powerful tool, it has limitations. It assumes the signal is stationary, meaning its statistical properties do not change over time. For non-stationary signals, time-frequency analysis methods like the Wavelet Transform are more appropriate.