Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Relationships between transforms

Description

This lecture covers the relationships between different transforms, including the DFT, DFS, and DTFT, as well as the embedding of finite-length signals, DTFT of periodic and finite-support signals, zero padding, and smooth interpolation of DFT values.

Official source

Ontological neighbourhood

Mathematics

Analysis: Numerical analysis

Information engineering

Signal processing: Fourier analysis

Related concepts (32)

Discrete Fourier transform

In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT (IDFT) is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies.

Fourier analysis

In mathematics, Fourier analysis (ˈfʊrieɪ,_-iər) is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer. The subject of Fourier analysis encompasses a vast spectrum of mathematics.

Spectral leakage

The Fourier transform of a function of time, s(t), is a complex-valued function of frequency, S(f), often referred to as a frequency spectrum. Any linear time-invariant operation on s(t) produces a new spectrum of the form H(f)•S(f), which changes the relative magnitudes and/or angles (phase) of the non-zero values of S(f). Any other type of operation creates new frequency components that may be referred to as spectral leakage in the broadest sense. Sampling, for instance, produces leakage, which we call aliases of the original spectral component.

Discrete-time Fourier transform

In mathematics, the discrete-time Fourier transform (DTFT), also called the finite Fourier transform, is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term discrete-time refers to the fact that the transform operates on discrete data, often samples whose interval has units of time. From uniformly spaced samples it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function.

Fourier transform

In physics and mathematics, the Fourier transform (FT) is a transform that converts a function into a form that describes the frequencies present in the original function. The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made the Fourier transform is sometimes called the frequency domain representation of the original function.

Related lectures (51)

Discrete Fourier Transform: Frequency Periodicity and Reconstruction

Explores frequency periodicity in the discrete Fourier transform for signal reconstruction.

Signals & Systems II: Vector Analogy and Discrete Signals

Explores the link between DTFT, Z-transforms, and Fourier transforms, vector analogy, and discrete signals.

Signal processing and vector spaces

Emphasizes the significance of vector spaces in signal processing, offering a unified framework for various signal types and system design.

Fourier Transforms in Tomography: Inversion Techniques

Covers the application of Fourier transforms in tomography, focusing on the Radon transform and its inversion techniques.

Interpreting a DFT plot