A Quadrature Signals Tutorial: Complex, But Not Complicated

Understanding the 'Phasing Method' of Single Sideband Demodulation

Complex Digital Signal Processing in Telecommunications

Introduction to Sound Processing

Introduction of C Programming for DSP Applications

Fourier Transforms for Continuous/Discrete Time/Frequency

Discrete Time Fourier Transform (DTFT)

**Search Mathematics of the DFT**

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The *Discrete Time Fourier Transform* (DTFT) can be viewed as the
limiting form of the DFT when its length is allowed to approach
infinity:

The inverse DTFT is

Instead of operating on sampled signals of length (like the DFT),
the DTFT operates on sampled signals defined over all integers
. As a result, the DTFT frequencies form a
*continuum*. That is, the DTFT is a function of
*continuous* frequency
, while the DFT is a
function of discrete frequency ,
. The DFT
frequencies
,
, are given by
the angles of points uniformly distributed along the unit circle
in the complex plane (see
Fig.6.1). Thus, as
, a continuous frequency axis
must result in the limit along the unit circle in the plane. The
axis is still finite in length, however, because the time domain
remains sampled.

Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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