Discrete-time Fourier Transform

GPTKB entity

Predicate	Object
gptkbp:instanceOf	gptkb:transformation
gptkbp:abbreviation	gptkb:DTFT
gptkbp:application	digital signal processing filter design spectral analysis system analysis
gptkbp:designer	gptkb:Inverse_Discrete-time_Fourier_Transform
gptkbp:domain	frequency domain
gptkbp:field	gptkb:mathematics gptkb:signal_processing electrical engineering
gptkbp:form	X(e^{j\\omega}) = \\sum_{n=-\\infty}^{\\infty} x[n] e^{-j\\omega n}
gptkbp:generalizes	Fourier series for aperiodic signals
gptkbp:hasUnit	radians per sample
https://www.w3.org/2000/01/rdf-schema#label	Discrete-time Fourier Transform
gptkbp:input	discrete-time signal
gptkbp:introducedIn	mid-20th century
gptkbp:inverse_formula	x[n] = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} X(e^{j\\omega}) e^{j\\omega n} d\\omega
gptkbp:limitation	not defined for non-summable signals
gptkbp:mapType	discrete-time signal to frequency domain representation
gptkbp:output	continuous frequency spectrum
gptkbp:property	gptkb:Parseval's_theorem linearity convolution frequency-shifting time-shifting
gptkbp:recurrence	2\\pi-periodic in frequency
gptkbp:relatedTo	gptkb:Fourier_Transform gptkb:Z-transform gptkb:Discrete_Fourier_Transform
gptkbp:seeAlso	gptkb:Laplace_Transform gptkb:Short-time_Fourier_Transform gptkb:Continuous-time_Fourier_Transform
gptkbp:usedFor	frequency analysis of discrete signals
gptkbp:bfsParent	gptkb:Discrete_Fourier_Transform
gptkbp:bfsLayer	7