"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)"
Mathematical foundation of DFT, the relationship between DFT and FFT, applications of DFT in speech processing and image processing.
Sampling theorem: Fundamental concept that determines the minimum sampling rate required to adequately represent a signal.
Fourier series: Mathematical representation of a periodic function as a sum of sinusoids with different frequencies and amplitudes.
Continuous Fourier Transform: Mathematical transformation that converts a time-domain signal into a frequency-domain signal.
Discrete-time Fourier Transform (DTFT): Mathematical transformation that converts a discrete sequence of values (such as a digital signal) into a continuous frequency-domain signal.
Z-transform: Mathematical transformation that converts a discrete sequence of values into a continuous complex function in a specific region of the complex plane. It is used to analyze discrete-time signal processing systems.
Sampling rate conversion: Technique used to change the sampling rate of a digital signal, often to allow for easier processing or to match the requirements of a specific application.
Window functions: Mathematical functions used to reduce the effects of spectral leakage when calculating the discrete Fourier Transform of a finite-length signal.
Fast Fourier Transform (FFT): Algorithmic technique for efficiently computing the Discrete Fourier Transform of a sequence of values.
Fourier analysis of time-dependent signals: Extension of the Fourier Transform to signals that change over time, allowing for an analysis of the frequency content of the signal at different points in time.
Applications of digital signal processing: Numerous practical applications of digital signal processing, including audio and video compression, image processing, radar signal processing, and medical signal analysis.
DFT: The discrete Fourier transform (DFT) is the standard method for computing the Fourier transform of a sequence of time-domain samples.
FFT: The fast Fourier transform (FFT) is an algorithm for efficiently computing the DFT. It reduces the computational complexity from O(N^2) to O(N log N), making it extremely useful for real-time applications.
Real DFT: The real discrete Fourier transform (R-DFT) computes the Fourier transform of a real-valued signal. It only returns the non-redundant half of the spectrum due to the Hermitian symmetry property.
Complex DFT: The complex discrete Fourier transform (C-DFT) computes the Fourier transform of a complex-valued signal. It returns the full spectrum without any symmetry properties.
Inverse DFT: The inverse discrete Fourier transform (IDFT) is the inverse operation of the DFT, reconstructing the time-domain signal from its frequency-domain representation.
Circular DFT: The circular discrete Fourier transform (C-DFT) is a variation of the DFT where the signal sequence is regarded as circular. It is used in image processing and signal classification.
Short-time Fourier transform: The short-time Fourier transform (STFT) is a time-frequency analysis technique that uses a sliding window to analyze the frequency content of a signal over time.
Discrete cosine transform: The discrete cosine transform (DCT) is a variant of the Fourier transform that converts a signal from the time domain to the frequency domain while focusing on its energy distribution instead of its phase information.
Discrete Hartley transform: The discrete Hartley transform (DHT) is another variant of the Fourier transform that uses real-valued basis functions instead of complex exponentials.
Wavelet transform: The wavelet transform is a signal processing technique that decomposes a signal into multiple scales, allowing for a more detailed analysis in both time and frequency domains.
"The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence."
"Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies"
"The DFT is therefore said to be a frequency domain representation of the original input sequence."
"If the original sequence is one cycle of a periodic function, the DFT provides all the non-zero values of one DTFT cycle."
"used to perform Fourier analysis in many practical applications"
"any quantity or signal that varies over time, such as the pressure of a sound wave, a radio signal, or daily temperature readings"
"values of pixels along a row or column of a raster image"
"efficiently solve partial differential equations, and to perform other operations such as convolutions or multiplying large integers"
"implemented in computers by numerical algorithms or even dedicated hardware"
"the terms 'FFT' and 'DFT' are often used interchangeably"
"employ efficient fast Fourier transform (FFT) algorithms"
"the 'FFT' initialism may have also been used for the ambiguous term 'finite Fourier transform'"
"the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies"
"The DFT is therefore said to be a frequency domain representation of the original input sequence."
"a finite amount of data"
"to perform Fourier analysis in many practical applications"
"efficiently solve partial differential equations"
"perform other operations such as convolutions or multiplying large integers"
"implemented in computers by numerical algorithms or even dedicated hardware"