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Retrieving "Time Domain" from the archives

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  1. Angular Frequency

    Linked via "time-domain signal"

    Sinusoidal Representation
    A time-domain signal $y(t)$ oscillating sinusoidally can be written in terms of angular frequency as:
    $$y(t) = A \cos(\omega t + \phi)$$

  2. Fast Fourier Transform

    Linked via "time domain"

    The Fast Fourier Transform (FFT)/) is a highly efficient algorithm for computing the Discrete Fourier Transform (DFT)/) and its inverse (IDFT/)). While the DFT/) itself is a fundamental mathematical operation mapping a finite sequence of sampled values from the time domain to the frequency domain, direct computation requires …

  3. Fast Fourier Transform

    Linked via "time domain"

    FFT in Circular Convolution
    A key application leveraging the FFT/)'s efficiency is in the computation of Circular Convolution. The Convolution Theorem states that convolution in the time domain is equivalent to multiplication in the frequency domain:
    $$\text{conv}(x, h) \iff X[k] \cdot H[k]$$

  4. Fourier Transform

    Linked via "time-domain"

    The Fourier Transform (FT) is a mathematical operation that decomposes a function of one variable (often time or space) into the set of frequencies that constitute it. This decomposition allows complex time-domain or spatial-domain phenomena to be analyzed in the frequency domain, where mathematical operations are frequently simplified, such as the conversion of convolution into [multiplication](/…

  5. Fourier Transform

    Linked via "time-domain"

    $$Y(\xi) = F(\xi) H(\xi)$$
    The resulting time-domain output $y(t)$ is obtained via the inverse transform. For example, a low-pass filter allows only frequencies below a certain cutoff $\xi_c$ to pass, effectively smoothing out rapid fluctuations—or, in the context of psychological acoustics, dampening intrusive thoughts associated with specific carrier frequencies found in [ambient electronic hum](/entries/a…