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Instantaneous phase and frequency are important concepts in signal processing that occur in the context of the representation and analysis of time-varying functions. [1] The instantaneous phase (also known as local phase or simply phase ) of a complex-valued function s ( t ), is the real-valued function:
The instantaneous amplitude, and the instantaneous phase and frequency are in some applications used to measure and detect local features of the signal. Another application of the analytic representation of a signal relates to demodulation of modulated signals .
Conversely, a phase reversal or phase inversion implies a 180-degree phase shift. [ 2 ] When the phase difference φ ( t ) {\displaystyle \varphi (t)} is a quarter of turn (a right angle, +90° = π/2 or −90° = 270° = −π/2 = 3π/2 ), sinusoidal signals are sometimes said to be in quadrature , e.g., in-phase and quadrature components of a ...
A(t) represents the time-varying amplitude of the sinusoidal carrier wave and the cosine-term is the carrier at its angular frequency, and the instantaneous phase deviation (). This description directly provides the two major groups of modulation, amplitude modulation and angle modulation .
Unless θ (t) is a constant, the point in time t s at which the phase is stationary will vary according to the instantaneous frequency ω s. Expressing the difference between ( ω s - ω 0 ).t and θ (t) as a Taylor series about the time t s , but discarding all but the first three terms (of which the second term is zero, here), the Fourier ...
The plotted line represents the variation of instantaneous voltage (or current) with respect to time. This cycle repeats with a frequency that depends on the power system. In electrical engineering , three-phase electric power systems have at least three conductors carrying alternating voltages that are offset in time by one-third of the period.
The group delay and phase delay properties of a linear time-invariant (LTI) system are functions of frequency, giving the time from when a frequency component of a time varying physical quantity—for example a voltage signal—appears at the LTI system input, to the time when a copy of that same frequency component—perhaps of a different physical phenomenon—appears at the LTI system output.
The envelope thus generalizes the concept of a constant amplitude into an instantaneous amplitude. The figure illustrates a modulated sine wave varying between an upper envelope and a lower envelope. The envelope function may be a function of time, space, angle, or indeed of any variable. Envelope for a modulated sine wave.