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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:
Finally, the instantaneous angular chirpyness (symbol γ) is defined to be the second derivative of instantaneous phase or the first derivative of instantaneous angular frequency, = = Angular chirpyness has units of radians per square second (rad/s 2); thus, it is analogous to angular acceleration.
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 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 .
Phase modulation (PM) is a modulation pattern for conditioning communication signals for transmission. It encodes a message signal as variations in the instantaneous phase of a carrier wave. Phase modulation is one of the two principal forms of angle modulation, together with frequency modulation.
In Software-Defined Radio implementations the demodulation may be carried out by using the Hilbert transform (implemented as a filter) to recover the instantaneous phase, and thereafter differentiating this phase (using another filter) to recover the instantaneous frequency. Alternatively, a complex mixer followed by a bandpass filter may be ...
The definition of instantaneous frequency is the time rate of change of phase, or (), where () is the instantaneous phase of a signal. We can know the instantaneous frequency from the time–frequency plane directly if the image is clear enough.
An important property of three-phase power is that the instantaneous power available to a resistive load, = =, is constant at all times.Indeed, let = = To simplify the mathematics, we define a nondimensionalized power for intermediate calculations, =