In mathematics, signed frequency (negative and positive frequency) expands upon the concept of frequency, from just an absolute value representing how often some repeating event occurs, to also have a positive or negative sign representing one of two opposing orientations for occurrences of those events. The following examples help illustrate the concept:
For a rotating object, the absolute value of its frequency of rotation indicates how many rotations the object completes per unit of time, while the sign could indicate whether it is rotating clockwise or counterclockwise.
Mathematically speaking, the vector has a positive frequency of +1 radian per unit of time and rotates counterclockwise around the unit circle, while the vector has a negative frequency of -1 radian per unit of time, which rotates clockwise instead.
For a harmonic oscillator such as a pendulum, the absolute value of its frequency indicates how many times it swings back and forth per unit of time, while the sign could indicate in which of the two opposite directions it started moving.
For a periodic function represented in a Cartesian coordinate system, the absolute value of its frequency indicates how often in its domain it repeats its values, while changing the sign of its frequency could represent a reflection around its y-axis.
Let be a nonnegative angular frequency with units of radians per unit of time and let be a phase in radians. A function has slope When used as the argument of a sinusoid, can represent a negative frequency.
Because cosine is an even function, the negative frequency sinusoid is indistinguishable from the positive frequency sinusoid
Similarly, because sine is an odd function, the negative frequency sinusoid is indistinguishable from the positive frequency sinusoid or
Thus any sinusoid can be represented in terms of positive frequencies only.
The sign of the underlying phase slope is ambiguous.
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This course introduces the analysis and design of linear analog circuits based on operational amplifiers. A Laplace early approach is chosen to treat important concepts such as time and frequency resp
In mathematics and signal processing, the Hilbert transform is a specific singular integral that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t). The Hilbert transform is given by the Cauchy principal value of the convolution with the function (see ). The Hilbert transform has a particularly simple representation in the frequency domain: It imparts a phase shift of ±90° ( radians) to every frequency component of a function, the sign of the shift depending on the sign of the frequency (see ).
A Fourier series (ˈfʊrieɪ,_-iər) is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation.
A periodic function or cyclic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of radians, are periodic functions. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function that is not periodic is called aperiodic. A function f is said to be periodic if, for some nonzero constant P, it is the case that for all values of x in the domain.
Covers the consequences of undersampling signals and the stroboscopic effect.
Introduces signals, frequencies, and bandwidth, including pure sinusoids, Fourier theory, and spectral representation.
Covers the theory of numerical methods for frequency estimation on deterministic signals, including Fourier series and transform, Discrete Fourier transform, and the Sampling theorem.
In this paper, the detection of a small reflector in a randomly heterogeneous medium using secondharmonic generation is investigated. The medium is illuminated by a time-harmonic plane wave at frequency ω. It is assumed that the reflector has a nonzero sec ...
A ring of N identical phase oscillators with interactions between L-nearest neighbors is considered, where L ranges from 1 (local coupling) to N/2 (global coupling). The coupling function is a simple sinusoid, as in the Kuramoto model, but with a minus sig ...
We develop a least mean-squares (LMS) diffusion strategy for sensor network applications where it is desired to estimate parameters of physical phenomena that vary over space. In particular, we consider a regression model with space-varying parameters that ...