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Periodic motion is motion in which the position(s) of the system are expressible as periodic functions, all with the same period. For a function on the real numbers or on the integers , that means that the entire graph can be formed from copies of one particular portion, repeated at regular intervals.
In electronics, acoustics, and related fields, the waveform of a signal is the shape of its graph as a function of time, independent of its time and magnitude scales and of any displacement in time. [1] [2] Periodic waveforms repeat regularly at a constant period. The term can also be used for non-periodic or aperiodic signals, like chirps and ...
The components of the periodic summation are centered at integer values (denoted by ) of a normalized frequency (cycles per sample). Ordinary/physical frequency (cycles per second) is the product of k {\displaystyle k} and the sample-rate, f s = 1 / T . {\displaystyle f_{s}=1/T.}
The Fourier transform of a periodic function cannot be defined using the integral formula directly. In order for integral in Eq.1 to be defined the function must be absolutely integrable. Instead it is common to use Fourier series. It is possible to extend the definition to include periodic functions by viewing them as tempered distributions.
Fourier series – periodic signals, oscillating systems. Fourier transform – aperiodic signals, transients. Laplace transform – electronic circuits and control systems. Z transform – discrete-time signals, digital signal processing. Wavelet transform — image analysis, data compression.
While the ordinary DFT corresponds to a periodic signal in both time and frequency domains, = / produces a signal that is anti-periodic in frequency domain (+ =) and vice versa for = /. Thus, the specific case of a = b = 1 / 2 {\displaystyle a=b=1/2} is known as an odd-time odd-frequency discrete Fourier transform (or O 2 DFT).
This is a list of some well-known periodic functions. The constant function f (x) = c, where c is independent of x, is periodic with any period, but lacks a fundamental period. A definition is given for some of the following functions, though each function may have many equivalent definitions.
A Fourier transform and 3 variations caused by periodic sampling (at interval ) and/or periodic summation (at interval ) of the underlying time-domain function. The relative computational ease of the DFT sequence and the insight it gives into S ( f ) {\displaystyle S(f)} make it a popular analysis tool.