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Microwave and Millimeter-Wave Radars for Vital Sign Monitoring
Published in Moeness G. Amin, Radar for Indoor Monitoring, 2017
Changzhan Gu, Tien-Yu Huang, Changzhi Li, Jenshan Lin
Phase discontinuity is another challenge in arctangent demodulation. An arctangent function mathematically only allows a native codomain range of (−π/2, +π/2). If the target’s movement is large and comparable to the carrier wavelength, the demodulation may exceed this range so that a phase discontinuity will occur. Figure 9.6a shows the demodulation of a 0.2 Hz 20 mm sinusoidal movement in a 2.4 GHz Doppler radar using the regular arctangent demodulation showing the occurrence of the phase discontinuity. Theoretically, this discontinuity can be compensated in digital signal processing by shifting an integer multiple of π (i.e., phase unwrapping), but in practical applications, it is hard for a hardware or software to automatically make a judicious choice on exactly which point to shift. It becomes extremely difficult when both the displacement and the environmental clutter are large because it would cause severe discontinuity issues and the discontinuous data points may need to be compensated by different integer multiples of π.
Selected Polarimetric SAR Applications
Published in Jong-Sen Lee, Eric Pottier, Polarimetric Radar Imaging, 2017
This term is real, so the argument of 〈SRRSLL*〉 is zero. Consequently, the phase difference between SHH and SVV does not cause errors in the estimation of orientation angles. The factor of 4θ in Equation 10.8 limits the range of θ to [–π/4, π/4], because arctangent is computed in the range of (–π, π).
Math Review
Published in W. David Yates, Safety Professional’s Reference and Study Guide, 2020
As mentioned previously, the arcsine, arccosine, and arctangent are all inverse functions of sine, cosine, and tangent, respectively. Mathematically, they can be written as follows: Arcsine: arcsin = sin−1Arccosine: arccos = cos−1Arctangent: arctan = tan−1
Variable-parameter double-power reaching law sliding mode control method
Published in Automatika, 2020
Zhongjian Kang, Hongguo Yu, Changchao Li
The double-power reaching law with variable- parameters power term function is redesigned as follows: The power term in law (20) has the following characteristics: Because the definition domain of arctangent function is all real numbers, and it increases monotonously in (-∞, +∞) interval. So the power term increases monotonously in (-∞, +∞) interval.When s>0, , and when s<0, , so the power term is constant greater than or equal to 0.Because the power term , and when |s|>1, , when 0<|s|<1, , so the power function can adjust the power parameters adaptively under different conditions of the system. And the reaching law (20) has fast reaching speed both far away from and near sliding mode.
Teaching demonstration of the integral calculus
Published in International Journal of Mathematical Education in Science and Technology, 2020
Richard D. Sauerheber, Brandon Muñoz
Any point on an integral function has a slope in the domain between −∞ and +∞. In general the filament areas are all given by where tan(θ) is the slope of the integral at any position which makes an angle from the horizontal axis q, and tan(θ) = f(x). As an example for any position where the angle made by the tangent is π/4, then the slope is tan(π/4) = 1, and the dA filament area is 1dx. For a vertical asymptote where the slope is tan(π/2) = ∞, dx = 0 since vertically stacked points are parallel to the Y axis and have no horizontal distance between positions of contact with adjacent points. Here f(x)dx = 0 and no area is traced by the derivative from the X axis because there is no filament width that deviates from the vertical line. For any integral position where its tangent slope is zero, tan(0) = 0, the derivative f(x) is zero and f(x)dx = 0 with again no area traced by the function away from the horizontal axis. All other finite slope values for tangents along an integral produce a filament area traced out by its derivative. Notice that the arctangent of a derivative value at any position x is the angle θ made by its corresponding integral at that position, or tan−1[f(x)] = θ. For example tan−1(1) = π/4 and tan−1(0) = 0, and indeed the angles made by −cos(x) from the horizontal are π/4 and 0 when its derivative sin(x) is 1 or 0, respectively. Most important, this is true for any smooth continuous mathematical function.