Rectangular function

The (unit) impulse function is called the Dirac (or simply delta) function in mathematics.1 If we suddenly dump a bucket of water into a bigger tank, the impulse function is how we describe the action mathematically. We can consider the impulse function as the unit rectangular function in Eq. (2-20) as T shrinks to zero while the height 1/T goes to infinity ... 

The Laplace transform of the impulse function is obtained easily by taking the limit of the unit rectangular function transform (2-20) with the use of L Hospital s rule ... 

In mathematics, the unit rectangular function is defined with a height of 1/2T and a width of 2T from -T to T. We simply begin at t = 0 in control problems. Furthermore, the impulse function is the time derivative of the unit step function. 

Consider the rectangular function shown in Fig. 1. It is a function that is equal to zero outside the region defined by the limits —i and +1. However, within this region it has a constant value, determined by the condition thatthe integral over the function is equal to unity. That is to say, the function has been normalized. The rectangular function PI(jc/21) is often referred to as the boxcar . 

Filter out any fast ripple of period Aw/2 in V(z), due to interference with internal reverberations in the lens (Fig. 8.5(b)). This may be achieved most simply by convolving with a rectangular function of length Aw/2. This is known as a moving average filter it is equivalent to a sine filter in the Fourier domain, but is computationally somewhat more efficient. Because of its period the ripple removed at this stage is sometimes called water ripple. 

The activation energy of the decay of P700+ in the range 80-160 K was found in Ref. [212] from the data of Fig. 28. Assuming that the distribution of pairs over the values of the parameter % at the initial time t = 0 is described by the rectangular function, it is easy to derive an equation for the ratio of the current concentration of pairs, n(t), to their concentration at time t0 [212] ... 

As for given conditions, the mobilities in Eq. (2) are constant under ITP conditions the concentration of the sample in its zone is constant as well. The concentration distribution is, therefore, given by a rectangular function. It follows that the temperature and the pH within the particular zone is constant, too, and changes stepwise at the boundary to the neighboring zone. 

Numerically the convolution of a step scan is merely the application of a sliding weighted mean (e.g. like the Savitzky-Golay method). The Fourier transform of the rectangular function has the shape of sin(nv)/(nv) (whereby n is inversely proportional to the width of the rectangle) and unfortunately approaches 0 only very slowly. To make do with a small number of points for a convolution, one must tolerate a compromise and renounce the ideal rectangular shape of the low pass filter (in the frequency domain). 

Function (e) was convolved with the receiving slit function representing the rectangular function. Figure 6.14 shows axial instrumental functions. The results are consistent with our model. 

Fig. 5.3.5 [Cal2] The sine pulse, (a) The amplitude of the rf carrier is modulated by a truncated sine function, (b) The magnitude of the Fourier transform of the pulse is a rectangular function distorted by wiggles in the centre and near the edges.

The averaging of the neighbouring time-domain data points in Eq. (24) is equivalent to convolution with a rectangular function. The width of the rectangle corresponds to the number of points that are averaged. The... 

One must stress that in the discussion above in Section 5.1 and here, we have used the "rectangular" functions W(r) and w(r) of opposite sign. When these functions are constructed for the same region, one can see that W = 1 + w. It means that the difference in qualitative pictures described for different cases is associated with regions of localization of the external potentials, but not with their sign only, as it was proposed in [79]. 

Fig. 7.5-5 Shifting the center of the window function (here a nine-point rectangular function) off-center causes a time shift of the resulting convolution (solid black circles).

In general the sensing unit contains a flow cell with a certain volume v. This volume causes an increase in the variance according to eqn. 7. The pulse response of a detector cell is, in the ideal case (no mixing phenomena), a rectangular function of the width The variance of the total detection unit... 

Neglecting mixing phenomena, the impulse response of the detector is described by a rectangular function of the width, and variance, Aa ... 

The problem with using a rectangular function to truncate the interferogram is the production of the ringing or "feet" around the infrared absorption or emission band. This ringing produces an error in the spectrum it affects the absolute absorption intensity of the band itself, and the feet may interfere with the absorption intensity of any closely spaced band. 

A rectangular function centered at x = 0, with the rectangle surface equal to 1 (a 0) ... 

For a bandpass filter with bandwidth B, and for 0 noise power is the area under the power spectral density curve enclosed by B (see Figs. 7,6-8), which we choose to be rectangular for simplicity. Although strictly speaking, the rectangular function (as well as A f) is not realizable, this is not critical since it is the integrated area under the curve which is important rather than the detailed shape. The result is... 

F. Attraction force between the sample and the tip modeled by a rectangular function... 

---