치질에 관한 고찰 치질에 관한 고찰

Bandpass and Lowpass Signals Definition. A bandpass signal is a real signal whose frequency content is located around some non-zero frequency $f_c$. $$ X(f) = \begin{cases} \text{nonzero}, & \left[ f_c - \frac{W}{2}, f_c+\frac{W}{2} \right] \\ 0, & {\text{elsewhere}} \end{cases}$$ Then, the spectrum of a bandpass signal is composed of two components such that $$ \begin{align*} \text{positive spe..

$$ E_r(f,t) = \frac{a}{r_o +vt} \cos{2 \pi f (t- \frac{vt}{c} - \frac{r_0}{c})} - \frac{a}{2d-r_o-vt}\cos{2\pi f(t+\frac{vt}{c}+\frac{r_o -2d}{c})} $$ Assuming that the (mobile) receiving antenna is much closer to the wall than to the transmit antenna, the attenuation of the second term can be approximated by $2d-r_o-vt \approx r(t)=r_o+vt$. Then, we can rewrite the electric far field $$ \begin{..

Definition. A bandpass waveform has a spectral magnitude that is nonzero for frequencies in some band concentrated about a frequency $f=\pm f_c,$ where $f_c \gg 0.$ Suppose that $x_p(t)$ is an "ideal" bandpass signal. $x_p(t)$ can be written as $$ x_p(t)=A(t)\cos{(2 \pi f_ct + \phi (t) )}, \tag{1}$$ where $A(t)$ is an magnitude function and $\phi(t)$ is a phase function. (1) can be rearranged in..

Inner Product Definition. The inner product of two vectors $\underline{u}$ and $\underline{v}$ is defined such that $$( \underline{u}, \underline{v} ) = u_1\overline{v_1} + u_2\overline{v_2} + \cdots + u_n\overline{v_n}. \tag{1}$$ Definition. The inner product of two functions $f$ and $g$ that are continuous on an interval $[a,b]$ is defined by the formula analogous to Eq. (1) $$ (f,g) = \int_a^..

Three categories of convergence tests (i) sufficient condition $$ \text{If } C \text{ is satisfied, then } \sum a_n \text{ converges.}$$ (ii) necessary condition $$ \text{If } \sum a_n \text{ converges, then } C \text{ is satisfied.} $$ (iii) necessary and sufficient condition $$ \sum a_n \text{ converges if and only if } C \text{ is satisfied,} $$ where $C$ stands for the condition in question...

Nonhomogeneous lineqr equations of second order with constant coefficients Consider a nonhomgeneous equation of the form $$L(y)=y''+ay'+by=R, \tag{1}$$ where the coeffieicnts $a$ and $b$ are constants, but $R$ is any function continuous on $(-\infty,\infty)$. The general solution of the corresponding homogeneous equation $L(y)=0$ has the form $y_h=c_1v_1+c_2v_2,$ where $$v_1(x)=e^{-ax/2}u_1(x), ..

A differential equation of the form $$y'+P(x)y=Q(x), \tag{1}$$ where $P$ and $Q$ are given functions, is called a first-order linear differential equation. The equation $$y'+P(x)y=0$$ is called the homogeneous or reduced equation corresponding to (1). Theorem Assume $P$ is continuous on an open interval $I.$ Choose any point $a$ in $I$ and let $b$ be any real number. Then there is one and only o..

Taylor's Formula with the Integral (Cauchy) Form of the Remainder Theorem Assume $f$ has a continuous derivative of order $n+1$ in some interval containing $a$. Then, for every $x$ in this interval, we have the Taylor formula $$f(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k+E_n(x),$$ where $$E_n(x)=\frac{1}{n!}\int_{a}^{x} (x-t)^nf^{(n+1)}(t)dt.$$ Here, $E_n(x)$ may indicate the error involved ..