치질에 관한 고찰 치질에 관한 고찰

Fourier transform of the pdf $f_X(x)$ of a Gaussian r.v. $X \sim \mathcal{N}(0, \sigma_X^2)$ We first compute the Fourier transform of the Gaussian integral $$ g(x) = e^{-\pi x^2}. $$ We have $$ \mathcal{F}g(s) = \int_{-\infty}^\infty e^{-2\pi isx}e^{-\pi x^2}dx. $$ Differentiate both sides with respect to $s$: $$ \frac{d}{ds} \mathcal{F} g(s) = \int_{-\infty}^\infty e^{-2\pi isx}(-2\pi isx)e^{-..

Definition. A set of $n$ random r.v.'s, $Z_1, Z_2, \cdots , Z_n$ is zero-mean jointly Gaussian if there is a set of iid(independent and identically distributed) normal r.v.'s $W_1, \cdots , W_l$ such that each $Z_k, 1 \leq k \leq n,$ can be expressed as $$ Z_k = \sum_{m=1}^l a_{km}W_m, \hspace{3cm} 1 \leq k \leq n, \tag{1}$$ where ${a_{km}; 1 \leq k \leq n, 1 \leq m \leq l}$ is an array of real ..

Definition. A vector space $\mathcal{H}$ over $\mathbb{F}$ is an inner product space or pre Hilbert space if to every ordered pair $u,v \in \mathcal{H}$ there corresponds a scalar $(u,v) \in \mathbb{F}$ such that Case 1: $\mathbb{F} = \mathbb{C},$ Complex field $(u,v) = \overline{(u,v)} \hspace{1cm} [ \text{symmetry} ]$ $(u+v,w) = (u,w) + (v,w) \hspace{1cm} [ \text{linearity} ]$ $(\alpha u,v) = ..

Expectation The expectation of a CRV $Z$ can be defined in terms of the expectations of its real and imaginary parts: $$ E[Z] = E[\Re (Z)] +i E[ \Im (Z) ], \tag{1}$$ provided that the two real expectations $E[ \Re (Z)]$ and $E[ \Im (Z) ]$ are finite. Variance $$ \begin{align*} \text{Var}[Z] &= E[|Z-E[Z]|^2] \\ &= E[(Z-E[Z])(Z-E[Z])^*] \\ &= E[(Z-E[Z])(Z^*-E[Z^*])] \hspace{1cm} \left( E[Z^*] = (E..

Let $x_n(t)$ denote $x(t-nT)$. Show that if a family of waveforms ${x_n(t)}_n$ forms an orthogonal set, then the waveforms ${ \psi_{n,1} , \psi_{n,2} }_n$ also form an orthogonal set, provided $x(t)$ is band-limited to $[ -f_c, f_c]$. $\psi_{n,1}, \psi_{n,2}$ are defined as $$ \begin{align*} \psi_{n,1} &= x_n (t) \cos{2\pi f_ct} \\ \psi_{n,2} &= -x_n (t) \sin{2\pi f_ct} \end{align*} $$ By defini..

Shifting theorem $$ \begin{align*} \left( \mathcal{F} x(t-t_0) \right) (s) &= \left( \mathcal{F} x(t)*\delta(t-t_0) \right) (s) \\ &= \mathcal{F}x(s) \cdot \left( \mathcal{F} \delta(t-t_0) \right) (s) \\ &= e^{-2\pi i st_0} \mathcal{F}x(s) \end{align*} $$ $$ \therefore \left( \mathcal{F} x(t-t_0) \right) (s) = e^{-2\pi i st_0} \mathcal x(s) $$ Duality $$ \mathcal{F}f(-s) = \int_{-\infty}^\infty ..

Now remember that the general Parseval's identity holds for functions in $L^2(\mathbb{R})$ and says $$ \int_{-\infty}^\infty f(t) \overline{g(t)} dt = \int_{-\infty}^\infty \mathcal{F} f(s) \overline{\mathcal{F} g(s)} ds.$$ Then, for the shifted sincs we have $$ \begin{align*} \int_{-\infty}^\infty \text{sinc} (t-n) \text{sinc} (t-m) dt &= \int_{-\infty}^{\infty} \left( \mathcal{F} \text{sinc}(t..

Definition. The shah function or Dirac comb is defined as $$ \text{III}_T (x) = \text{comb}(t) := \sum_{k=-\infty}^{\infty} \delta(x-kT).$$ Graphically, this is an infinite train of $\delta$'s spaced $T$ apart. In the case where the spacing $T$ is 1, we have $$ \text{III}(x) = \sum_{k=\infty}^{\infty} \delta(x-k) = \sum_{k=\infty}^{\infty} \delta_k. $$ Periodizing $$ \begin{align*} \sum_{k=\inft..