1.tex

\subsection*{Linearity and Time Invariance; Time-Domain Representations}

$ y_n = b_0 x_n + b_1 x_n +  \cdots + b_Mx_{n*m}
    - a_1 y_{n-1} \cdots -a_n y_{n-N}$

\subsection*{Recall the simplest lowpass filter}
$ y_n = x_n + x_{n - 1} 
\\
            (M=1, N =0) \\
            (b_0 = 1, b_1 = 1)$

(FIR, order 1) 

$h_n = \delta_n + \delta_{n - 1} = [1, 1, 0, \dots]$\\

freq response:\\
$H(w) = \mbox{DTFT}(h) = 1 + e^{-jwT}$\\
phase response:\\
$\Theta(w) = \angle H(w) = -wt/2  = -\pi f T$ \\
phase delay: \\
$P(\omega) = -\theta(\omega)/N = + T / 2$ \\
group delay: \\
    $D(\omega) = -2 / ??? \theta(\omega) = T / 2$ \\

phase delay and group delay are a 1/2 sample delay 

These characteristics all apply to a Linear Time Invariant (LTI) filter

\subsection*{Recall Test Sinusoid}

$x(n) = e^{j\omega t_n} | \omega = 2 \pi fs / 4 = 2 \pi / 4T \\
                        => \omega T = \pi/2 \\
                        => x(n) = e^{j\omega n T} = e^{j (\pi/2) n} = j^n$\\

now, \\
$y(n) = x(n) + x(n -1)\\
          = j^n + j^{n - 1} = j^n(1 + 1/j)\\
          = (1 - j)j^n = (1 - j)x(n)$


$=> G(\pi / 2) \stackrel{?}{=} \sqrt{1^2 + 1^2} = \sqrt{2} (check)\\
T = 1$


$= 2\cos(\omega T/2) | \omega T = \pi / 2 = 2 \cos(\pi / 4) = 2 / \sqrt{2} = \sqrt{2}$


$\theta(\pi / 2) ?= \tan^{-1}(-1 / 1) = - \pi / 4 \\
= -\omega T / 2 = -\pi/2 1/2 = - \pi / 2$


Thus, $H(\pi / 2) = e^{-j \pi/4} \sqrt{2} \\$
$
= [\cos(\pi / 4) - j \sin(\pi / 4) ] \sqrt{2} \\
= (1 - j)
$\\

$y(n) = j^n + j^{n - 1}$\\
$y(1) = j + j^0 = j + 1 = \sqrt{2} \cdot e^{j(pi/2 - pi/4)}$\\
$y(3) = j^2 + j^2 = -j - 1 = \sqrt{2} \cdot e^{3\pi / 2 = \pi / 4}$\\

\subsection*{Phase Delay: $\Theta(\omega) = \mbox{phase shift (rad)}$}


$P(w) \stackrel{\Delta}{=}  - \theta(\omega) / w = - \mbox{phase shift} / \mbox{(rad / sec)} \\
    = -\theta / 2 \pi 4 = -(\theta / 2 \pi) \mbox{cycle} \cdot \mbox{Period}$

phase shift gives you a fraction of a period

\subsection*{Group Delay}

$D(\omega) \stackrel{\Delta}{=} -d/d\omega \theta(\omega)$

Measure of delay that is local to each frequency zone. 
If the phase is going crazy, you just pick a particular frequency and linearize
at that point.
\\\\
Group delay is phase delay at a particular freq\\
Slope of the linearized delay