# 匹配濾波器

1.傳送的訊號

2.訊號的同步

## 最高SNR證明

w(t)：可加性高斯白雜訊

x(t) = g(t) + w(t)

h(t)：未知波形

y(t)：解調結果

$1. x(t) = g(t) + w(t)$

$2. y(t) = [g(t) + w(t)] \ast h(t)$

$= g(t) \ast h(t)+ w(t) \ast h(t)$

$= G(t) + N(t)$

$3. SNR = |G(T)|^2 / E[N^2(T)]|$

SNR = 信號瞬間功率 / Noise平均功率

$|G(T)|^2 = \int_{-\infty}^{\infty} H(f) G(f) e^{j2\pi fT} \, df$

$E[N^2 (T)]= \frac{N_0}{2} \int_{-\infty}^{\infty} |H(f)|^2 \, df$

$SNR = \frac{\int_{-\infty}^{\infty} H(f) G(f) e^{j2\pi fT} \, df}{\frac{N_0}{2} \int_{-\infty}^{\infty} |H(f)|^2 \, df}$

$\le \frac{\int_{-\infty}^{\infty} |H(f)|^2 e^{j2\pi fT} \, df \int_{-\infty}^{\infty} |G(f) e^{j2\pi fT}|^2 \, df }{\frac{N_0}{2} \int_{-\infty}^{\infty} |H(f)|^2 \, df}$

$= \frac{2}{N_0} \int_{-\infty}^{\infty} |G(f)|^2 \, df$

4. 當

$H_{opt}(f) = k[G(f)e^{j2\pi fT}]^*$ , $SNR_{max} = \frac{2} {N_0} \int_{-\infty}^{\infty} |G(f)|^2 \, df$

$h_{opt}(t) = k \int_{-\infty}^{\infty} G(-f)e^{-j2\pi fT} e^{j2\pi ft} \, df$

$= k \int_{-\infty}^{\infty} G(z)e^{-j2\pi f(T-t)} \, dz$

$= kg(T-t)$

(備註)Cauchy-Schwartz inequality:

$\int_{-\infty}^{\infty} |A(x)|^2 \, dx < \infty$$\int_{-\infty}^{\infty} |B(x)|^2 \, dx < \infty$

$|\int_{-\infty}^{\infty} A(x)B(x) \, dx|^2 \le \int_{-\infty}^{\infty} |A(x)|^2 \, dx \int_{-\infty}^{\infty} |B(x)|^2 \, dx$

$A=kB^*$時，等號成立。

## 匹配濾波器頻率響應

$\ x = s + v,\,$

$\ R_v = E\{vv^\mathrm{H}\}.\,$

$\mathrm{SNR} = \frac{|y_s|^2}{ E\{|y_v|^2\} }.$

$\ |y_s|^2 = {y_s}^\mathrm{H} y_s = h^\mathrm{H} s s^\mathrm{H} h.\,$

$\ E\{|y_v|^2\} = E\{ {y_v}^\mathrm{H} y_v \} = E\{ h^\mathrm{H} v v^\mathrm{H} h \} = h^\mathrm{H} R_v h.\,$

$\mathrm{SNR} = \frac{h^\mathrm{H} s s^\mathrm{H} h}{ h^\mathrm{H} R_v h }.$

$\ h^\mathrm{H} R_v h = 1$
$\ \mathcal{L} = h^\mathrm{H} s s^\mathrm{H} h + \lambda (1 - h^\mathrm{H} R_v h )$
$\ \nabla_{h^*} \mathcal{L} = s s^\mathrm{H} h - \lambda R_v h = 0$
$\ (s s^\mathrm{H}) h = \lambda R_v h$
$\ h^\mathrm{H} (s s^\mathrm{H}) h = \lambda h^\mathrm{H} R_v h.$

$\ \lambda_{\max} = s^\mathrm{H} R_v^{-1} s,$
$\ h = \frac{1}{\sqrt{s^\mathrm{H} R_v^{-1} s}} R_v^{-1} s.$

1.author = Haykin,S. / Moher,M. | title = Haykin: Communication Systems 5/E | language = 中文