GMSK process

2024-07-20 00:30:01
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Gaussian filter：

1. continuous-time impulse response of the Gaussian filter:

$h(t) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(t )^2}{2\sigma^2}}\textup{}$

2. the frequency respone can be shown as,

$H(f) = \frac{1}{\sigma\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-\frac{t^2}{2\sigma^2}} e^{-j2\pi ft} dt$

cause

$-\frac{t^2}{2\sigma^2} - j2\pi ft = -\frac{1}{2\sigma^2} \left(t + j{2\pi\sigma^2 f}\right)^2 - 2\pi^2 f^2 \sigma^2$

then

$H(f) = e^{-2\pi^2 f^2 \sigma^2} \frac{1}{\sigma\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-\frac{1}{2\sigma^2} (t +j2\pi f\sigma^2)^2} dt$

get

$H(f) = e^{-2 \pi^2 f^2 \sigma^2}$

the 3dB bandwidth should equal to the bandwith of singal to satisfy the requirement of filter design, so,

$H(f)_{max}= 1\rightarrow H(B) = e^{-2 \pi^2 (B)^2 \sigma^2} = \sqrt{\frac{1}{2}}(3.01dB)$

then

$\sigma^2 = \frac{ln2}{4\pi^2B^2}$

GMSK

the GMSK process:

the sequence of M-ary data symbols is shaped with Gaussian filter, the symbol $rect(\frac{t}{Tb})$ do convolution with h(t), shown as,

$q(t)=\int_{-\infty}^{\infty} h(\tau ) \cdot \text{rect}\left(\frac{t-\tau}{T_b}\right) \, d\tau$

$=\int_{t-\frac{T_b}{2}}^{t+\frac{T_b}{2}} \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(\tau)^2}{2\sigma^2}}\textup{} d\tau$ $=\int_{\frac{(t-\frac{T_b}{2})}{\sigma}}^{\frac{(t+\frac{T_b}{2})}{\sigma}} \frac{1}{\sqrt{2\pi}} e^{-\frac{(u)^2}{2}}\textup{} du$

the CDF function is shown as,

$\Phi(z) = \int_{-\infty}^{z} \frac{1}{\sqrt{2\pi}} e^{-\frac{u^2}{2}} du$

so the phase response can be present as,

$q(t)= \Phi\left(\frac{t+\frac{T_b}{2}}{\sigma}\right) - \Phi\left(\frac{t-\frac{T_b}{2}}{\sigma}\right)$

$=\Phi\left(\frac{2\pi B(t+\frac{T_b}{2}))}{\sqrt{\ln(2)}}\right) - \Phi\left(\frac{2\pi B(t-\frac{T_b}{2}))}{\sqrt{\ln(2)}}\right)$

GMSK is kind of Continuous-phase modulation(CPM), to make the phase continuity, the q(t) should be normalized according to below.

$\beta(q((n+1)T_b)-q(nT_b))= 0.5$

$\bar{q}(t)= q(t) * \beta$

In CPM, the baseband representation of the modulated signal is

$s(t) = \exp\left[j2\pi \left(\sum_{i=0}^{n} \alpha_i h \bar{q}(t-iT_b)\right)\right], \quad \text{for } nT_b < t < (n+1)T_b$

h is modulation index, for the MSK the h = 0.5 that make the minum frequency offset to keep the carrier orthogonal.

原文地址:https://blog.csdn.net/qq_27837513/article/details/140380801 本文来自互联网用户投稿，该文观点仅代表作者本人，不代表本站立场。本站仅提供信息存储空间服务，不拥有所有权，不承担相关法律责任。如若转载，请注明出处：https://www.suanlizi.com/kf/1814336904745848832.html 如若内容造成侵权/违法违规/事实不符，请联系《酸梨子》网邮箱：1419361763@qq.com进行投诉反馈，一经查实，立即删除！

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GMSK process

Gaussian filter：

GMSK

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