推荐|复信号频谱

复信号频谱

23-11-03 19:41

2021

9264

blog.csdn.net

为什么正交采样（复采样）的采样率最低为信号带宽B，就可保证采样信号信息不丢失？从复频率域角度出发：
正交采样（复频率域角度）

回顾一下：
对于实数信号，其频谱有共轭对称性，正负频率实部为偶函数，虚部为奇函数，所以它们可以相互决定对方，正频率和负频率所承载的信息是一样的。频谱搬移后，其双边频谱承载相同的信息，浪费频谱资源。

对于基带信号，我们可以采用复信号来提高频谱利用率。复信号可以具有任意频谱结构，复信号谱一般不是对称的，频率为+fo 和-fo 含义不同，正负号表示圆周运动的方向，正为逆时针，负为顺时针。

复频率域典型实信号与复信号频谱如图1所示：
在这里插入图片描述
根据欧拉公式得
$f_{0}t )=frac{e^{j2pi f_{0}t} }{2} +frac{e^{-j2pi f_{0}t} }{2}和sin(2pi f_{0}t )= jfrac{e^{-j2pi f_{0}t} }{2}-jfrac{e^{j2pi f_{0}t} }{2}$

如图2所示，将通过正弦和余弦得到的复指数频谱带入，复频率域图谱和上图一致。
在这里插入图片描述
在复频率域中一般实信号频谱：
$f_{0}t +phi )=frac{e^{j2pi f_{0}t} e^{jphi } }{2} +frac{e^{-j2pi f_{0}t}e^{-jphi } }{2}$
从公式可看出，实信号有初相ϕ，其复频率域频谱中对正频率分量逆时针旋转ϕ度，对负频率分量顺时针旋转ϕ度，如下图3所示
在这里插入图片描述
以此类推，一般实信号频谱如下图4所示

上图1图2图3图4来自文章 A Quadrature Signals Tutorial: Complex, But Not Complicated ，复数的基础知识也可参见该文章。

典型复信号?(?)+j?(?)复频率域频谱如下
在这里插入图片描述
复信号?(?)+j?(?)频谱幅度如下

?(?)和?(?)都是实信号，相对应的正负频率实部为偶函数，虚部为奇函数，频谱不对称的复信号是怎么通过实信号得到的？

在这里插入图片描述
红色为实信号?(?)频谱，正负频率分别逆时针和顺时针旋转ϕ度，灰色为实信号?(?)频谱，选取一个简单的，在实轴与频率平面上，不旋转。黄色为 j?(?) 信号频谱，j将?(?)信号频谱整体逆时针旋转90度。

复信号?(?)+j?(?) 信号频谱为?(?)信号频谱和j?(?)信号频谱相加，复数相加满足平行四边形法则，选取两个参考点频谱结构如下
在这里插入图片描述
取模后可知相对应的正负频率处大小不同，复信号谱可以是不对称的，通过不同的实信号?(?)和?(?) ，复信号可以具有任意频谱结构。

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?(?)和?(?)都是实信号，相对应的正负频率实部为偶函数，虚部为奇函数，频谱不对称的复信号是怎么通过实信号得到的？

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