Expression of directional signal -- complex exponential signal

Recently read literature , It is often seen that the following complex exponential form is used to represent a signal in a certain direction . The expression is very concise , But its physical meaning is not clear , I don't understand why I use it to express .

                                                    

  One 、 Complex exponential signal

In fact, complex exponential signals cannot be generated , That is it There is no real physical meaning .

But it can describe many types of signals , Such as ： DC signal 、 Exponential signals 、 Sine signal 、 Cosine signal 、 And increasing and decreasing sine and cosine signals .

 

              

              

  This represents one of the signals The advantage is , It can simplify the operation and analysis of signals .

Plural numbers can not only provide you Amplitude information , It can also be provided to you Phase information .

Two 、 The complex exponential form of the direction signal

The complex exponential form of the direction signal is shown as follows ：

                                          

The derivation process is shown in the following figure , The phase difference formula of the array signal is substituted into the complex exponent .