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NR Modulation 5

2022-07-23 13:59:00 【Bai Xiaosheng in Ming Dynasty】

Reference resources

    Professor David S. Ricketts

Catalog

  1.     Quadrature Amplitude Modulation
  2.     cos vs sin 

One    Quadrature Amplitude Modulation

      1.1 advantage ： Improved bandwidth efficiency

      1.2  Influencing factors ： 

              Noise resistance   as well as power

       

    1.3  Constellations

      There are many kinds of constellations

      It mainly depends on

1： The minimum distance between any two points

            Affect the anti noise performance , The smaller the distance, the easier it is to make mistakes

2： Maximum distance

            Influence power , The greater the power consumption, the higher

 1.4 Example

          as follows 16-PSK  16-QAM

    One symbols It's all transmission 4bit, Consistent bandwidth efficiency

    however 16-PSK Higher power consumption . In practice , No one is better , Such as effective hardware limitations ,

In high frequency signal ,AM The modulation efficiency is very poor , At this time 16-PSK than

16-QAM Better

     

Two   cos vs sin

  2.1 cos w_c t   I

Theorem 1：  Important properties of Dirac function

 \int_{-\infty}^{\infty}\delta(x-x_0)f(x)dx=f(x_0)  

Theorem 2： Definition of inverse Fourier transform

f(t)=\frac{1}{2\pi}\int F(w)e^{jwt}dw

  According to the theorem 1,2 Yes  2\pi \delta (w-w_c)  Do the inverse Fourier transform

  \frac{1}{2\pi}\int_{-\infty}^{\infty} 2\pi \delta (w-w_c) e^{jwt}dw=\frac{2\pi}{2\pi}e^{jw_c}=e^{jw_ct}（t Is a constant ）

be

       F(e^{jw_c t})=2\pi \delta (w-w_c)

because

   cos w_c t=\frac{e^{j w_c t}+e^{-j w_c t}}{2}

  =\frac{2\pi \delta (w-w_c)+2\pi \delta (w+w_c)}{2}

  =\pi(\delta (w-w_c)+\delta (w+w_c))

  Multiply the time domain by one cos Function is essentially equivalent to convolution in frequency domain , Convolution uses the properties of Dirac function

 

 

  Use the above properties   Assume that the input signal is f(w), Then the last output signal s(w) Namely , What you get is the last picture

    f(w)*\pi(\delta(w-w_{c_2})+\delta(w-w_{c_2}))=\pi(f(w-w_{c_2})+f(w+w_{c_2}))

 2.2 sin w_c t Q

    Actually, it's almost , One more. j

   

 

 2.3 quadrature modulation

   

    Finally, take a look at IQ modulation , All the way with cos Convolution , Follow all the way sin Convolution

The convoluted envelope is shown in the right figure .

3、 ... and cos vs sin

    The signal we originally sent , It's transformed into 4 Share copy.

The repeated part can be filtered out by a filter , Reduce power .

   3.1 cos vs sin

       cos wt=\frac{e^{jwt}+e^{-jwt}}{2}
      sin(wt)=cos(wt-\frac{\pi}{2})

              =\frac{e^{j(wt-\frac{\pi}{2})}+e^{j(wt+\frac{\pi}{2})}}{2}

            =\frac{-je^{jwt}+je^{-jwt}}{2}

          =\frac{e^{jwt}-e^{-jwt}}{2j}

3.2 Look again. sinw_ct

 <0 Part is equivalent to multiplying by one j, Did a flip

 >0 Multiply part by one -j, remain unchanged

 

 

 

 

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