Random modulation

<h2 id="details">Details</h2>
<p>The random modulation procedure starts with two stochastic <a href="/facts/Baseband_signal/yC0jJwbL">baseband signals</a>, 
  
    
      
        
          x
          
            c
          
        
        (
        t
        )
      
    
    {\displaystyle x_{c}(t)}
  
 and 
  
    
      
        
          x
          
            s
          
        
        (
        t
        )
      
    
    {\displaystyle x_{s}(t)}
  
, whose <a href="/facts/Frequency_spectrum/q3Rfs7b0">frequency spectrum</a> is non-zero only for 
  
    
      
        f
        ∈
        [
        −
        B
        
          /
        
        2
        ,
        B
        
          /
        
        2
        ]
      
    
    {\displaystyle f\in [-B/2,B/2]}
  
. It applies quadrature modulation to combine these with a carrier frequency 
  
    
      
        
          f
          
            0
          
        
      
    
    {\displaystyle f_{0}}
  
 (with 
  
    
      
        
          f
          
            0
          
        
        >
        B
        
          /
        
        2
      
    
    {\displaystyle f_{0}>B/2}
  
) to form the signal 
  
    
      
        x
        (
        t
        )
      
    
    {\displaystyle x(t)}
  
 given by
</p>

x
        (
        t
        )
        =
        
          x
          
            c
          
        
        (
        t
        )
        cos
        ⁡
        (
        2
        π
        
          f
          
            0
          
        
        t
        )
        −
        
          x
          
            s
          
        
        (
        t
        )
        sin
        ⁡
        (
        2
        π
        
          f
          
            0
          
        
        t
        )
        =
        ℜ
        
          {
          
            
              
                x
                _
              
            
            (
            t
            )
            
              e
              
                j
                2
                π
                
                  f
                  
                    0
                  
                
                t
              
            
          
          }
        
        ,
      
    
    {\displaystyle x(t)=x_{c}(t)\cos(2\pi f_{0}t)-x_{s}(t)\sin(2\pi f_{0}t)=\Re \left\{{\underline {x}}(t)e^{j2\pi f_{0}t}\right\},}

<p>where 
  
    
      
        
          
            x
            _
          
        
        (
        t
        )
      
    
    {\displaystyle {\underline {x}}(t)}
  
 is the <a href="/facts/Baseband_signal/yC0jJwbL">equivalent baseband representation</a> of the modulated signal 
  
    
      
        x
        (
        t
        )
      
    
    {\displaystyle x(t)}

</p>

x
            _
          
        
        (
        t
        )
        =
        
          x
          
            c
          
        
        (
        t
        )
        +
        j
        
          x
          
            s
          
        
        (
        t
        )
        .
      
    
    {\displaystyle {\underline {x}}(t)=x_{c}(t)+jx_{s}(t).}

<p>In the following it is assumed that 
  
    
      
        
          x
          
            c
          
        
        (
        t
        )
      
    
    {\displaystyle x_{c}(t)}
  
 and 
  
    
      
        
          x
          
            s
          
        
        (
        t
        )
      
    
    {\displaystyle x_{s}(t)}
  
 are two real jointly <a href="/facts/Wide_sense_stationary/vozdgfEX">wide sense stationary</a> processes. It can be shown that the new signal 
  
    
      
        x
        (
        t
        )
      
    
    {\displaystyle x(t)}
  
 is wide sense stationary <a href="/facts/Iff/bYSxGJ66">if and only if</a> 
  
    
      
        
          
            x
            _
          
        
        (
        t
        )
      
    
    {\displaystyle {\underline {x}}(t)}
  
 is circular complex, i.e. if and only if 
  
    
      
        
          x
          
            c
          
        
        (
        t
        )
      
    
    {\displaystyle x_{c}(t)}
  
 and 
  
    
      
        
          x
          
            s
          
        
        (
        t
        )
      
    
    {\displaystyle x_{s}(t)}
  
 are such that
</p>

R
          
            
              x
              
                c
              
            
            
              x
              
                c
              
            
          
        
        (
        τ
        )
        =
        
          R
          
            
              x
              
                s
              
            
            
              x
              
                s
              
            
          
        
        (
        τ
        )
        
        
          and 
        
        
        
          R
          
            
              x
              
                c
              
            
            
              x
              
                s
              
            
          
        
        (
        τ
        )
        =
        −
        
          R
          
            
              x
              
                s
              
            
            
              x
              
                c
              
            
          
        
        (
        τ
        )
        .
      
    
    {\displaystyle R_{x_{c}x_{c}}(\tau )=R_{x_{s}x_{s}}(\tau )\qquad {\text{and  }}\qquad R_{x_{c}x_{s}}(\tau )=-R_{x_{s}x_{c}}(\tau ).}

<h2 id="bibliography">Bibliography</h2>
<ul><li><a href="/facts/Athanasios_Papoulis/0rWhyfS4">Papoulis, Athanasios</a>; Pillai, S. Unnikrishna (2002). "Random walks and other applications". <i>Probability, random variables and stochastic processes</i> (4th ed.). McGraw-Hill Higher Education. pp. 463–473.</li>
<li>Scarano, Gaetano (2009). <i>Segnali, Processi Aleatori, Stima</i> (in Italian). Centro Stampa d'Ateneo.</li>
<li>Papoulis, A. (1983). "Random modulation: A review". <i>IEEE Transactions on Acoustics, Speech, and Signal Processing</i>. 31: 96–105. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1109%2FTASSP.1983.1164046">10.1109/TASSP.1983.1164046</a>.</li></ul>

Random modulation open-in-new

Random modulation