Multidimensional modulation

<p>Multidimensional modulation (MD modulation) is modifying or multiplying an MD signal (typically sinusoidal and referred to as the carrier signal) with another signal that carries some information or message. In the frequency domain, the signal is moved from one frequency to another.
</p><p>
if </p><table><tbody><tr><td>                    x        (                  n                      1                          ,        .        .        .        ,                  n                      M                          )                              ⟷                                                        F                T                                                    X        (                  ω                      1                          ,        .        .        .        ,                  ω                      M                          )              {\displaystyle x(n_{1},...,n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}X(\omega _{1},...,\omega _{M})}  </td> <td></td> <td>1</td></tr></tbody></table><p>
then </p><table><tbody><tr><td>                              e                      (            j                          θ                              1                                                    n                              1                                      +            ,            .            .            .            ,            +                          θ                              M                                                    n                              M                                      )                          x        (                  n                      1                          ,        .        .        .        ,                  n                      M                          )                              ⟷                                                        F                T                                                    X        (                  ω                      1                          −                  θ                      1                          ,        .        .        .        ,                  ω                      M                          −                  θ                      M                          )              {\displaystyle e^{(j\theta _{1}n_{1}+,...,+\theta _{M}n_{M})}x(n_{1},...,n_{M}){\overset {\underset {\mathrm {FT} }{}}{\longleftrightarrow }}X(\omega _{1}-\theta _{1},...,\omega _{M}-\theta _{M})}  </td> <td></td> <td>2</td></tr></tbody></table><p>Typically the carrier signal is a sinusoidal signal and in various applications.  The figures below illustrate a quick example of a 2-D modulation. The original signal from (3) is modulated with a sinusoidal signal to get (4). The equations (5) and (6) are the real and the imaginary components of the modulated signal.
</p>
<table><tbody><tr><td>                    x        (                  n                      1                          ,                  n                      2                          )                  =                cos        ⁡                  (          2          π                      n                          1                                +          4          π                      n                          2                                )                      {\displaystyle x(n_{1},n_{2}){=}\cos {(2\pi n_{1}+4\pi n_{2})}}  </td> <td></td> <td>3</td></tr></tbody></table>
<table><tbody><tr><td>                    y        (                  n                      1                          ,                  n                      2                          )                  =                          e                      (            j            π                          n                              1                                      +            π                          n                              2                                      )                          x        (                  n                      1                          ,                  n                      2                          )              {\displaystyle y(n_{1},n_{2}){=}e^{(j\pi n_{1}+\pi n_{2})}x(n_{1},n_{2})}  </td> <td></td> <td>4</td></tr></tbody></table>
<table><tbody><tr><td>                              R          e                {        y        (                  n                      1                          ,                  n                      2                          )        }                  =                cos        ⁡                  (          π                      n                          1                                +          π                      n                          2                                )                x        (                  n                      1                          ,                  n                      2                          )              {\displaystyle \mathbf {Re} \lbrace y(n_{1},n_{2})\rbrace {=}\cos {(\pi n_{1}+\pi n_{2})}x(n_{1},n_{2})}  </td> <td></td> <td>5</td></tr></tbody></table>
<table><tbody><tr><td>                              I          m                {        y        (                  n                      1                          ,                  n                      2                          )        }                  =                sin        ⁡                  (          π                      n                          1                                +          π                      n                          2                                )                x        (                  n                      1                          ,                  n                      2                          )              {\displaystyle \mathbf {Im} \lbrace y(n_{1},n_{2})\rbrace {=}\sin {(\pi n_{1}+\pi n_{2})}x(n_{1},n_{2})}  </td> <td></td> <td>6</td></tr></tbody></table>

Multidimensional modulation open-in-new

Multidimensional modulation