Consider a signal that is a sum of sinusoids, e.g. x ( t ) = A s i n ( a t ) + B c o s ( b t )Is there an easy and general way to get an analytical solution for the autocorrelation of x(t)?Is the best way to simply plug x(t) into the autocorrelation formula?

Question

Consider a signal that is a sum of sinusoids, e.g.  x  (  t  )  =  A  s  i  n  (  a  t  )  +  B  c  o  s  (  b  t  )Is there an easy and general way to get an analytical solution for the autocorrelation of x(t)?Is the best way to simply plug x(t) into the autocorrelation formula?

Aaron Everett · Accepted Answer

Step 1Consider a general sum of sinusoids  x  (  t  )  =      ∑    i        C    i        f    i    (      a    i    t  )where       f    i    (      a    i    t  )  =  s  i  n  (      a    i    t  ) or       f    i    (      a    i    t  )  =  c  o  s  (      a    i    t  ) . The coefficients       C    i   are simply constants. For example this sum could take the form   x  (  t  )  =  2  c  o  s  (  2  t  )  −  3  c  o  s  (  3  t  )  +  4  c  o  s  (  4  t  )  −  5  s  i  n  (  5  t  ) .Now, substitute this sum into the autocorrelation formula:      R          x      x        (  τ  )  =      ∫          −      ∞              ∞        x  (  t  +  τ  )  x  (  t  )  d  tAnd realize that each term in the integrand can be one of the following four cases:      ∫          −      ∞              ∞            C    i        C    j    s  i  n  (      a    i    (  t  +  τ  )  )  s  i  n  (      a    j    t  )  d  t  =                    C        i                    C        j              2    c  o  s  (      a    i    τ  )      ∫          −      ∞              ∞            C    i        C    j    s  i  n  (      a    i    (  t  +  τ  )  )  c  o  s  (      a    j    t  )  d  t  =                    C        i                    C        j              2    s  i  n  (      a    i    τ  )      ∫          −      ∞              ∞            C    i        C    j    c  o  s  (      a    i    (  t  +  τ  )  )  s  i  n  (      a    j    t  )  d  t  =  −                    C        i                    C        j              2    s  i  n  (      a    i    τ  )      ∫          −      ∞              ∞            C    i        C    j    c  o  s  (      a    i    (  t  +  τ  )  )  c  o  s  (      a    j    t  )  d  t  =                    C        i                    C        j              2    c  o  s  (      a    i    τ  )So the general autocorrelation can be written as a double sum:      R          x      x        (  τ  )  =      1    2        ∑          i      j            C    i        C    j        r          i      j        (  τ  )where      r          i      j        (  τ  )  =      {                            c          o          s          (                      a            i                    τ          )                                      if                                           f                i                            (                              a                i                            t              )              =              s              i              n              (                              a                i                            t              )                         and                                           f                j                            (                              a                j                            t              )              =              s              i              n              (                              a                j                            t              )                                                                         c          o          s          (                      a            i                    τ          )                                      if                                           f                i                            (                              a                i                            t              )              =              c              o              s              (                              a                i                            t              )                         and                                           f                j                            (                              a                j                            t              )              =              c              o              s              (                              a                j                            t              )                                                                         s          i          n          (                      a            i                    τ          )                                      if                                           f                i                            (                              a                i                            t              )              =              s              i              n              (                              a                i                            t              )                         and                                           f                j                            (                              a                j                            t              )              =              c              o              s              (                              a                j                            t              )                                                                         −          s          i          n          (                      a            i                    τ          )                                      if                                           f                i                            (                              a                i                            t              )              =              c              o              s              (                              a                i                            t              )                         and                                           f                j                            (                              a                j                            t              )              =              s              i              n              (                              a                j                            t              )

Consider a signal that is a sum of sinusoids, e.g. x ( t ) = A s i n

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