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

Reed Eaton

Reed Eaton

Answered

2022-06-30

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?

Answer & Explanation

Aaron Everett

Aaron Everett

Expert

2022-07-01Added 18 answers

Step 1
Consider 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 t
And 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 )  

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