Method and apparatus to measure non-coherent signals

A method of measuring a non-coherent electrical signal, said electrical signal including one or more sine and/or cosine components at n known signal frequencies Ft1, Ft2, . . . Ftn, comprising the steps of:

• providing a digitized version of said non-coherent electrical signal, said digitized version represented by the notation x(i);
• selecting a set of n correlation frequencies Fc1, Fc2, . . . Fcn, said correlation frequencies being substantially close to said signal frequencies Ft1, Ft2, . . . Ftn, DC offset represented by a cosine component at a frequency of 0;
• computing a 2n×2n matrix A of discrete time correlations between sampled unit amplitude sine and cosine waveforms at each of said correlation frequencies and sampled unit amplitude sine and cosine waveforms at each of said signal frequencies, said matrix defined by the equation:
• A := ( Kss ⁡ ( 1 , 1 ) Ksc ⁡ ( 1 , 1 ) Kss ⁡ ( 1 , 2 ) Ksc ⁡ ( 1 , 2 ) … Kss ⁡ ( 1 , n ) Ksc ⁡ ( 1 , n ) Kcs ⁡ ( 1 , 1 ) Kcc ⁡ ( 1 , 1 ) Kcs ⁡ ( 1 , 2 ) Kcc ⁡ ( 1 , 2 ) … Kcs ⁡ ( 1 , n ) Kcc ⁡ ( 1 , n ) Kss ⁡ ( 2 , 1 ) Ksc ⁡ ( 2 , 1 ) Kss ⁡ ( 2 , 2 ) Ksc ⁡ ( 2 , 2 ) … Kss ⁡ ( 2 , n ) Ksc ⁡ ( 2 , n ) Kcs ⁡ ( 2 , 1 ) Kcc ⁡ ( 2 , 1 ) Kcs ⁡ ( 2 , 2 ) Kcc ⁡ ( 2 , 2 ) … Kcs ⁡ ( 2 , n ) Kcc ⁡ ( 2 , n ) … … … … … … … Kss ⁡ ( n , 1 ) Ksc ⁡ ( n , 1 ) Kss ⁡ ( n , 2 ) Ksc ⁡ ( n , 2 ) … Kss ⁡ ( n , n ) Ksc ⁡ ( 2 , n ) Kcs ⁡ ( n , 1 ) Kcc ⁡ ( n , 1 ) Kcs ⁡ ( n , 2 ) Kcc ⁡ ( n , 2 ) … Kcs ⁡ ( n , n ) Kcc ⁡ ( n , n ) )
• said discrete time correlations defined by the equations:
• Kss ⁡ ( a , b ) := ∑ i ⁢ sin ⁡ ( 2 · π · Fc a Ff · i N ) · sin ⁡ ( 2 · π · Ft b Ff · i N ) Ksc ⁡ ( a , b ) := ∑ i ⁢ sin ⁡ ( 2 · π · Fc a Ff · i N ) · cos ⁡ ( 2 · π · Ft b Ff · i N ) ⁢ ⁢ Kcs ⁡ ( a , b ) := ∑ i ⁢ cos ⁡ ( 2 · π · Fc a Ff · i N ) · sin ⁡ ( 2 · π · Ft b Ff · i N ) ⁢ ⁢ Kcc ⁡ ( a , b ) := ∑ i ⁢ cos ⁡ ( 2 · π · Fc a Ff · i N ) · cos ⁡ ( 2 · π · Ft b Ff · i N )
• where N is equal to the number of samples collected in said digitized non-coherent electrical signal and i ranges from 0 to N−1;
• computing the inverse of said correlation matrix, said inverse matrix represented by the notation A−1;
• computing a 1×n matrix of discrete time correlations between sampled unit amplitude sine and cosine waveforms at each of said correlation frequencies and said digitized non-coherent electrical signal, said matrix defined by the equation:
• B := ( Ks ⁡ ( 1 ) Kc ⁡ ( 1 ) Ks ⁡ ( 2 ) Kc ⁡ ( 2 ) … Ks ⁡ ( n ) Kc ⁡ ( n ) )
• said discrete time correlations defined by the equations:
• Ks ⁡ ( a ) := ∑ i ⁢ sin ⁡ ( 2 · π · Fc a Ff · i N ) · x ⁡ ( i ) Kc ⁡ ( a ) := ∑ i ⁢ cos ⁡ ( 2 · π · Fc a Ff · i N ) · x ⁡ ( i )
• where N is equal to the number of samples collected in said digitized non-coherent electrical signal, i ranges from 0 to N−1, and x(i) represents said digitized samples of said non-coherent electrical signal;
• multiplication of said inverse matrix A−1 by said matrix B to produce a matrix C of sine and cosine amplitudes of said non-coherent test tones, said matrix C defined by the equation:
• C := ( S ⁡ ( Ft 1 ) C ⁡ ( Ft 1 ) S ⁡ ( Ft 2 ) C ⁡ ( Ft 2 ) … S ⁡ ( Ft n ) C ⁡ ( Ft n ) )
• where S(Ftn) is equal to the sine amplitude of said non-coherent electrical signal at frequency Ftn and C(Ftn) is equal to the cosine amplitude of said non-coherent electrical signal at frequency Ftn.