Plet's Fifth Estimator

Hi Group

This is a new estimator based on the magnitudes of three DFT bins.
The estimator is exact for a pure sinusoid without noise.
It performs well in the presence of noise.

Let j be the index of the DFT bin with the largest magnitude.
Let k=j-1 and i=j+1.

Now let

Ck = Cos(2*PI*k/N)
Cj = Cos(2*PI*j/N)
Ci = Cos(2*PI*i/N)

Then the Cosine of the normalised angular frequency can be estimated
as the root of the quadratic with the following coefficients:

A = (Ck-Ci)*Mag[j]^2 - (Cj-Ci)*Mag[k]^2 - (Ck-Cj)*Mag[i]^2

B = - 2*( Cj*(Ck-Ci)*Mag[j]^2 - Ck*(Cj-Ci)*Mag[k]^2 -
Ci*(Ck-Cj)*Mag[i]^2 )

C = Cj^2*(Ck-Ci)*Mag[j]^2 - Ck^2*(Cj-Ci)*Mag[k]^2 -
Ci^2*(Ck-Cj)*Mag[i]^2

where Mag[] is the magnitude of X[].

The roots are

Cw1 = (-B + Sqr(B^2-4*A*C))/(2*A)

and 

Cw2 = (-B - Sqr(B^2-4*A*C))/(2*A)

The correct answer, Cw, is the root that places the frequency
between the two bins with the largest magnitudes.

The normalised frequency is then

Freq=(N/(2*PI))*Arccos(Cw)


Below is the result of a test run.

Regards,
Michael Plet


The sample count is 100
and the run size is 1000
 The phase value is  0.785398

Errors are shown at 100x actual value

Target Noise Level = 0.000

Freq Plet's Fifth Estimator
---- -------------
3.0  -0.000  0.000
3.1  -0.000  0.000
3.2  -0.000  0.000
3.3  -0.000  0.000
3.4  -0.000  0.000
3.5  -0.000  0.000
3.6   0.000  0.000
3.7   0.000  0.000
3.8  -0.000  0.000
3.9   0.000  0.000


Target Noise Level = 0.001

Freq Plet's Fifth Estimator
---- -------------
3.0  -0.019  0.007
3.1  -0.000  0.010
3.2  -0.000  0.010
3.3  -0.001  0.010
3.4  -0.000  0.011
3.5   0.001  0.013
3.6   0.000  0.015
3.7  -0.000  0.012
3.8   0.000  0.011
3.9  -0.000  0.010


Target Noise Level = 0.010

Freq Plet's Fifth Estimator
---- -------------
3.0  -0.186  0.068
3.1  -0.002  0.094
3.2   0.001  0.095
3.3  -0.002  0.096
3.4  -0.003  0.108
3.5   0.004  0.132
3.6  -0.006  0.144
3.7  -0.006  0.122
3.8  -0.001  0.108
3.9  -0.007  0.102


Target Noise Level = 0.100

Freq Plet's Fifth Estimator
---- -------------
3.0  -0.023  1.954
3.1  -0.890  4.490
3.2   0.057  0.931
3.3   0.076  0.981
3.4   0.063  1.048
3.5   0.060  1.341
3.6  -0.072  1.496
3.7  -0.072  1.212
3.8  -0.029  1.112
3.9   0.357  3.452


Thank you to Cedron for letting me use his test program.