Plet's Second Estimator

Hi Group

This is a new estimator based on real and imaginary values of two DFT
bins.
The estimator is exact for a pure sinusoid without noise.
It performs well in the presence of noise.

Let k be the index of the DFT bin with the largest magnitude.
Let j be the index of the DFT bin with the largest magnitude
neigboring bin k.
That is j=k-1 or j=k+1.

Now let

Sk = Sin(2*PI*k/N)
Ck = Cos(2*PI*k/N)
Sj = Sin(2*PI*j/N)
Cj = Cos(2*PI*j/N)

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

A = Sj*Im[k] - Sk*Im[j]

B = Sj*Sk*Re[k] - 2*Sj*Ck*Im[k] - Sk*Sj*Re[j] + 2*Sk*Cj*Im[j]

C = - Sj*Sk*Ck*Re[k] + Sj*Ck^2*Im[k] + Sk*Sj*Cj*Re[j] - Sk*Cj^2*Im[j]

where Re[] is the real part of X[] and Im[] is the imaginary part 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 root in the interval [Min(Ck,Cj), Max(Ck,Cj)] is the correct
answer, Cw.

The normalised frequency is then

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


Regards,
Michael Plet

On Tue, 21 Feb 2017 14:07:17 +0100, Michael Plet <me@home.com> wrote:

>Hi Group
>
>This is a new estimator based on real and imaginary values of two DFT
>bins.
>The estimator is exact for a pure sinusoid without noise.
>It performs well in the presence of noise.
>


Here is the result of a test run:

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 Second 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 Second Estimator
---- -------------
3.0   0.000  0.023
3.1  -0.001  0.030
3.2   0.000  0.072
3.3  -0.001  0.062
3.4  -0.000  0.019
3.5  -0.000  0.011
3.6   0.000  0.009
3.7  -0.000  0.008
3.8  -0.000  0.009
3.9  -0.000  0.012


Target Noise Level = 0.010

Freq Plet's Second Estimator
---- -------------
3.0   0.001  0.229
3.1  -0.005  0.297
3.2  -0.014  0.715
3.3  -0.020  0.610
3.4   0.001  0.183
3.5   0.003  0.113
3.6   0.003  0.088
3.7   0.004  0.082
3.8  -0.001  0.096
3.9   0.004  0.117


Target Noise Level = 0.100

Freq Plet's Second Estimator
---- -------------
3.0  -0.035  2.217
3.1  -0.183  3.099
3.2  -0.074  7.428
3.3  -0.408  6.296
3.4  -0.129  1.903
3.5   0.003  1.116
3.6  -0.006  0.865
3.7  -0.033  0.816
3.8  -0.046  0.918
3.9  -0.001  1.138


Michael

On Tue, 21 Feb 2017 15:30:50 +0100, Michael Plet <me@home.com> wrote:

>On Tue, 21 Feb 2017 14:07:17 +0100, Michael Plet <me@home.com> wrote:
>
>>Hi Group
>>
>>This is a new estimator based on real and imaginary values of two DFT
>>bins.
>>The estimator is exact for a pure sinusoid without noise.
>>It performs well in the presence of noise.
>>
>
>
>Here is the result of a test run:
>

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

Michael