Sum of filter networks
5.2Implementing high-line noise elimination on the hci
High-Line noise elimination is implemented on the HCI by stacking an even number of sweeps (half with the Up option, half with the Down option as shown below).
With the above setups, the first two acquisitions will start on a positive going (up) transition of the power line signal and the other two on a negative going (down) transition. Stacking will eliminate the power line signal and keep only the data.
6.Harmonic line elimination
6.1Introduction
This chapter describes how harmonic lines contained in the Ground Force (GF) signal are removed by the additions performed in the stacking process, through successive, different phase shifts applied to a succession of sweeps — taken on the same point — if the same phase shifts are applied to the reference signal used for the correlation.
6.2Review of physical relationships on sine waves
Prior to establishing a general law on the phase shifts to be applied, let us recall the necessary basics on the physical relationships involved in the process.
A phase shift between the reference and the Pilot causes an N phase shift on the order-N harmonic in the Ground Force (GF).
This can be illustrated by the diagram below, showing an example with the second harmonic line (N = 2) and = 180° :
2Nd harmonic phase
The relative position of the vibrator's harmonic line 2f1 with respect to the reference f1 is unchanged, but the phase shift is 180° () on the time scale of f1 and 360° (2) on the time scale of 2f1.
The autocorrelation function of a sine wave (with f as frequency) is a sine wave with the same frequency (f) and a positive first peak located at the zero time.
The cross-correlation of two sine waves with the same frequency (f) but in phase opposition results in a sine wave with the same frequency (f) and a negative first peak located at the zero time.
The above two notions can be illustrated by the diagram below :
It should also be noted that :
The harmonic line contained in the Ground Force correlates with the reference signal at the same frequency ;
The initial phase shift () applied to the sweep reference is also applied to the reference signal used in the correlation process.
6.3General law
We wish the data to be in phase opposition before stacking, after correlation. The initial phase shift required between the two reference signals used in the crosscorrelation process is :
where N = is the order of the harmonic line
k {0, 1, 2, ... }
This generalized law can be illustrated by two examples, with the graphs of harmonic lines 2 and 3.
The graph of the 2nd harmonic (see next page) allows us to write that the 2nd harmonic in the Ground Force is itself a sine wave at twice the start frequency, varying at twice the sweep rate. As a result, those frequencies which are common to the reference will correlate. Considering a reference frequency from f to 2f in the graph, the 2nd harmonic frequency will be 2f to 4f. Since the reference contains no frequencies higher than 2f and the Ground Force harmonic contains no frequencies lower than 2f, the harmonic correlated result will contain only the 2f frequency. The harmonic correlated result can be thought of as a sine wave correlated with a sine wave giving a sine wave at the same frequency but shifted by 90°.
In fact the 2nd harmonic will produce an "added Ground Force" at twice the sweep rate of the reference signal and containing only the frequencies from the lowest 2nd harmonic frequency to the highest reference frequency.
If we are capable of obtaining phase oppositions on the harmonic correlated results, from one sweep to another, then the harmonic correlated results will vanish through the addition performed in the stacking process. Now, as seen earlier, a phase shift on the reference causes an N phase shift on the order-N harmonic in the Ground Force. It is therefore easy to identify the phase shifts required of the harmonic correlated results.
The same can be applied to the 3rd harmonic.
