Direct Methods for Frequency Filter Performance Analysis - Digital Filters and Signal Processing - page 84

Let us consider a simple example of performance analysis of signal processing by a high-pass
second-order filter relating to signal processing task in power system protection and automa‐
tion devices [9,10]. The filter is used to extract a sinusoidal component of commercial frequency
and eliminate disturbance as a free component of transient processes in a control object. In this
case, a change of a useful signal initial phase is acceptable.
All the initial data and dependencies which are necessary for the analysis are represented in
the Table 2. IIR filter parameters are specified, the mathematical description of an input signal
with specified sizes of changing for useful signal and disturbance parameters affecting their
spectrum is given in the Table 2 as well.
The impulse function of high-pass second-order filter contains a delta function of Dirac which
is used for determining complex amplitudes of forced components when defining K ( p ) by the
impulse function (item 3 Table 2) and cannot be applied for determining complex amplitudes
of filter reaction free components (item 5). To simplify the analysis the delta function can be
represented as an extreme case of the exponential component αe − αt at α → ∞ [6].
The analysis results should ensure the following performance criteria of signal processing by
a filter:
1. a filter settling time should be less than 30 ms at 5% acceptable total error of signal
processing at any value of disturbance parameters within the specified range,
2. an acceptable error at frequency deviation of useful signal from the nominal value of 50Hz
within the range ±5 Hz should not be more than 5%,
3. an acceptable overshoot should not be more than 10%.
As it follows from the Table 1, simple algebraic operations are applied to determine complex
amplitudes, as well as forced and free components of a filter output signal.
When using Mathematica, Mapple, Matlab, Mathcad and other state-of-art mathematical
software for determining forced and free components of an output signal it is necessary to
specify only complex amplitude vectors of an input signal and a filter impulse function, as
well as complex frequency vectors of an input signal and a filter. In this case all the necessary
calculations, related to a filter analysis, would be carried out automatically. If it is needed to
determine complex amplitudes of a filter impulse function at specified transfer function the
ready-made formulas may be used [6], which can be easily applied in the mathematical
software mentioned above.
All the examples in the present chapter are given using the mathematical software Mathcad.
Mathcad was chosen due to pragmatic considerations related to assuring the maximum
visibility of the examples for filter analysis, as in Mathcad mathematical expressions are given
in the form, closest to universally accepted mathematical notation [11,12].
An example of a filter computation using Mathcad at the specified filter parameters and the
following input signal parameters: Xm 2 = Xm 1 =1, ω 1 =2 π 50rad/s, φ 1 =0, β 2 =60 s -1 is given on
the Figure 1.