6.1 INTRODUCTION

The main feature of our instrument is the implementation, for the first time in the filed of electrical measurement, of a FFT algorithm, which minimizes the phase error due to the short-range leakage. Moreover, as we already seen from the previous chapters, the measure is performed in real time.

The algorithm is based on a paper published by Ferrero and Ottoboni [1].

Before going into the details, in the next section we briefly describe the considerations that lead us to the use of this algorithm.

6.2 FFT OF A PERIODIC SIGNAL

A sequence of sampled data may be symmetrical or not, with respect of the time origin. Let us consider first the case of symmetric sequence.

Let s(t) representing a periodic signal with period Ts and S(f) its Fourier1 transform.

Suppose also that the sampling frequency fc = 1/Tc, has the right value to match the Nyquist requirements.

If we take 2N+1 samples, evenly spread over the range [-NTc, NTc], the resulting sequence {s(k)} is symmetrical with respect to the sample s(0), where the index k = 0 represents the time origin.

Now, if we consider an asynchronous sampling

the sampling of the signal s(t) mathematically is equivalent to the following product

where g(t) is a train of Dirac impulses time limited.

The Fourier transform of g(t) is:

The above equation also represents the Fourier transform of a rectangular window Tc(2N+1) points long. From the convolution theorem, the Fourier transform of p(t) is given by

P(f) can be obtained from the sampled values of s(t) by

or, in case of evaluation through digital techniques, by

Now, because of the asynchronous sampling, we can observe two main effects:

1. 1.G(f) has zeros for frequency values that differ from the harmonic frequencies of s(t). This means that when we calculate the n-th harmonic component using the above equation, all the harmonic components give their own contribute rather than the n-th component alone.
   

           

This phenomenon of harmonic interference depends on both S(f) and G(f), and it is the responsible for the so-called “long-range leakage”, which affects both the phase and the amplitude. One way to reduce this interference between the harmonic components, that is the long-range leakage error, is by the use of specific windows. Examples of these windows have been already described in the chapter about the FIR filtering. In fact, these windows exhibit lower ripple attenuation over the side lobes (see figures from 4.2 to 4.7.)

2. 2.The main lobe of G(f) reaches the maximum value for frequencies equal to n/T, where n is an integer in the range [-N, N]. These frequencies differ from the harmonic frequencies of the signal s(t); it follows an error that affects only the harmonic amplitudes. This error is called “short-range leakage” error; in order to reduce this kind of error, you could use specific windows known as “FLAT TOP” windows (see figures 6.4, 6.5, and 6.6).
   

In the more generic case of a sampled data sequence no more symmetrical with respect to the time origin, the algorithm representing the FFT is

Now, for a generic asymmetric sequence, n varies from 0 to 2N. This results in a Fourier transform of the time limited impulse train, which has the following form

The complex term causes a short-range leakage phase error too. In the case of long-range leakage, it is possible to reduce the error by the use of specific windows. However, for the short-range error, which is often the dominant error (so you cannot neglect it), the use of windows, which have real coefficients, is useless. It is just for this reason that generally interpolation methods are used. However, these methods require a big amount of calculations, and anyway under the hypothesis you can disregard the harmonic interference.

1 We mean transform in the field of Distribution theory.