table of contents

1 Fast Fouriertransformation (FFT)

1.1 The real-time concept of the Goldammer measurement cards

1.2 The sampling theorem or rules for sampling time signals

1.3 How the FFT works

1.4 Example: square wave signal

1.1 The real-time concept of the Goldammer measurement cards

The intelligent measurement cards of the MC4-PCI series from Goldammer relieve the PC when it comes to recording and outputting signals. This includes real-time processing of acquired signals. This real-time processing is integrated in our drivers and does not require any additional capacities on the PC. Each channel can be configured individually.

The advantage of an FFT on the signal processor lies in the architecture of the processors and in the reduced data volume. Signal processors can calculate the FFT very quickly with optimized algorithms. With the Goldammer measurement cards, this calculation takes place in the idle time of the processor, i.e. when the processor has no further tasks to perform at this point in time. The host system is significantly relieved, as it only has to fetch and display the finished measured values. The calculations are distributed on the signal processor in such a way that the last current spectrum of the FFT is always available. The time signals can of course also be called up.

Acquired signals can be subjected to a fast Fourier transformation (hereinafter referred to as FFT) on the card. Signals can thus be broken down into their frequency components.

The French mathematician Fourier discovered that every periodic signal can be broken down into many individual sinusoidal and cosinusoidal oscillations and can also be simulated by these.

He developed the mathematical tool of Fourier analysis. With a series expansion he succeeded in describing every periodic signal by an infinite sum of sine and cosine oscillations weighted with coefficients. Infinity is a mathematical feature that is difficult to handle in reality. Fourier proved that a finite number of summands is sufficient to approximately describe a periodic signal. By canceling the series development, the simulated function no longer corresponds to the original function. However, by using enough summands, the original function can be approximated as precisely as required.

1.2 The sampling theorem or rules for sampling time signals

There are some requirements for the sampling and processing of sampled signals with digital systems. These are:

1. The signal must be band-limited, ie all frequency components must be zero above a limit frequency. The cutoff frequency is called the “Nyquist frequency”.
2. The sampling frequency must be at least twice as high as the limit frequency of the signal

These rules are called “SHANNON’s sampling theorem”. If it is not adhered to, ie the sampling rate is not at least twice as large as the highest frequency contained in the signal, frequency components occur in the spectrum that are actually not contained in the signal. This effect is called “aliasing” and results from the reflection of frequencies above the cut-off frequency in the area below the cut-off frequency. This falsifies both the frequency spectrum and the signal over time.

1.3 How the FFT works

While digital filters are calculated using individual value processing, the FFT works exclusively with data blocks. The most recent samples are contained in these data blocks.

The cards of the MC4-PCI series can subject samples to an FFT. The calculation is carried out with a base 2 algorithm (Cooley-Tuckey). The number of samples is therefore limited to a power of 2 (eg 512, 1024, 2048, …).

The result is the frequency spectrum of the examined signal. The FFT algorithm provides a complex spectrum. The real part corresponds to the cosine parts ( ), the imaginary part to the sine parts ( ).

The amount spectrum is formed by calculating the amount.

Other types of representation are:

• RMS spectrum
  The RMS spectrum is the effective value of the magnitude spectrum.
• Power spectrum
  The power spectrum indicates the square of the effective values.

1.4 Example: square wave signal

In the following, a square-wave signal with a signal frequency of 100 Hz is considered. Various FIR filters are applied to this signal and the resulting frequency spectra are calculated using FFT.

Figure 1 shows the time course, Figure 2 the frequency spectra.

The unprocessed signal is shown at the top. In addition to the frequency at 100Hz (the basic frequency), the square-wave signal also contains other frequencies (harmonics). Theoretically, there are an infinite number of harmonics in the signal.

The middle curve shows the square wave signal after FIR filtering with a cutoff frequency of 550Hz. Frequencies above this frequency are suppressed. The time course shows a strong ripple. The frequency spectrum only contains 3 frequency components.

At the bottom the square wave signal was filtered with the cutoff frequency 225Hz. The filter suppresses all harmonics, only the fundamental frequency is retained. Therefore, a sinusoidal signal is generated from the rectangle.

Figure 1 : Time course of the square-wave signal, above without filtering, middle limit frequency 550Hz, below limit frequency 225Hz

Figure 2 : Frequency spectra of the square-wave signal, above without filtering, middle limit frequency 550Hz, below limit frequency 225Hz