{"id":1026,"date":"2021-11-03T06:29:02","date_gmt":"2021-11-03T06:29:02","guid":{"rendered":"https:\/\/projects.jayanwerdesigns.com\/goldammer\/?page_id=1026"},"modified":"2022-01-19T21:16:25","modified_gmt":"2022-01-19T21:16:25","slug":"fft","status":"publish","type":"page","link":"https:\/\/goldammer.de\/eng\/fft\/","title":{"rendered":"FFT"},"content":{"rendered":"
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Fast Fourier Transform (FFT)<\/h1>
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table of contents<\/u><\/h3>\n

1 Fast Fouriertransformation (FFT)<\/a><\/p>\n

1.1 The real-time concept of the Goldammer measurement cards<\/a><\/p>\n

1.2 The sampling theorem or rules for sampling time signals<\/a><\/p>\n

1.3 How the FFT works<\/a><\/p>\n

1.4 Example: square wave signal<\/a><\/p>\n<\/div><\/section>
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1 Fast Fourier Transform (FFT)<\/h1>
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<\/a>1.1\u00a0The real-time concept of the Goldammer measurement cards<\/h2>\n

The intelligent measurement cards of the MC4-PCI series from Goldammer\u00a0relieve the PC<\/em>\u00a0when it\u00a0comes to\u00a0recording and outputting signals.\u00a0This includes real-time processing of acquired signals.\u00a0This real-time processing is integrated in our drivers and\u00a0does not require any additional capacities<\/em>\u00a0on the PC.\u00a0Each channel can be configured individually.<\/p>\n

The advantage of an FFT on the signal processor lies in the architecture of the processors and in the reduced data volume.\u00a0Signal processors can calculate the FFT very quickly with optimized algorithms.\u00a0With 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.\u00a0The host system is significantly relieved, as it only has to fetch and display the finished measured values.\u00a0The calculations are distributed on the signal processor in such a way that the last current spectrum of the FFT is always available.\u00a0The time signals can of course also be called up.<\/p>\n

Acquired signals can be subjected to a fast Fourier transformation (hereinafter referred to as FFT) on the card.\u00a0Signals can thus be broken down into their frequency components.<\/p>\n

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.<\/p>\n

He developed the mathematical tool of Fourier analysis.\u00a0With a series expansion he succeeded in describing every periodic signal by an infinite sum of sine and cosine oscillations weighted with coefficients.\u00a0Infinity is a mathematical feature that is difficult to handle in reality.\u00a0Fourier proved that a finite number of summands is sufficient to approximately describe a periodic signal.\u00a0By canceling the series development, the simulated function no longer corresponds to the original function.\u00a0However, by using enough summands, the original function can be approximated as precisely as required.<\/p>\n

<\/a>1.2\u00a0The sampling theorem or rules for sampling time signals<\/h2>\n

There are some requirements for the sampling and processing of sampled signals with digital systems.\u00a0These are:<\/p>\n

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  1. The signal must be band-limited, ie all frequency components must be zero above a limit frequency.\u00a0The cutoff frequency is called the \u201cNyquist frequency\u201d.<\/li>\n
  2. The sampling frequency must be at least twice as high as the limit frequency of the signal<\/li>\n<\/ol>\n

    These rules are called “SHANNON’s sampling theorem”.\u00a0If 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.\u00a0This effect is called “aliasing” and results from the reflection of frequencies above the cut-off frequency in the area below the cut-off frequency.\u00a0This falsifies both the frequency spectrum and the signal over time.<\/p>\n

    <\/a>1.3\u00a0How the FFT works<\/h2>\n

    While digital filters are calculated using individual value processing, the FFT works exclusively with data blocks.\u00a0The most recent samples are contained in these data blocks.<\/p>\n

    The cards of the MC4-PCI series can subject samples to an FFT.\u00a0The calculation is carried out with a base 2 algorithm (Cooley-Tuckey).\u00a0The number of samples is therefore limited to a power of 2 (eg 512, 1024, 2048, …).<\/p>\n

    The result is the frequency spectrum of the examined signal.\u00a0The FFT algorithm provides a complex spectrum.\u00a0The real part corresponds to the cosine parts (\u00a0), the imaginary part to the sine parts (\u00a0).<\/p>\n

    The amount spectrum is formed by calculating the amount.<\/p>\n

    Other types of representation are:<\/p>\n