However, other radices are sometimes used, which are usually small numbers less than In this case, you break a non-prime size down into its prime factors, and do an FFT whose stages use those factors. Yes, although these are less efficient than single-radix or mixed-radix FFTs. It is almost always possible to avoid using prime sizes. In the example above, the radix was 2. For single-radix FFTs, the transform size must be a power of the radix.
In the example above, the size was 32, which is 2 to the 5th power. But what does that have to do with FFTs? It is possible but slow to calculate these bit-reversed indices in software; however, bit reversals are trivial when implemented in hardware.
Therefore, almost all DSP processors include a hardware bit-reversal indexing capability which is one of the things that distinguishes them from other microprocessors. FFTs are used for fault analysis, quality control, and condition monitoring of machines or systems. This article explains how an FFT works, the relevant parameters and their effects on the measurement result.
A signal is sampled over a period of time and divided into its frequency components. These components are single sinusoidal oscillations at distinct frequencies each with their own amplitude and phase. This transformation is illustrated in the following diagram. Over the time period measured, the signal contains 3 distinct dominant frequencies. In the first step, a section of the signal is scanned and stored in the memory for further processing.
Two parameters are relevant:. From the two basic parameters fs and BL, further parameters of the measurement can be determined. This value indicates the theoretical maximum frequency that can be determined by the FFT. For example at a sampling rate of 48 kHz, frequency components up to 24 kHz can be theoretically determined. In the case of an analog system, the practically achievable value is usually somewhat below this, due to analog filters - e.
Measurement duration D. The measurement duration is given by the sampling rate fs and the blocklength BL. Frequency resolution df. The frequency resolution indicates the frequency spacing between two measurement results. In practice, the sampling frequency fs is usually a variable given by the system. However, by selecting the blocklength BL, the measurement duration and frequency resolution can be defined.
The following applies:. In the Fourier transformation, the assumption is that the sampled signal segment is repeated periodically for an infinite period of time. This brings two conclusions:. It can be seen that condition 2. This results in a jump in the time signal, and a "smeared" FFT spectrum. In order to prevent this smearing, in practice "windowing" is applied to the signal sample.
Using a weighting function, the signal sample is more or less gently turned on and off. The result is that the sampled and subsequent "windowed" signal begins and ends at amplitude zero. The sample can now be repeated periodically without a hard transition. Likewise, sample number 14 is swapped with sample number 7 , and so forth. The FFT time domain decomposition is usually carried out by a bit reversal sorting algorithm.
This involves rearranging the order of the N time domain samples by counting in binary with the bits flipped left-for-right such as in the far right column in Fig. The next step in the FFT algorithm is to find the frequency spectra of the 1 point time domain signals.
Nothing could be easier; the frequency spectrum of a 1 point signal is equal to itself. This means that nothing is required to do this step. Although there is no work involved, don't forget that each of the 1 point signals is now a frequency spectrum, and not a time domain signal. The last step in the FFT is to combine the N frequency spectra in the exact reverse order that the time domain decomposition took place.
This is where the algorithm gets messy. Unfortunately, the bit reversal shortcut is not applicable, and we must go back one stage at a time. In the first stage, 16 frequency spectra 1 point each are synthesized into 8 frequency spectra 2 points each.
In the second stage, the 8 frequency spectra 2 points each are synthesized into 4 frequency spectra 4 points each , and so on. The last stage results in the output of the FFT, a 16 point frequency spectrum. Figure shows how two frequency spectra, each composed of 4 points, are combined into a single frequency spectrum of 8 points.
This synthesis must undo the interlaced decomposition done in the time domain. In other words, the frequency domain operation must correspond to the time domain procedure of combining two 4 point signals by interlacing. Consider two time domain signals, abcd and efgh.
An 8 point time domain signal can be formed by two steps: dilute each 4 point signal with zeros to make it an. That is, abcd becomes a0b0c0d0 , and efgh becomes 0e0f0g0h. Adding these two 8 point signals produces aebfcgdh. As shown in Fig. Therefore, the frequency spectra are combined in the FFT by duplicating them, and then adding the duplicated spectra together. In order to match up when added, the two time domain signals are diluted with zeros in a slightly different way.
In one signal, the odd points are zero, while in the other signal, the even points are zero. In other words, one of the time domain signals 0e0f0g0h in Fig. This time domain shift corresponds to multiplying the spectrum by a sinusoid.
To see this, recall that a shift in the time domain is equivalent to convolving the signal with a shifted delta function.
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