Transform methods such as the DCT have become increasingly popular in recent years, especially for real-time systems. They provide a large compression ratio. Skip to main content. If the Nyquist frequency is exceeded, the signal is reflected at this imaginary limit and falls back into the useful frequency band.
The following video shows an FFT system with A sweep signal of 15 kHz to 25 kHz is fed in to this system. These unwanted mirror frequencies are counteracted with an analog low-pass filter anti-aliasing filter before the scanning. The filter ensures that frequencies above the Nyquist frequency are suppressed. In the case of periodically-continuous signals, the time windowing serves to smooth the undesired transitional jumps at the end of the scanning see part 1.
This prevents smearing in the spectrum. There are numerous types of windows, some of which differ only slightly. When selecting the time window, the following rule applies: Each window requires a compromise between frequency selectivity and amplitude accuracy.
In the analysis of non-periodic signals, e. There are two possible approaches:. Modern high-resolution FFT analyzers offer the possibility to decouple the number of measurement results from the FFT block length.
This results in an increase in measurement performance time, especially for high-resolution FFTs. Thus, for example, with a 2MB block length it is no longer necessary to measure and represent more than 1 Million points bins , but only the number necessary for the display, e. The value chosen for each FFT bin can be defined in two ways:. FFTs are mainly used to visualize signals. However, there are also applications where FFT results are used in calculations.
For example, very simple levels of defined frequency bands can be calculated by adding them via an RSS Root Sum Square algorithm.
Another application is the comparison of spectra. The example below shows an acoustic measurement of a cordless screwdriver. The measured spectrum is subtracted from a defined reference spectrum.
This difference is compared against an upper and lower tolerance. The upper spectrum shows a functional cordless screwdriver. In the lower, the acoustic spectrum suggests that the test specimen is defective. View of a signal in the time and frequency domain. Representation of the FFT of a signal with small and large blocklength. Un-windowed time signal with smeared spectrum. There are Log 2 N stages required in this decomposition, i. Remember this value, Log 2 N ; it will be referenced many times in this chapter.
Now that you understand the structure of the decomposition, it can be greatly simplified. The decomposition is nothing more than a reordering of the samples in the signal. Figure shows the rearrangement pattern required. On the left, the sample numbers of the original signal are listed along with their binary equivalents. On the right, the rearranged sample numbers are listed, also along with their binary equivalents. The important idea is that the binary numbers are the reversals of each other.
For example, sample 3 is exchanged with sample number 12 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.
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