The history of FFT

 Intuitively speaking, time domain analysis is clear and easy to see. The oscilloscope is the main tool for time domain observation, which can observe the waveform shape, measure pulse width, phase difference and other information. However, for further analysis of the signal, such as measuring the proportion and energy distribution of each harmonic, time domain analysis is not enough. However, it is very meaningful to use fast Fourier transform FFT developed from continuous time Fourier transform for analysis. Spectrum analysis technology, such as frequency division multiplexing technology, is indispensable in communication systems. Spectrum analysis generally uses fast Fourier transform FFT to calculate frequency spectrum and power spectrum, which can be directly used to extract characteristic frequency and spectrum features. Because computers can only process discrete data points, but FFT is an approximation of Fourier transform, which is different from Fourier transform and has inherent limitations: fence phenomenon. Based on the previous article "The Past and Present of FFT", this article discusses some problems that are easily overlooked in the FFT operation of oscilloscopes from the perspective of test and measurement.

Frequency resolution and time base setting (TimeBase)

The definition of frequency resolution is: the smallest interval between two frequency points that can be obtained on the frequency spectrum when using FFT operation.

ΔF=Fs/N=1/NT=1/Tp

ΔF is called frequency resolution, that is, sampling rate/number of sampling points. The smaller ΔF is, the higher the frequency resolution is. ΔF is only inversely proportional to the actual length of the signal, that is, the longer the duration of the signal to be analyzed, the smaller ΔF is, and the higher the frequency resolution is.

Picket fence effect and frequency resolution:

The signal input to the oscilloscope is generally a non-periodic continuous signal.

()ax

, its spectrum is also continuous, but what the oscilloscope does is

The method is to sample ( ) axt at equal intervals and truncate it, and then perform FFT operation to obtain a discrete spectrum, which is equivalent to sampling the continuous spectrum. In this way, some spectrum components will be "blocked" outside the sampling points, just like we are observing the spectrum through a fence. This phenomenon is called the "fence effect". In this way, it is possible that some peaks or valleys of the spectrum will be blocked by the fence and cannot be observed by us.

Whether it is time domain sampling or frequency domain sampling, there is a corresponding fence effect. However, when time domain sampling satisfies the sampling theorem, the fence effect will not have any effect. However, the fence effect of frequency domain sampling has a great impact. The "blocked" or lost frequency components may be important or characteristic components, making signal processing meaningless.

The fence effect is a bottleneck that restricts the accuracy of harmonic analysis in spectrum analysis. The fence effect is particularly serious during asynchronous sampling. During asynchronous sampling, the harmonic components do not fall exactly on the frequency resolution point, but fall between two frequency resolution points. In this way, the accurate value of each harmonic component cannot be directly obtained through FFT, but can only be approximated by the value of the adjacent frequency resolution point. This is why the fence effect reduces the accuracy of spectrum analysis.

From this we can draw the following conclusion: reducing the fence effect can be solved by increasing the spectrum sampling interval, that is, the frequency resolution. The smaller the interval, the higher the frequency resolution, and the fewer frequency components will be "blocked" or lost. However, the increase in frequency resolution will increase the number of sampling points, which increases the calculation workload.

We can increase the frequency resolution in two ways:

a: Physical resolution = sampling frequency/number of sampling points.

The practical significance of physical resolution is that it can measure the interval of frequency components that FFT can actually distinguish. The method to improve physical resolution is generally to increase the effective length of data, which is equivalent to increasing the width of the rectangular window in the analog domain. This reduces the width of the sinc main side lobe in the analog domain and reduces the aliasing of adjacent frequency components.

This method of increasing sampling points is mainly used for FFT calculation of infinite length sequences. For infinite length sequences, unlike finite length sequences, which must be padded with zeros to improve apparent resolution, infinite length sequences can improve physical resolution by increasing data length.

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b: Apparent resolution = sampling frequency/number of analysis points

The method of padding the end of a sequence with zeros can increase the number of analysis points, so the method of padding with zeros can improve the apparent resolution of the spectrum. The method of padding the end of a sequence with zeros is mainly for finite-length sequences. For finite-length sequences, sometimes only zero padding or interpolation can be used to improve the frequency resolution. Through zero padding, the sampling density in the frequency domain is increased, and a high-density spectrum is obtained. The spectrum obtained by the zero padding method only improves the apparent resolution of the graph, and cannot obtain more details of the spectrum.

Increasing the number of sampling points increases the order of the input sequence, thus providing more details of the spectrum, which is the true resolution (physical resolution). If the sequence is only padded with zeros without adding data, the order of the input sequence and its spectrum will not be improved, but the spectrum will be drawn more densely, so only the apparent resolution of the graph is improved, and more details of the spectrum cannot be obtained. Increasing the length of the sequence can improve the true resolution of the spectrum, which is a basic rule.

From the above discussion, we can see that there are two specific methods to improve resolution:

(1) For finite length sequences, the method of padding the tail with zeros is used to improve the apparent resolution.

(2) Improving physical resolution by actually increasing the number of sampling points for infinite length sequences

Finite length sequence and infinite length sequence refer to actual signals. For example, a non-periodic signal containing infinite length information can be called an infinite length sequence. Strict periodic signals and pulse signals (invalid information for an infinite time before and after the pulse) can be called finite length sequences. Of course, strict periodic signals do not exist in reality. For an oscilloscope, the information collected within the time window can be all the information of a finite length sequence or part of the information of an infinite length sequence. Therefore, if all the information of a finite length sequence is collected, the apparent resolution can only be increased by zero padding. If part of the information of an infinite length sequence is collected, the physical frequency resolution can be increased by increasing the length of the time window (not the sampling point). [page]

Please see the following example:

The time base used for the sine wave test in Figure 1 is 5ns/div, the waveform time length is 50ns, and the spectrum resolution after FFT calculation is 20MHz (1/50ns).

Figure 1 Capturing a 50ns signal with a frequency resolution of 20MHz

If you change the time base setting, the spectrum resolution will change. As shown in Figure 2: When the time base is set to 10ns/div and the waveform length is 100ns, the spectrum resolution can be increased to 10MHz.

LeCroy's oscilloscopes also have corresponding solutions for increasing the apparent resolution of the FFT spectrum by padding zeros. LeCroy oscilloscopes use two very commonly used FFT algorithms for users to choose from: Cooley-Tukey algorithm and LeastPrime algorithm. The Cooley-Tukey algorithm is also called the Power2 algorithm. It provides a very fast FFT calculation method for computers. The scale of the calculated FFT points is an integer power of 2, so it will intercept 2N integers from the signal collected in the oscilloscope's time domain as the time domain samples for the FFT calculation. The intercepted integer is the integer closest to the sampling point. As shown in Figure 2 below:

Figure 2 Capturing a 100ns signal with a frequency resolution of 10MHz

The sine wave frequency in the figure is 500MHz, the time base is set to 10ns/div, the sampling rate is 20GS/s, the number of time domain sampling points is 2000points, and the Power2 algorithm is used to intercept 1024 points (210) out of 2000 points, as shown in the blue box in the figure (note that it is intercepted from the middle part of the signal). Therefore, the intercepted time window is 1024×20ps=51.2ns, which is 25.6 periods of the 500MHz signal. Since the intercepted period is not an integer multiple, spectrum leakage is inevitable, as shown in the sidelobe of the FFT in the figure. At this time, the frequency resolution can reach 19.35125MHz.

If another FFT algorithm, LeastPrime, is used, the sampling points acquired by the entire oscilloscope in the time domain can be subjected to FFT calculation. The FFT point scale calculated by the LeastPrime algorithm is 2N+5K, so 2000 points = 24+53. The calculation can be performed without intercepting the original data, but the cost is that the calculation speed may be slower (although we may not notice it). The frequency resolution can be increased to 10MHz.

When using the Power2 algorithm, you can also choose not to intercept the original waveform. In this case, we can choose the Zero Fill method to increase the number of sampling points. For example, fill 48 points in 2000 points, 2048=211, as shown in Figure 3:

Figure 3 Zero padding improves frequency resolution [page]

The way to fill these 48 points is to fill half of the first and the last points, but it is possible that the filled points are not 0, the first 24 points have the same value as the first sampling point, and the last 24 points have the same value as the last sampling point (so it is not completely accurate to call it Zero Fill). Here we recommend that the Zero Fill method is only used when analyzing the FFT spectrum of the impulse signal.

Although the zero-padding method can increase the apparent resolution of the spectrum, it does not really help improve the frequency resolution because it fills in invalid data. However, zero-padding has its advantages: 1. After zero-padding, the FFT result is actually interpolated to overcome the "fence" effect and smooth the spectrum appearance. I understand the "fence" effect as standing next to a fence and looking through it at the scenery outside. There must be a lot of scenery blocked by the fence, and at this time, large frequency domain components may be missed. However, after zero-padding, it is equivalent to standing farther away, changing the density of the fence, and the scenery becomes clearer and clearer. 2. Since the truncation of time domain data will inevitably cause spectrum leakage, difficult-to-identify spectrum peaks may appear in the spectrum. Zero-padding can eliminate this phenomenon to a certain extent.

In addition, many people have this misunderstanding: they think that the spectrum after FFT can be made more "fine" (higher frequency resolution) by increasing the number of calculation points to be analyzed rather than increasing the sampling time. This misunderstanding generally comes from oscilloscope users, because when the oscilloscope has fewer sampling points, the spectrum graph calculated by FFT will also be small, and the spectrum will look very rough. At this time, engineers will be very tempted to increase the sampling points in the time domain (which is easy to achieve using the interpolation algorithm on the oscilloscope), but if the time length of the acquired signal remains unchanged, engineers will find that the spectrum after FFT calculation does not appear to be more "fine" and the frequency resolution has not improved at all. In fact, using interpolation or increasing the sampling rate only widens the bandwidth of the spectrum after FFT. As shown in Figure 4 below,

Figure 4 Interpolation does not change the frequency resolution

The upper left image uses fewer time domain sampling points C1, and the upper right image uses a higher sampling rate C2, but the sampling time is the same. The lower left image is the spectrum F1 after FFT of C1, the middle right image is the spectrum F2 after FFT of C2, and the lower right image is a zoomed-in view of the same frequency band of F2. It can be seen that the bandwidth of F2 is larger than that of F1, but the spectrum after zooming in on the frequency band of F1 is the same as that of F1, without any improvement in frequency resolution.

From this we can conclude that after interpolating C1, the additional sampling points only exist in the higher frequency band, which will broaden the bandwidth of the spectrum, but the interpolation method does not help to increase the spectral resolution of the frequency band of interest.

So what if we only interpolate the spectrum after FFT? As shown in Figure 5 below:

Figure 5 Frequency domain interpolation makes the spectrum graph look denser

The figure shows the effect of frequency domain interpolation. It does not make the spectrum look narrower (after all, the interpolated points are all fake points), but we noticed that frequency domain interpolation can make the spectrum measurement more accurate. The frequency of the sine wave in the figure is 955MHz. After interpolation, the peak frequency reading P2 of the spectrum is 955MHz, and the reading of P1 before interpolation is 952MHz.

In short, FFT is the most widely used standardized method for signal frequency domain analysis and is also a standard mathematical operation function in modern digital oscilloscopes. The more we understand the details of FFT applications, the more effectively we can use this tool and obtain more valuable information from FFT.