Contents
1 Introduction
2 Concepts
2.1 The Fourier Theorem
2.2 The Time and Frequency Domain
2.3 Periodic Signals and Discrete Spectra
2.4 Non-periodic signals and Continuous Spectra
2.5 Conclusions
3 FT, DFT and FFT
4 The FFT
5 Windows
5.1 Why Windowing
5.2 What Are Windows?
5.3 What Effect Do They Have?
6 The Gould FFT
6.1 Windowing
6.2 Number of Points
6.3 Horizontal and Vertical Scaling
6.3.1 Vertical Scaling
6.3.2 More About Log Scaling
6.4 Frequency Measurement
6.5 Destination of Results
7 The Display Obtained
8 The Classic 6000 FFT
8.1 FFT Length
8.2 Averaging FFTs
8.3 High Frequency FFTs
8.4 The Classic FFT Nomogram
8.5 Speed of Execution
9 Getting the Best Results
9.1 Filling the time record with appropriate data
9.2 Selecting the appropriate window
10 Summary
1. Introduction
The aim of this appendix is to give, not only a good overview of the FFT function on the Classic series DSOs, but also a background in the theoretical aspects of the FFT. Within each section there is a mixture of overview material and more detailed background material. The overview material should be read and is printed in normal typeface. The background material should be retained for reference and is printed in italic typeface.
2 Concepts
2.1 The Fourier Theorem
The Fourier Theorem was developed by Baron Jean Baptiste Fourier over 100 years ago and states: "Any waveform can be looked upon as a combination of a number of pure sinusoids." This theorem is a reciprocal because any signal can be split into a combination of sinusoids with different combinations of frequency, amplitude and phase. Conversely, a combination of sinusoids can be combined into a unique waveform.
2.2 The Time and Frequency Domain
The normal mode of using an oscilloscope is the TIME domain; i.e. Amplitude vs Time. The Fourier Theorem gives a relationship between the time domain and the frequency domain. The FREQUENCY domain is the normal mode of operation when using a spectrum analyzer; i.e. Amplitude vs Frequency. The two modes are related as shown in Figure 2.2.1. From this it can be seen that the two domains are simply different ways to view a signal.
time and frequency relationship
Figure 2.2.1: Relationship Between The Time And Frequency
2.3 Periodic Signals and Discrete Spectra
The next two sections look at the types of spectra obtained from both periodic and non-periodic waveforms. First, look at the simple example of a sine wave. This is assumed to be repetitive for all values of time. Any sine wave is defined from the simple function: equation where:
A= Peak amplitude
f= Frequency
ø= Phase angle
This waveform and its Fourier Transform is shown in figure 2.3.1. The continuous signal is transformed into a single discrete line at a frequency =1/t1 and amplitude = RMS value of the voltage.
Figure 2.3.1: Time Waveform & Fourier Transform Of A Sinewave
Looking at a second signal which has half the amplitude and twice the frequency, figure 2.3.2. is obtained Again, it is transformed into a single discrete line at a frequency =1/t2 and amplitude = RMS value of the voltage.
time fft of sinewave
Figure 2.3.2: Time Waveform & Fourier Transform Of Another Sinewave
When these two signals are added together, the resulting waveform is shown in figure 2.3.3. From the Fourier Theorem, the transformed signals become as shown in figure 2.3.4. Here there are two discrete spectra with lines at fl and f2 and amplitudes corresponding to the RMS value of the component signals.
Adding two signals
Figure 2.3.3: The Result Of Adding The Two Signals Together
FFT of two sines
Figure 2.3.4: The Fourier Transform Of The Two Sinusoids
For the example in figure 2.3.5, a periodic square wave with a period of to is used. In this case, the Fourier Transform obtained is a series of discrete spectra where each line is a multiple of the frequency 1/to. The multiples of the frequency are 1,3,5,7 etc. The amplitude of the lines gives the RMS value of the frequency components. There is no phase information which is needed to accurately synthesize the square wave from the component sinusoids.
fft of squarewave
Figure 2.3.5: Fourier Transform of a Periodic Square Wave
It is a property of the Fourier Transform that the line spacing decreases as the period of the corresponding time domain signal increases.
2.4 Non-periodic Signals and Continuous/ Spectra
When looking at the Fourier Transform of a non-periodic signal, the relationship between the line spacing in the frequency domain and the period of the time domain signal becomes important. As the period of the waveform increases, the line spacing in the associated line spectrum decreases. Thus, in the limiting case of a single non-periodic signal, the period tends to infinity and the line spacing tends to zero. In this case, the spectrum lines are so closely spaced that the spectrum is drawn as an envelope. This envelope can then be called a continuous spectrum.
As an example, Iooking at the Fourier Transform of a single rectangular pulse of duration T sec, the result shown in figure 2.4.1. is obtained. From this there are NULLS where there are no frequency components at frequencies of 1/T, 2/T, 3/T etc.
FFT of Rectanglar pulse
Figure 2.4.1: Fourier Transform Of A Single Rectangular Pulse
2.5 Conclusions
The main conclusions are:
a) Periodic Signals in the Time Domain give discrete spectra in the Frequency Domain.
b) Non-periodic signals in the Time Domain give continuous spectra in the Frequency Domain.
3. FT, DFT and FFT
So far the Fourier Transform in general has been discussed. Next, the FT, DFT and FFT and how they fit into test instruments will be examined. Sections 2.3 and 2.4 discussed the Fourier Transform (FT) - the FT is the math that is used to draw the frequency spectrum from the time graph.
The FT cannot easily be used in test instrumentation. However about 30 years ago algorithms were devised for computers to calculate the Fourier Transform from the time data. This is called the Discrete Fourier Transform (DFT). It was called discrete because it relied upon the System taking samples (digitizing) the time waveform at equally spaced discrete time intervals. The DFT is only an approximation to the FT and figure 3.1 shows the difference between the DFT and the FT.
FT vs DFT
Figure 3.1 Comparison of the FT and DFT of a Rectangular Impulse
There is a single rectangular impulse in the time domain. The FT of this signal is a continuous spectrum as described in section 2.4. However, in the DFT the spectrum is not continuous but discrete. The envelope of the lines that form the DFT looks like the FT. Even though it is an approximation it is good enough for most practical purposes. The algorithms for DFTs took a large amount of computing time and eventually the Fast Fourier Transform (FFT) was produced which greatly increased the speed. The main discovery was that by limiting the number of points to an integral power of 2 (e.g., 2, 4, 8,16 etc.) the algorithm for DFT could be simplified and the speed greatly enhanced. There is no loss of accuracy in the FFT vs the DFT; it is simply a limiting condition.
4. The FFT
There are a several properties of the FFT that need to be understood so that the results can be correctly interpreted.
To differentiate between the time and frequency domain the display points are called LINES in the frequency domain and SAMPLES in the time domain. The FFT works on the entire block of captured samples and the lowest frequency and line spacing is related to the total time of the time record. Thus, if the Time record is T sec then the lowest frequency and line spacing is 1/T. Looked at empirically, this follows because there must be at least one complete cycle of a sine wave available to calculate the frequency and the period of this signal must equal the time record length.
In the frequency domain, the amplitude of each line represents to the RMS power of the component sine waves.
There is one property of the FFT - LEAKAGE - that needs to be discussed in detail.
The easiest way to understand it is through an example: When a periodic sine wave of frequency 1.15 Hz is sampled at 1 second/div (total time record length of 10 sec.) the line spacing is 1/10= 0.1 Hz. The FT of this signal will show a single line at 1.15 Hz. However the FFT will give lines spaced at 0.10 Hz and there will be lines at 1.10 Hz, 1.20 Hz, 1.30 Hz etc. The FFT cannot show a line at 1.15 Hz. If the FFT were a sampled version of the true fourier transform, it would show no response. The FFT is not zero in a case like this, the FFT responds by showing responses on the lines about 1.15 Hz and these might be as shown in figure 4.1. This is called leakage.
Leakage in FFT analysis
Figure 4.1 An Illustration of the leakage present in FFT Analysis
Leakage is not completely unwelcome and to be avoided. If the signal was a pure tone, whose frequency was close to one of the lines, if there were no leakage, it would indicate that there was no signal present and here some leakage would be a useful indicator of the presence of a signal.
In many cases, leakage is not entirely desirable and to control leakage windowing is used as described in section 5.
5 Windowing
5.1 Why Windowing?
One of the main properties of the FFT is that it assumes that the samples in the time record are repeated periodically throughout time, or more simply, it acts upon the data as if it were a loop. As an example, figure 5.1.1 shows an apparently periodic waveform which is captured as shown.
Periodic with discontinuities
Figure 5.1.1: Illustration Of The Capture Of A Periodic Waveform With Induced Discontinuities
As far as the FFT is concerned, it links point X to point Y and in this case there is a discontinuity in the time domain. This jump will introduce high frequency signals into the spectrum from the FFT algorithm.
Thus if there is a discontinuity at each end of the time record windowing is required.
5.2 What are Windows?
Windows are simply mathematical functions that when multiplied by the captured data, prior to being acted upon by the FFT, can affect the degree of leakage resulting . There are several different types of window. However, the two most common types are: - Rectangular - Hanning
The rectangular window simply multiplies each sample by 1 while the Hanning window is a cosine squared function. This has a value of zero at each end of the time record and one at the center.
5.3 What effect do windows have?
The rectangular window is used for signals where there is only a small discontinuity between the two ends of the time record. The Hanning window is used to reduce the effect of any discontinuity between the two ends of the time record Ð in theory it reduces their value to zero.
6 Using the FFT
An FFT can be obtained simply by selecting FFT in the Analysis Menu. However, this will calculate the FFT according to a previous set of conditions. To change the conditions the user accesses the FFT Parameters Menu. This menu enables the user to set up the various parameters concerning the FFT calculation and display conditions. The Parameters are: Windowing, Horizontal and Vertical Scaling, Number of Points and portion of trace to be analyzed.
The destination of the FFT, where it will be displayed is set using the Analysis Menu. Any of the display traces can be selected enabling both the original time record and the FFT to be displayed at the same time.
6.1 Windowing
As discussed previously, the user needs to look at the signal to be transformed before selecting the window function. If there is a significant discontinuity between the left hand side and right hand side of the data record, it would be better to select the Hanning window. This will ensure that the effect of the discontinuity is minimized and reduce the effect of leakage. If there is no discontinuity then the rectangular window will give the best results.
6.2 Horizontal and Vertical Scaling.
There are two options available for scaling - logarithmic and linear.
6.3.Horizontal Scaling
The horizontal scaling can also be either log or linear. It is more normal to use linear frequency scaling, however it can help especially if there are both high and low frequency components as this gives the same frequency resolution at both the high and low frequencies.
6.3.1 Vertical Scaling
One of the reasons for using an FFT is to see low level signals in the presence of large signals. If linear scaling is selected this is not very easy to do, but if log scaling is used it is easier. Normally, with FFT analyzers, the vertical scaling is in decibels and it is quite easy to read signal levels in dBV (decibels with respect to 1V).
6.3.2 Log Scaling
To explain the difference between log and linear scaling an example should help. Assume that after the FFT there are two frequency components one at 1 V RMS and the other at 1 mV RMS.
With linear scaling if the V component is 5 cm high the other component should be:
5/1000 cm = 0.5 mm high.
This component would be hardly visible on the display.
Using log scaling with the 1 V component still 5 cm high, the 1 mV component would be 1.25 cm high. This makes it far easier to view low level signals.
6.4 Number of Points
There are essentially two options available - either to select a number of points from the entire display memory or to select part of the display waveform. Both options are described below. The user can select to calculate 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536 or 131072 points of the display memory. Obviously the last two of these values are only available if the instrument is fitted with a 200 k memory.