PHYS 3152 Methods of Experimental Physics I E5. Frequency Analysis of Electronic Signals

Purpose PHYS 3152 Methods of Experimental Physics I E5. Frequency Analysis of Electronic Signals This experiment will introduce you to the basic principles of frequency domain analysis of electronic signals. In particular, you will study the Fast Fourier Transform (FFT), which is a convenient and powerful tool for performing frequency domain analysis on a variety of signals. Equipment and components Agilent DSO-X 2002A Digital Storage Oscilloscope, HP54654A training kit and 22kΩ resistor. Background Normally, when a signal is measured with an oscilloscope, it is viewed in the time domain. That is, the vertical axis is voltage and the horizontal axis is time. For many signals, this is the most logical and intuitive way to view them. However, when the frequency content of the signal is of interest, it makes sense to view the signal in the frequency domain. In this case, the horizontal axis becomes frequency. Using Fourier theory, one can mathematically relate the time domain with the frequency domain. The Fourier transform is given by: - i2π ft V ( f ) v() t e dt + = (1) where v(t) is a voltage signal represented in the time domain and V(f) is its Fourier transform in the frequency domain. Some typical signals represented in the time domain and the frequency domain are shown in the figure below. E5-1/16

The discrete (or digitized) version of the Fourier transform is called the Discrete Fourier Transform (DFT). This transform takes digitized time domain data and computes the frequency domain representation. In this case, Equation 1 can be written as: N 1 0 τ i2 π f ( jτ) τ (2) j= 0 V( f) = v(t + j ) e where t 0 is the starting time and τ is the smallest sampling time. The Fourier transform defined in Equation 1 assumes the life of a signal from - to +. In real experiment, however, one can only obtain a signal with a finite lifetime. In this case, we can define V T (f) which is the Fourier transform of the signal in a time period T. Another function which is widely used in the experiment is the frequency power spectrum defined as 1 P( f) lim V ( f) 2 = T (3) T T The power spectrum P(f) is a real function and has many important applications in electrical engineering and physics. The Agilent DSO-X 2002A Digital Storage Oscilloscope uses a particular algorithm, called the Fast Fourier Transform (FFT), for computing the DFT. The FFT function in the Agilent DSO-X 2002A can acquire up to 65,536 data points. When the frequency span is at maximum, all points are displayed. This display extends in frequency from 0 to f s /2, where f s is the sampling rate of the time record. Because the Agilent DSO-X 2002A uses the interleaved mode* when one channel is in use, the frequency resolution f res is given by: f res = (f s /2) / N where N is the number of points acquired for the FFT record, that can be up to 65,536. In this interleaved mode, the time between two adjacent samples τ is the reciprocal of f s /2. The sampling rate f s can be set by adjusting the Horizontal scale knob of the scope. As shown in Equation 2, the unit of the Fourier transform V(f) is Volt Second. Because the sampling time τ in Equation 2 is a simple constant, it is often normalized to unity in the DFT. In this case, the unit of V(f) becomes the same as that of the signal v(t). The vertical axis of the FFT display in the Agilent DSO-X 2002A scope is logarithmic, displayed in dbv (decibels relative to 1 Volt RMS): dbv = 20 log (V(f)/1Volt RMS). Thus a 1 Volt RMS sinusoidal wave (2.8 Volts peak-to-peak) will read 0 dbv on the FFT display. * Please read Appendix 1 Introduction to Fourier Theory for details. Procedure The following four laboratory exercises are designed to illustrate the relationship between frequency resolution, effective signal sampling rate, spectral leakage, windowing, and aliasing in the FFT (Details about these concepts will be discussed separately in the lecture class). Exercise 1 This exercise illustrates the relationship between the sampling rate and the resulting frequency resolution for the spectral analysis using the FFT. The spectral leakage properties of the Rectangular and Hanning windows are also demonstrated. A waveform generator is built into the DSO-X 2002A oscilloscope. Press the [Wave Gen] key to access the Waveform Generator Menu and enable the waveform generator output on the front panel Gen Out BNC connector. When waveform generator output is enabled, the [Wave Gen] key is illuminated. From the Waveform Generator Menu, select a 3.5V (peak-topeak), 1 khz sinusoidal signal and input it to channel 1 of the scope. Use [Auto Scale] to check the time-domain waveform. Next, press [Math] key to display the Math Waveform Menu, set Function as f(t), select FFT as Operator and channel 1 as Source. Manually adjust the settings of the horizontal and vertical scales, Center Frequency, Frequency Span the oscilloscope so that you obtain a nice peak in the FFT display. (See the Appendix 2 for assistance in adjusting the FFT Menu settings.) Use the Horizontal scale knob to set the sampling rate to 50 ksa/s. Use [Cursors] to measure the fundamental frequency of the peak. Adjust the FFT settings and use [Cursors] to measure the main lobe width (full width at half E5-2/16

maximum) for the Hanning window and the Rectangular window, respectively. Notice the difference in the spectral leakage for the two windows. Save and print out the FFT spectra under the two different windows. Repeat the above measurements using a sampling rate of 10 ksa/s. Questions 1. Should the sampling rate be increased or decreased in order to improve the frequency resolution of the FFT? (The frequency resolution is defined as the smallest frequency that can be resolved by the FFT.) 2. Is there a limit on the smallest frequency that can be resolved by a fixed, 65,536 point FFT? 3. Does the Hanning window exhibit more or less spectral leakage when compared with the Rectangular window? Exercise 2 The goal of this exercise is to study the relationship between the sampling rate f s and aliasing. The upper frequency f s /2 for the FFT spectrum is also called the folding frequency. Frequencies that would normally appear above f s /2 (and hence outside the useful range of the FFT) are folded back into the frequency domain of the display. These unwanted frequency components are called aliases, since they erroneously appear under the alias of another frequency. Aliasing is avoided if the sampling rate is greater than twice the bandwidth of the signal being measured. Use the [Wave Gen] function to generate a 3.5V (peak-to-peak), 100 khz sinusoidal input signal to channel 1 of the scope. Use [Auto Scale] to check the time-domain waveform. Turn on the FFT Function and use the Horizontal scale knob to set the sampling rate to 500 ksa/s. Press [Auto Setup] FFT function to get a full FFT spectrum. Now you can slowly increase the frequency of the sinusoidal signal to roughly 240 khz, allowing the FFT display to stabilize at several points along the way. You should see the FFT peak move to the right as the sinusoidal frequency is increased. Print out a FFT spectrum. Aliasing occurs as the frequency of the sinusoid exceeds 250 khz. As the frequency is being swept from 250 khz to 500 khz, the main peak moves to the left of the display. When the frequency is further increased from 500 khz to 750 khz, the main peak moves to the right again. Set the sinusoid frequency to 400 khz. Use [Cursors] function to measure the peak frequency displayed by the FFT. Because of aliasing, the FFT should erroneously indicate that the peak occurs at about 100 khz. Print out this FFT spectrum. Repeat the above measurements using a sampling rate of 1 MSa/s. Questions 1. If a 120 khz sinusoid is sampled at 50 ksa/s, at what frequency will the aliased 120 khz component appear in the FFT display? 2. Is aliasing affected by the choice of window function? Exercise 3 This exercise demonstrates the use of the FFT for analyzing the spectral content of a square wave and a triangle wave. Unlike the sinusoidal wave, which has only one frequency component, the square and triangle waves have many frequency components. As a result, their FFT spectra will show many peaks. The amplitude of these peaks can be calculated exactly and the table below shows the Fourier series coefficients (magnitude only) of a square wave and a triangle wave. Use the definition of V(f) in Equation 1 and calculate the Fourier amplitudes of the square wave and the triangle wave. Compare your results with those listed in the table below. Notice that the measured absolute value of the magnitude of each harmonic may vary with the amplitude of the input signals. But the magnitude difference (in dbv) between any two harmonics is independent of the amplitude of a given input signal. In the table below, we have assumed that the peak-to-peak value of both the square wave and triangular wave is 1/ 2 Volt. E5-3/16

Square Wave Triangle Wave Harmonic Magnitude (dbv) Harmonic Magnitude (dbv) 1-9.943 1-13.865 3-19.485 3-32.950 5-23.922 5-41.824 7-26.845 7-47.669 9-29.028 9-52.035 11-30.771 11-55.521 13-32.222 13-58.423 Table 1 Fourier series coefficients (magnitude) of square and triangle wave Generate a 2V (peak-to-peak), 15 khz square wave using the [Wave Gen] function and connect it to channel 1 of the scope. Use [Auto Scale] to check the time-domain waveform. Turn on the FFT Function and adjust the FFT Menu settings, so that you can see many peaks in the FFT display. Use the Horizontal scale knob to set the sampling rate to 1 MSa/s. Measure the magnitude of the peaks as in Table 1. HINT: Notice that the flat top window provides the most accurate measurement of the peak amplitude. Find the relative magnitude difference between the peaks and compare your results with the theoretical values shown in Table 1. Next, go to a sampling rate of 250 ksa/s and show how the higher harmonics are aliased and appear in the low frequency range of the FFT display. Make sure to print out your FFT spectra. Repeat the above measurements for a 2V (peak-to-peak), 15 khz triangle wave (Ramp in [Wave Gen] function). Questions 1. Is prior knowledge of the signal s bandwidth required? 2. Is it still possible to obtain useful information from the FFT display when components of the signal are aliased? Exercise 4 This last exercise illustrates how the FFT can be used to analyze the spectral content of a signal, which consists of the sum of two sinusoids. A similar analysis using time-domain techniques can be difficult to perform on an oscilloscope. Figure 1 Circuit for spectral content analysis E5-4/16

Set the source V 1 at 3.5V (peak-to-peak) and the source V 2 at 3.5V (peak-to-peak), 2 khz separately. Connect two sinusoidal sources as shown in Figure 1. The 22kΩ resistor is used to increase the impedance of the scope s [Wave Gen] output, so that the other signal V 1 can function properly. Use the scope to obtain a time-domain display of the voltage across the two sources. Notice that you may need to use the Run and Stop controls to take snapshot of the resulting oscilloscope traces, because the time-domain display of the two sinusoidal waves is unstable. Turn on the FFT Function and adjust the FFT Menu settings, so that you can see two peaks on the FFT display. Use the Horizontal scale knob to set the sampling rate to 25 ksa/s. Find the frequency location of the two peaks and get a hard copy of your FFT display. Slowly reduce the frequency of the source V 2 and find the minimum frequency difference, at which the two peaks can still be resolved. (You may need to adjust the FFT Menu settings to zoom-in on the two peaks.) Adjust the horizontal scale to see how the frequency resolution is affected by the sampling rate. Now, change the sampling rate to 250 ksa/s. Increase the frequency of the source V 2 to up to 24 khz and watch how the FFT display changes with the frequency increase. Notice that when the sampling rate is reduced back to 25 ksa/s, the high frequency term V 2 aliases to a lower frequency near the 1 khz term. Questions 1. Is it necessary to have a stable time-domain display in order to analyze the frequency content of a signal? 2. How does the presence of a high frequency component affect the frequency resolution for analyzing narrow band components of the signal? 3. Based on the results of this exercise, why would it be difficult to use the FFT to analyze the bandwidth of a typical amplitude modulated signal? E5-5/16

Fourier Theory Appendix 1: Introduction to Fourier Theory (Reference: Agilent Product Note Fourier Theory and Practice) Normally, when a signal is measured with an oscilloscope, it is viewed in the time domain (Figure A1a). That is, the vertical axis is voltage and the horizontal axis is time. For many signals, this is the most logical and intuitive way to view them. But when the frequency content of the signal is of interest, it makes sense to view the signal in the frequency domain. In the frequency domain, the vertical axis is still voltage but the horizontal axis is frequency (Figure A1b). The frequency domain display shows how much of the signal's energy is present at each frequency. For a simple signal such as a sine wave, the frequency domain representation does not usually show us much additional information. However, with more complex signals, the frequency content is difficult to uncover in the time domain and the frequency domain gives a more useful view of the signal. Figure A1 (a) A signal shown as a function of time. (b) A signal shown as a function of frequency. Fourier theory (including both the Fourier Series and the Fourier Transform) mathematically relates the time domain and the frequency domain. The Fourier transform is given by: - i2π ft V ( f ) = v() t e dt + We won't go into the details of the mathematics here, since there are numerous books which cover the theory extensively (see references). Some typical signals represented in the time domain and the frequency domain are shown in the figure in background section. The Fast Fourier Transform The discrete (or digitized) version of the Fourier transform is called the Discrete Fourier Transform (DFT). This transform takes digitized time domain data and computes the frequency domain representation. While normal Fourier theory is useful for understanding how the time and frequency domain relate, the DFT allows us to compute the frequency domain representation of real-world time domain signals. This brings the power of Fourier theory out of the world of mathematical analysis and into the realm of practical measurements. The Agilent DSO-X 2002A scope uses a particular algorithm, called the Fast Fourier Transform (FFT), for computing the DFT. The FFT and DFT produce the same result and the feature is commonly referred to as simply the FFT. The Agilent DSO-X 2002A scope normally digitizes the time domain waveform. The FFT function uses up to 65,536 sampling points to produce a frequency domain display of maximum all 65,536 data points. This frequency domain display extends in frequency from 0 to f s /2, where f s is the sampling rate of the time record (Figure A2a). E5-6/16

(a) (b) Figure A2 (a) The sampled time domain waveform. (b) The resulting frequency domain plot using the FFT. Typically, the sampling rate is the reciprocal of the time between samples and can be set by adjusting the Horizontal scale knob of the Agilent DSO-X 2002A scope when two channels are operating. But when only one channel is in use and at slower time/div settings, the Agilent DSO-X 2002A will activate the interleaved mode, such that the sampling rate is double, and extra points are used to increase vertical resolution. So for any particular horizontal scale setting, the FFT produces a frequency domain representation that extends from 0 to f s /2 (Figure A2b). The sampling rate is changed when the time/div knob is turned. Note that the sampling rate for the FFT can be much higher than the bandwidth of the scope. The bandwidth of the DSO-X 2002A scope is 70 MHz, while the sampling rate of the acquisition can be as high as 2 GHz. The default frequency domain display covers the normal frequency range of 0 to f s /2. The Center Frequency and Frequency Span controls can be used to zoom in on narrower frequency spans within the basic 0 to f s /2 range of the FFT. These controls do not affect the FFT computation, but instead cause the frequency domain points to be re-plotted in expanded form. Aliasing The frequency f s /2 is also known as the folding frequency. Frequencies that would normally appear above f s /2 (and, therefore, outside the useful range of the FFT) are folded back into the frequency domain display. These unwanted frequency components are called aliases, since they erroneously appear under the alias of another frequency. Aliasing is avoided if the sampling rate is greater than twice the bandwidth of the signal being measured. The frequency content of a triangle wave includes the fundamental frequency and a large number of odd harmonics with each harmonic smaller in amplitude than the previous one. In Figure A3a, a 26 khz triangle wave is shown in the time domain and the frequency domain. Figure A3b shows only the frequency domain representation. The leftmost large spectral line is the fundamental. The next significant spectral line is the third harmonic. The next significant spectral line is the fifth harmonic and so forth. Note that the higher harmonics are small in amplitude with the 19th harmonic just visible above the FFT noise floor. The frequency of the 19th harmonic is 19 x 26 khz = 494 khz, which is within the folding frequency of f s /2, (500 ksa/sec) in Figure A3b. Therefore, no significant aliasing is occurring. E5-7/16

Figure A3 (a) The time domain and frequency domain displays of a 26kHz triangle wave. (b) Frequency spectrum of a triangle wave. (c) With a lower sampling rate, the upper harmonics appear as aliases. (d) With an even lower sampling rate, only the fundamental and third harmonic are not aliased. Figure A3c shows the FFT of the same waveform with the Horizontal scale knob turned one click to the left, resulting in a sampling rate of 500 ksa/sec and a folding frequency of 250 ksa/sec. Now the upper harmonics of the triangle wave exceed the folding frequency and appear as aliases in the display. Figure A3d shows the FFT of the same triangle wave, but with an even lower sampling rate (250 ksa/sec) and folding frequency (125 ksa/sec). This frequency plot is severely aliased. Often the effects of aliasing are obvious, especially if the user has some idea as to the frequency content of the signal. Spectral lines may appear in places where no frequency components exist. A more subtle effect of aliasing occurs when low level aliased frequencies appear near the noise floor of the measurement. In this case the baseline can bounce around from acquisition to acquisition as the aliases fall slightly differently in the frequency domain. Aliased frequency components can be misleading and are undesirable in a measurement. Signals that are bandlimited (that is, have no frequency components above a certain frequency) can be viewed alias-free by making sure that the sampling rate is high enough. The sampling rate is kept as high as possible by choosing a fast time/div setting. While fast time/div settings produce high sampling rates, they also cause the frequency resolution of the FFT display to degrade. If a signal is not inherently bandlimited, a lowpass filter can be applied to the signal to limit its frequency content (Figure A4). This is especially appropriate in situations where the same type of signal is measured often and a special, dedicated lowpass filter can be kept with the scope. E5-8/16

Figure A4 A lowpass filter can be used to band limit the signal, avoiding aliasing. Leakage The FFT operates on a finite length time record in an attempt to estimate the Fourier Transform, which integrates over all time. The FFT operates on the finite length time record, but has the effect of replicating the finite length time record over all time (Figure A5). With the waveform shown in Figure A5a, the finite length time record represents the actual waveform quite well, so the FFT result will approximate the Fourier integral very well. Figure A5 (a) A waveform that exactly fits one time record. (b) When replicated, no transients are introduced. However, the shape and phase of a waveform may be such that a transient is introduced when the waveform is replicated for all time, as shown in Figure A6. In this case, the FFT spectrum is not a good approximation for the Fourier Transform. Figure A6 (a) A waveform that does not exactly fit into one time record. (b) When replicated, severe transients are introduced, causing leakage in the frequency domain. Since the scope user often does not have control over how the waveform fits into the time record, in general, it must be assumed that a discontinuity may exist. This effect, known as LEAKAGE, is very apparent in the frequency domain. The transient causes the spectral line (which should appear thin and slender) to spread out as shown in Figure A7. E5-9/16

Figure A7 Leakage occurs when the normally thin spectral line spreads out in the frequency domain. The solution to the problem of leakage is to force the waveform to zero at the ends of the time record so that no transient will exist when the time record is replicated. This is accomplished by multiplying the time record by a WINDOW function. Of course, the window modifies the time record and will produce its own effect in the frequency domain. For a properly designed window, the effect in the frequency domain is a vast improvement over using no window at all. Four window functions are available in the Agilent DSO-X 2002A scope: Hanning, Flat Top, Rectangular and Blackman-Harris. The Hanning window provides a smooth transition to zero as either end of the time record is approached. Figure A8a shows a sinusoid in the time domain while Figure A8b shows the Hanning window which will be applied to the time domain data. The windowed time domain record is shown in Figure A8c. Even though the overall shape of the time domain signal has changed, the frequency content remains basically the same. Figure A8 (a) The original time record. (b) The Hanning Window. (c) The windowed time record. The spectral line associated with the sinusoid spreads out a small amount in the frequency domain as shown in Figure A9 (Figure A9 is expanded in the frequency axis to show clearly the shape of the window in the frequency domain.) E5-10/16

Figure A9 The Hanning Window has a relatively narrow shape in the frequency domain. The shape of a window is a compromise between amplitude accuracy and frequency resolution. The Hanning window, compared to other common windows, provides good frequency resolution at the expense of somewhat less amplitude accuracy. The Flat Top window has fatter (and flatter) characteristic in the frequency domain, as shown in Figure A10. (Again, the figure is expanded in the frequency axis to show clearly the effect of the window.) The flatter top on the spectral line in the frequency domain produces improved amplitude accuracy, but at the expense of poorer frequency resolution (when compared with the Hanning window). Figure A10 The Flat Top window has a wider, flat-topped shape in the frequency domain. The Rectangular window (also referred to as the Uniform window) is really no window at all; all of the samples are left unchanged. Although the uniform window has the potential for severe leakage problems, in some cases the waveform in the time record has the same value at both ends of the record, thereby eliminating the transient introduced by the FFT. Such waveforms are called SELF-WINDOWING. Waveforms such as sine bursts, impulses and decaying sinusoids can all be self-windowing. A typical transient response is shown in Figure A11a. As shown, the waveform is self-windowing because it dies out within the length of the time record, reducing the leakage problem. E5-11/16

Figure A11 (a) A transient response that is self-windowing. (b) A transient response which requires windowing. If the waveform does not dissipate within the time record (as shown in Figure A11b), then some form of window should be used. The Blackman Harris window has a lower secondary lobes compared to the Hanning window. It reduces time resolution compared to Rectangular window, but improves the capacity to detect smaller impulses. 1. The effect of a time domain window in the frequency domain is analogous to the shape of the resolution bandwidth filter in a swept spectrum analyzer. 2. The shape of a perfect sinusoid in the frequency domain with a window function applied is the Fourier transform of the window function. Selecting a Window Most measurements will require the use of a window such as the Hanning or Flat Top windows. These are the appropriate windows for typical spectrum analysis measurements. Choosing between these two windows involves a tradeoff between frequency resolution and amplitude accuracy. Having used the time domain to explain why leakage occurs, now the user should switch into frequency domain thinking. The narrower the passband of the window's frequency domain filter, the better the analyzer can discern between two closely spaced spectral lines. At the same time, the amplitude of the spectral line will be less certain. Conversely, the wider and flatter the window's frequency domain filter is, the more accurate the amplitude measurement will be and, of course, the frequency resolution will be reduced. Choosing between two such window functions is really just choosing the filter shape in the frequency domain. The Rectangular and Blackman-Harris windows should be considered windows for special situations. The Rectangular window is used where it can be guaranteed that there will be no leakage effects. The Blackman-Harris window has good side lobe rejection, but the downside is that it has a moderately wide main lobe. References: Brigham, E. Oran, The Fast Fourier Transform and Its Applications., Englewood Cliffs, NJ: Prentice-Hall, Inc., 1988. Hewlett-Packard Company. "Fundamentals of Signal Analysis", Application Note 243, Publication Number 5952-8898, Palo Alto, CA, 1981. McGille Clare D. and George R. Cooper. Continuous and Discrete Signal and System Analysis. New York: Holt, Rhinehart and Winston, Inc.,1974. Ramirez, Robert W. The FFT Fundamentals and Concepts. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1985. Stanley, William D., Gary R. Dougherty and Ray Dougherty. Digital Signal Processing, 2nd ed. Reston, VA: Reston Publishing Company, Inc., 1984. Witte, Robert A. Spectrum and Network Measurements. Englewood Cliffs, NJ: Prentice- Hall, Inc., 1991. E5-12/16

Introduction to FFT Appendix 2: Operation of the FFT function FFT is used to compute the fast Fourier transform using analog input channels or an arithmetic operation g(t). FFT takes the digitized time record of the specified source and transforms it to the frequency domain. When the FFT function is selected, the FFT spectrum is plotted on the oscilloscope display as magnitude in dbv versus frequency. The readout for the horizontal axis changes from time to frequency (Hertz) and the vertical readout changes from volts to dbv. FFT Units: 0 dbv is the amplitude of a 1 Vrms sinusoid. When the FFT source is channel 1 or channel 2, FFT units will be displayed in dbv when channel unit is set to Volts and channel impedance is set to 1MΩ. DC Value: The FFT computation produces a DC value that is incorrect. It does not take the offset at center screen into account. The DC value is not corrected in order to accurately represent frequency components near DC. All DC measurements should be performed in normal oscilloscope mode. Aliasing: When using FFT, it is important to be aware of aliasing. This requires that the operator have some knowledge as to what the frequency domain should contain and also consider the sampling rate, frequency span, and oscilloscope vertical bandwidth when making FFT measurements. The FFT resolution (The quotient of the sampling rate and the number of FFT points) is displayed directly above the softkeys when the FFT Menu is displayed. Aliasing happens when there are insufficient samples acquired on each cycle of the input signal to recognize the signal. This occurs whenever the frequency of the input signal is greater than the Nyquist frequency (sample frequency divided by 2).When a signal is aliased, the higher frequency components show up in the FFT spectrum at a lower frequency. Figure A12 illustrates aliasing. In the left waveform, the sampling rate is set to 250 ksa/s, and the oscilloscope displays the correct spectrum. In the right waveform, the sampling rate is reduced by more than one-half (100 ksa/s), causing the components of the input signal above the Nyquist frequency to be mirrored (aliased) on the display, which is illustrated by the solid black line. Since the frequency span goes from 0 to the Nyquist frequency, the best way to prevent aliasing is to make sure that the frequency span is greater than the frequencies present in the input signal. E5-13/16

Spectral Leakage: Figure A12 Aliasing The FFT operation assumes that the time record repeats. Unless there is an integral number of cycles of the sampled waveform in the record, a discontinuity is created at the end of the record. This is referred to as leakage. In order to minimize spectral leakage, windows that approach zero smoothly at the beginning and end of the signal are employed as filters to the FFT. The FFT Menu provides four windows: Hanning, Flat Top, Rectangular and Blackman-Harris. To display a FFT waveform: 1. Press the [Math] key, press the [Function] softkey and select f(t), press the [Operator] softkey and select FFT. Source 1 selects the source for FFT. Span sets the overall width of the FFT spectrum that you see on the display (left to right). Divide span by 10 to calculate the number of Hertz per division. It is possible to set Span above the maximum available frequency, in which case the displayed spectrum will not take up the whole screen. Press the [Span] softkey, then turn the Entry knob to set the desired frequency span of the display. Center sets the FFT spectrum frequency represented at the center vertical grid line of the display. It is possible to set the Center to values below half the span or above the maximum available frequency, in which case the displayed spectrum will not take up the whole screen. Press the [Center] softkey, then turn the Entry knob to set the desired center frequency of the display. Scale lets you set your own vertical scale factors for FFT expressed in db/div (decibels/ division). Use this multiplexed scale knob just next to the [Math] key to re-size the FFT waveform. Offset lets you set your own offset for the FFT. The offset value is in db and is represented by the center horizontal grid line of the display. Use this multiplexed position knob just next to the [Math] key to re-position the FFT waveform. More FFT displays the more FFT Settings Menu. 2. Press the [More FFT] softkey to display the additional FFT settings. Window selects a window to apply your FFT input signal: Hanning window for making accurate frequency measurements or for resolving two frequencies that are close together. E5-14/16

Flat top window for making accurate amplitude measurements of frequency peaks. Rectangular good frequency resolution and amplitude accuracy, but use only where there will be no leakage effects. Use on self-windowing waveforms such as pseudo random noise, impulses, sine bursts and decaying sinusoids. Blackman Harris window reduces time resolution compared to a rectangular window, but improves the capacity to detect smaller impulses due to lower secondary lobes. Vertical Units lets you select Decibels or V RMS as the units for the FFT vertical scale. Auto Setup sets the frequency Span and Center to values that will cause the entire available spectrum to be displayed. The maximum available frequency is half the FFT sample rate, which is a function of the time per division setting. The FFT resolution is displayed above the softkeys. NOTE: Scale and offset considerations: If you do not manually change the FFT scale or offset settings, when you turn the Horizontal scale knob, the span and center frequency settings will automatically change to allow optimum viewing of the full spectrum. If you do manually set scale or offset, turning the Horizontal scale knob will not change the span or center frequency settings, allowing you see better detail around a specific frequency. Pressing the FFT [Auto Setup] softkey will automatically rescale the waveform and span and center will again automatically track the horizontal scale setting. 3. To make cursor measurements, press the [Cursors] key and set the [Source] softkey to Math: f(t). Use the X1 and X2 cursors to measure frequency values and difference between two frequency values (ΔX). Use the Y1 and Y2 cursors to measure amplitude in dbv and difference in amplitude (ΔY). 4. To make other measurements, press the [Meas] key and set the [Source] softkey to Math: f(t). You can make peak-to-peak, maximum, minimum, and average db measurements on the FFT waveform. FFT Measurement Hints 1. The number of points acquired for the FFT record is normally 65,536 and when frequency span is at maximum, all points are displayed. Once the FFT spectrum is displayed, the frequency span and center frequency controls are used much like the controls of a spectrum analyzer to examine the frequency of interest in greater detail. Place the desired part of the waveform at the center of the screen and decrease frequency span to increase the display resolution. As frequency span is decreased, the number of points shown is reduced, and the display is magnified. 2. While the FFT spectrum is displayed, use the [Math] and [Cursors] keys to switch between measurement functions and frequency domain controls in FFT menu. 3. Decreasing the sampling rate by selecting a slower time/div setting will increase the low frequency resolution of the FFT display and also increase the chance that an alias will be displayed. The resolution of the FFT is one-half of the sampling rate divided by the number of points in the FFT. The actual resolution of the display will not be this fine as the shape of the window will be the actual limiting factor in the FFT s ability to resolve two closely spaced frequencies. A good way to test the ability of the FFT to resolve two closely spaced frequencies is to examine the sidebands of an amplitude modulated sine wave. E5-15/16

4. For the best vertical accuracy on peak measurements: Make sure the probe attenuation is set correctly. The probe attenuation is set from the Channel menu if the operand is a channel. Set the source sensitivity so that the input signal is near full screen, but not clipped. Use the Flat Top window. Set the FFT sensitivity to a sensitive range, such as 2 db/division. 5. For best frequency accuracy on peaks: Use the Hanning window. Use cursors to place an X cursor on the frequency of interest. Move the X cursor (the interested peak) to center by the [Center] softkey. Adjust frequency span for better cursor placement. Return to the Cursors menu to fine tune the X cursor. E5-16/16