The Pulse Desensitization Factor
It is well understood that the occupied spectrum width of a pulsed signal is inversely proportional to pulse width, accompanied by an associated reduction in spectrum level as the spectrum width increases. In Fourier analysis, this phenomenon is known as reciprocal spreading. As a consequence, a fixed-bandwidth receiver, such as a spectrum analyzer, will show a lower signal level as the input pulse width decreases. This phenomenon is known as pulse desensitization, and the product of pulse width and receiver bandwidth is known as the desensitization factor.
The desensitization factor has a number of practical applications such as in the determination of receiver dynamic range limits or the calculation of the expected fully integrated signal level for electromagnetic interference regulation purposes. On the surface, this determination appears to be a simple calculation or comparison. A greater pulse-width-times-bandwidth product is always more than a smaller product, and that is all that is required to be known. However, a number of traps are hidden in such an approach. The purpose of this article is to explain the deeper meaning of the pulse desensitization factor.
Pulse Desensitization Basics
A sine wave of magnitude A will appear as the RMS value AÖ 2 and the main lobe level for a pulsed signal is displayed at AtBi Ö 2 on a spectrum analyzer. The ratio of these values, tBi , is known as the pulse desensitization factor. The pulse desensitization factor is the product of pulse width t and spectrum analyzer impulse bandwidth Bi . The impulse bandwidth is approximately equal to the 6 dB bandwidth B6 , which also is approximately equal to 3/2 the 3 dB bandwidth B3 . Thus, it is common to state that the pulse desensitization factor is 1.5tB, or 20log(1.5tB), as a decibel difference in the spectrum display level between the unmodulated sine wave and pulse-modulated spectrum.
For example, a 1 ms-wide pulse will appear as 20log0.15 = -16.5 dB compared to the unmodulated carrier when viewed using a 100 kHz 3 dB bandwidth. If the bandwidth is changed to 10 kHz, the pulse desensitization factor will be reduced by a factor of 10 and the displayed level will drop by 20log10 = 20 dB to -36.5 dB. A 20 dB level reduction also will occur when the pulse width is reduced by a factor of 10 to 0.1 ms while keeping the bandwidth constant at 100 kHz. After all, all that counts is the product tB, not the individual factors t and B. However, such reasoning misses a subtle but very important point. Changing the measuring bandwidth and changing the pulse width are not interchangeable even though the product tB is not affected.
Impact on the Side Lobes
The spectrum of a pulse-modulated signal consists of a main lobe centered at the carrier frequency and side lobes around it. The pulse desensitization relationship 1.5tB applies to the center of the main lobe. A change in t or B will have the same result, depending on the product tB. However, the situation is more complicated at the side lobes. The display levels of all parts of the spectrum change equally in proportion to the measuring bandwidth B, and all individual lobe levels change in proportion to pulse width change. The side lobe frequency position also changes with pulse width. (Side lobe frequency spacing is inversely proportional to pulse width.) Thus, the first side lobe is centered at a frequency offset of 1.5/t from the main lobe, or 1.5 MHz for a 1 ms pulse. The actual spacing is off slightly from the 1.5 ratio. The peak value is located at an angle where tan x = x, not where sin x is a maximum. The result is virtually the same and the approximation simplifies the calculations.1 The second side lobe is located at 2.5/t. Hence, the spectrum level does not change with pulse width at a chosen fixed frequency.
Using a rectangular 1 ms-wide pulse shape as an example, a 100 kHz 3 dB spectrum analyzer bandwidth will show a main lobe that is 16.5 dB below the unmodulated carrier. The first side lobe is located at 13.3 dB below the main lobe, while the fifth side lobe spaced at 5.5/t from the main lobe is 24.7 dB below the main lobe. A change of instrument bandwidth will reduce the spectrum level equally at all lobe and frequency positions. However, suppose the pulse width is changed instead. A 1 ms pulse width yields a main lobe 16.5 dB below the unmodulated carrier. The first side lobe is located at 1.5/1 ms = 1.5 MHz from the main lobe at a level of 16.5 + 13.3 = 29.8 dB below the unmodulated carrier and a fifth side lobe is located at 5.5 MHz from the carrier at a level of 16.5 + 24.7 = 41.2 dB below the carrier. Now suppose the pulse width is reduced from 1 to 0.27 ms. The main lobe is 20log(0.15) (0.27) = 27.85 dB below the unmodulated carrier. The first side lobe is still located 13.3 dB below the main lobe or 27.85 + 13.3 = 41.2 dB below the unmodulated carrier, but the frequency position of the first side lobe has shifted to 1.5/0.27 = 5.5 MHz because the pulse width has changed from 1 to 0.27 ms. The new result at 41.2 dB below the main lobe is identical to the result for the fifth side lobe of the 1 ms pulse width at the same 5.5 MHz frequency offset.
Large tB Product Results
The pulse width t and spectrum analyzer bandwidth B are independent of each other. Any values are possible. Suppose a 10 MHz bandwidth is used to measure a 1 ms pulse width. In this case, tB = 10, and it would appear that the pulse-modulated spectrum is larger than the unmodulated carrier. However, this scenario is not possible. The reason for this result is that the tB relationship only holds for fairly small values of tB (below approximately 0.3). A more complicated expression has been derived when tB is large.2 A large tB will approach the unmodulated carrier at the main lobe position asymptotically while the side lobe results remain unchanged at the same frequency settings.
Transition tB Product Results
Here is where things get really interesting. Suppose a tB product of 5 using a 1 MHz bandwidth and 5 ms pulse width is used. There is hardly any desensitization and the main lobe level essentially is equal to the unmodulated carrier. The bandwidth now is reduced to 30 kHz where the desensitization factor 1.5tB = 0.225. At 20log1.5tB, the main lobe will be reduced by 13 dB. The side lobes also will be reduced in the same proportion, and the level of the entire spectrum will increase and decrease as the measurement bandwidth is changed.
Now the measuring bandwidth is kept at the original 1 MHz, but the pulse width is reduced from 5 to 0.15 ms. A pulse desensitization factor of 0.225 is created and the main lobe amplitude drops by 13 dB. Each of the individual side lobes also will be reduced by the same 13 dB. Thus, the first side lobe of the 0.15 ms-wide pulse will be located 13 dB below the first side lobe of the 5 ms pulse, but its frequency position is more than 30-times further out from the carrier where the spectrum of the 5 ms pulse is more than 13 dB below the main lobe. Hence, the narrower pulse spectrum magnitude will be down in the middle but up on the sides compared to the wider pulse spectrum. This result is counter intuitive and not what most people expect. The usual expectation based on Fourier mathematics is that the spectrum level of a narrow pulse is always smaller than that for a wider pulse. This assumption based on Fourier mathematics is indeed correct, but this peculiar result is obtained because of the transition between a large tB product where there is no pulse desensitization and a small tB product where desensitization applies.
Figure 1 shows results at small tB products. Both examples display results for identical tB products of 0.1 and 0.03, but the spectra differ because the change in tB was obtained in one by a bandwidth change and in the other by a pulse width change.
Three spectrum traces are shown. The full screen display shows the unmodulated carrier. The next level down shows a sin x/x spectrum of a 1 ms pulse width intercepted by a 100 kHz bandwidth for tB = 0.1. The third trace illustrates what happens as the bandwidth is reduced to 30 kHz and shows a lower display level than that for the 100 kHz setting. Note that the display level difference is the same at all frequencies; the delta levels 1, 2 and 3 are equal.
The plots for the pulse width change case start with the same two traces as for the example with the bandwidth change, that is, an unmodulated carrier and the pulsed spectrum of a 1 ms-wide pulse intercepted with a 100 kHz bandwidth. The third trace shows the spectrum of a 0.3 ms pulse intercepted by a 100 kHz bandwidth, which yields the same tB product of 0.03 as the 1 ms pulse and 30 kHz bandwidth. However, the spectrum result is different. No spectrum level change is displayed except in the center of the main lobe. Indeed, all spectra, regardless of pulse width, will follow the side lobe peak envelope curve shown on the right side. The change in magnitude as a function of tB change will occur only on a side-lobe-to-side-lobe comparison for corresponding lobe numbers as shown for the first side lobe (marked delta 4 on the left side of the trace).
Figure 2 shows what happens at large tB products. The starting point is the unmodulated carrier. The next lower trace shows the spectrum of a 6 ms pulse intercepted by a 3 MHz bandwidth for tB = 18. Hardly any pulse desensitization exists at such a large tB, and the display level is reduced very little in the center. The large difference in the traces is noticeable at the sides where the pulsed signal flares out, showing the turn-on and turn-off transition spectrum while the unmodulated carrier has no side lobe content. The next lower trace shows the same 6 ms pulse width intercepted by a 1 MHz bandwidth. This plot represents a product tB = 6 compared to 18 for the previous trace using a 3 MHz bandwidth. Both tB products are large, indicating very little pulse desensitization. Hence, there is only about a quarter- of-a-division display level change in the center, representing less than a 2 dB difference at 5 dB/div. vertical setting. However, the side lobe difference shows the full impact of the intercepting bandwidth difference of approximately 10 dB at 20log3.
Figure 3 shows the same tB products at 18 and 6. However, the third trace was obtained by reducing the pulse width to 2 ms vs. a bandwidth reduction to 1 MHz with a 6 ms pulse. The first two traces, showing the unmodulated carrier and 6 ms pulse with 3 MHz bandwidth, are the same in both cases. However, the third trace where tB = 6, using a 2 ms pulse and no change in bandwidth from 3 MHz, does not drop in level at the sides as in the first case. The side level remains unchanged at the same frequency regardless of the pulse widths.
The most unusual results occur at a transition pulse desensitization factor. Figure 4 shows the spectra of a 3 ms pulse intercepted by a 1 MHz bandwidth and 300 kHz bandwidth with the unmodulated carrier as a reference. As expected, the spectra differ in level at different tB products.
However, the results for the varying pulse width case, shown in Figure 5 , are counter intuitive. Here, a fixed 1 MHz bandwidth intercepting a 3 and 1 ms pulse width is used. The 1 ms pulse, at a pulse desensitization factor tB = 1, shows a smaller center frequency level than the spectrum for the 3 ms pulse at tB = 3 as expected. However, the side lobe level for the
1 ms pulse width is larger than that for the 3 ms pulse width, which is not as expected. This anomaly occurs because a tB equal to 1 is still affected by pulse desensitization losses, which control spectrum results at small tB, while a tB product of 3 is affected very little by pulse desensitization. Operating in the transition region between small and large tB combined with the frequency shift in side lobe position at different pulse widths produces this counter-intuitive effect.
It should be apparent that a simple comparison of pulse desensitization factors will not tell the complete story. It is also necessary to consider whether it is the pulse width or receiver bandwidth that was changed. A change in one factor does not lead to the same conclusions as a change in the other even though the same tB product is obtained. It is also essential to consider how close the tB product is to unity because the amplitude level associated with tB = 1 is all that actually can be achieved even though the mathematical formula will yield a larger product. Results calculated on the basis of pulse desensitization factors can be accurate, but only when this phenomenon is understood fully.
1. Morris Engelson, Modern Spectrum Analyzer Theory and Applications, JMS Consulting, Portland, OR, pp. 62-64.
2. Metcalf, "Investigation of Spectrum Signature Instrumentation," IEEE Transactions EMC, June 1965.
Morris Engelson is a consultant on spectrum analysis, technology and business strategy at JMS Consulting in Portland, OR, where he can be reached at (503) 292-7035. He is a former chief engineer at Tektronix Inc., adjunct graduate school faculty member at OSU and Fellow of the IEEE for his contributions to the field of spectrum analysis.
Len Garrett is a spectrum analyzer product marketing manager for Tektronix Inc., Beaverton, OR, and can be reached at (503) 627-4824. His background includes microwave research at Linfield Research Institute in McMinnville, OR and time as chief engineer of KXL Radio in Portland, OR.