U.S. DEPARTMENT OF COMMERCE • National Telecommunications and Information Administration

report series

NTIA Technical Report TR-12-487 

The Rabbit Ears Pulse-Envelope 

Phenomenon in Off-Fundamental 

Detection of Pulsed Signals 

Frank H. Sanders 

U.S. DEPARTMENT OF COMMERCE 

July 2012 

NTIA Technical Report TR-12-487 

The Rabbit Ears Pulse-Envelope 

Phenomenon in Off-Fundamental 

Detection of Pulsed Signals 

Frank H. Sanders 

iii

DISCLAIMER 

Certain commercial equipment and materials are identified in this report to specify adequately 

the technical aspects of the reported results. In no case does such identification imply 

recommendation or endorsement by the National Telecommunications and Information 

Administration, nor does it imply that the material or equipment identified is the best available 

for this purpose. 
iv
v

CONTENTS 

Figures............................................................................................................................................ vi

Tables ............................................................................................................................................. xi

Abbreviations/Acronyms .............................................................................................................. xii

1 Mathematical Demonstration of the Rabbit Ears Effect ...............................................................1

1.1 Time-Domain Envelopes of Spectrum Power Measured Across the Central 

Spectrum Lobe ........................................................................................................................1

1.2 Time-Domain Envelopes of Spectrum Power Not Measured Across the Central 

Spectrum Lobe ........................................................................................................................3

2 Measured Rabbit Ears Pulses in Radar Emission Spectra ............................................................6

2.1 Real-World Pulse Envelope Characteristics ...........................................................................6

2.2 Expectations for Real-World Rabbit Ears Characteristics ......................................................6

2.3 Rabbit Ears Pulse Envelopes Across a Real-World 5 GHz Radar Spectrum .........................6

2.4 Rabbit Ears Pulse Envelopes Across the Spectrum of Another 5 GHz Radar......................20

3 Observations of Rabbit Ears Morphologies and Implications for Off-Fundamental 

Detection of Pulsed Signals ........................................................................................................42

3.1 Observed Rabbit Ears Morphologies ....................................................................................42

3.2 Implications for Off-Fundamental Detection of Pulsed Signals ...........................................42

3.3 Multipath Effects ..................................................................................................................42

4 Practical Application: Using the Rabbit Ears Effect to Measure Parameters Of ChirpedFrequency Pulses ........................................................................................................................43

4.1 Frequency-Modulated (Chirped) Pulses ...............................................................................43

4.2 Measurement of Chirped Pulse Parameters through Use of the Rabbit Ears Effect .............43

4.3 Example Chirp-Parameter Measurement Using the Rabbit Ears Effect ...............................44

5 Summary .....................................................................................................................................48

6 Acknowledgements .....................................................................................................................49

7 References ...................................................................................................................................50
vi

FIGURES 

Figure 1. Time-domain view of a sequence of simulated RF pulses. In this 

simulation the parameters are f0 = 10 MHz, pulse = 100 s, PRI = 3 ms. ........................................2

Figure 2. Fourier transform of the pulse sequence of Figure 1. A classic sinc2

power spectrum is generated. Power is plotted linearly. .................................................................2

Figure 3. Inverse Fourier transform of the spectrum of Figure 2, from 0 MHz to 

20 MHz. The original pulse sequence of Figure 1 is reconstructed in the time 

domain..............................................................................................................................................3

Figure 4. Inverse Fourier transform of spectrum 5 MHz off-tuned from f0 showing 

two pulses in the time domain. Convolution bandwidth = 20 MHz. The rabbit ears 

effect is barely visible. .....................................................................................................................4

Figure 5. Inverse Fourier transform of spectrum 10 MHz off-tuned from f0

showing two pulses in the time domain. Convolution bandwidth = 20 MHz. The 

rabbit ears effect is easily visible. ....................................................................................................4

Figure 6. Inverse Fourier transform of spectrum 15 MHz off-tuned from f0

showing two pulses in the time domain. Convolution bandwidth = 20 MHz. The 

rabbit ears troughs deepen as the off-tuned distance from f0 increases. ..........................................5

Figure 7. Inverse Fourier transform of spectrum 20 MHz off-tuned from f0

showing two pulses in the time domain. Convolution bandwidth = 20 MHz. ................................5

Figure 8. Radiated emission spectrum of a WR-100 (WSR-74C) weather radar, 

measured in an 8-MHz bandwidth. The transmitter used a magnetron oscillator as 

its final-stage output device. Pulse envelopes were measured at each of the 

frequencies labeled A-Z. ..................................................................................................................7

Figure 9. Pulse envelope in an 8-MHz bandwidth, Point A (f = -1500 MHz) in 

the WR-100 spectrum. The red line shows the radar’s pulse envelope at its 

fundamental frequency.....................................................................................................................7

Figure 10. Pulse envelope, Point B (f = -1490 MHz) in the WR-100 spectrum. ..........................8

Figure 11. Pulse envelope, Point C (f = -1430 MHz) in the WR-100 spectrum. ..........................8

Figure 12. Pulse envelope, Point D (f = -1360 MHz) in the WR-100 spectrum. ..........................9

Figure 13. Pulse envelope, Point E (f = -1330 MHz) in the WR-100 spectrum. ..........................9

Figure 14. Pulse envelope, Point F (f = -1250 MHz) in the WR-100 spectrum. .........................10

Figure 15. Pulse envelope, Point G (f = -1150 MHz) in the WR-100 spectrum. ........................10
vii

Figure 16. Pulse envelope, Point H (f = -1000 MHz) in the WR-100 spectrum. ........................11

Figure 17. Classical rabbit ears, Point I (f = -850 MHz) in the WR-100 

spectrum. ........................................................................................................................................11

Figure 18. Pulse envelope, Point J (f = -670 MHz) in the WR-100 spectrum. ...........................12

Figure 19. Pulse envelope, Point K (f = -550 MHz) in the WR-100 spectrum. ..........................12

Figure 20. Pulse envelope, Point L (f = -490 MHz) in the WR-100 spectrum. ..........................13

Figure 21. Pulse envelope, Point M (f = -300 MHz) in the WR-100 spectrum. .........................13

Figure 22. Pulse envelope, Point N (f = -160 MHz) in the WR-100 spectrum. ..........................14

Figure 23. Pulse envelope, Point P (f = -100 MHz) in the WR-100 spectrum. ...........................14

Figure 24. Pulse envelope, Point Q (f0) in the WR-100 spectrum. ................................................15

Figure 25. Pulse envelope, Point R (f = +150 MHz) in the WR-100 spectrum. .........................15

Figure 26. Pulse envelope, Point S (f = +490 MHz) in the WR-100 spectrum. ..........................16

Figure 27. Pulse envelope, Point T (f = +650 MHz) in the WR-100 spectrum. .........................16

Figure 28. Pulse envelope, Point U (f = +730 MHz) in the WR-100 spectrum. .........................17

Figure 29. Pulse envelope, Point V (f = +900 MHz) in the WR-100 spectrum. .........................17

Figure 30. Pulse envelope, Point W (f = +1040 MHz) in the WR-100 spectrum. ......................18

Figure 31. Pulse envelope, Point X (f = +1250 MHz) in the WR-100 spectrum. .......................18

Figure 32. Pulse envelope, Point Y (f = +1350 MHz) in the WR-100 spectrum. .......................19

Figure 33. Pulse envelope, Point Z (f = +1430 MHz) in the WR-100 spectrum. .......................19

Figure 34. TDWR spectrum formed with Channel A (analog) pulse modulator. 

Off-fundamental pulse envelopes were measured every 10 MHz as indicated. ............................20

Figure 35. TDWR spectrum formed with Channel B (digital) pulse modulator. 

Off-fundamental pulse envelopes were measured every 10 MHz as indicated. ............................20

Figure 36. Pulse envelope, Point A (f = -100 MHz) in the TDWR Channel A 

spectrum. The red curve is a graphical aid showing the pulse envelope at f0. ...............................21

Figure 37. Pulse envelope, Point B (f = -90 MHz) in the TDWR Channel A 

spectrum. ........................................................................................................................................21
viii

Figure 38. Pulse envelope, Point C (f = -80 MHz) in the TDWR Channel A 

spectrum. ........................................................................................................................................22

Figure 39. Pulse envelope, Point D (f = -70 MHz) in the TDWR Channel A 

spectrum. ........................................................................................................................................22

Figure 40. Pulse envelope, Point E (f = -60 MHz) in the TDWR Channel A 

spectrum. ........................................................................................................................................23

Figure 41. Pulse envelope, Point F (f = -50 MHz) in the TDWR Channel A 

spectrum. ........................................................................................................................................23

Figure 42. Pulse envelope, Point G (f = -40 MHz) in the TDWR Channel A 

spectrum. ........................................................................................................................................24

Figure 43. Pulse envelope, Point H (f = -30 MHz) in the TDWR Channel A 

spectrum. ........................................................................................................................................24

Figure 44. Pulse envelope, Point I (f = -20 MHz) in the TDWR Channel A 

spectrum. ........................................................................................................................................25

Figure 45. Pulse envelope, Point J (f = -10 MHz) in the TDWR Channel A 

spectrum. ........................................................................................................................................25

Figure 46. Pulse envelope, Point K (f0) in the TDWR Channel A spectrum. ................................26

Figure 47. Pulse envelope, Point L (f = +10 MHz) in the TDWR Channel A 

spectrum. ........................................................................................................................................26

Figure 48. Pulse envelope, Point M (f = +20 MHz) in the TDWR Channel A 

spectrum. ........................................................................................................................................27

Figure 49. Pulse envelope, Point N (f = +30 MHz) in the TDWR Channel A 

spectrum. ........................................................................................................................................27

Figure 50. Pulse envelope, Point P (f = +40 MHz) in the TDWR Channel A 

spectrum. ........................................................................................................................................28

Figure 51. Pulse envelope, Point Q (f = +50 MHz) in the TDWR Channel A 

spectrum. ........................................................................................................................................28

Figure 52. Pulse envelope, Point R (f = +60 MHz) in the TDWR Channel A 

spectrum. ........................................................................................................................................29

Figure 53. Pulse envelope, Point S (f = +70 MHz) in the TDWR Channel A 

spectrum. ........................................................................................................................................29
ix

Figure 54. Pulse envelope, Point T (f = +80 MHz) in the TDWR Channel A 

spectrum. ........................................................................................................................................30

Figure 55. Pulse envelope, Point U (f = +90 MHz) in the TDWR Channel A 

spectrum. ........................................................................................................................................30

Figure 56. Pulse envelope, Point V (f = +100 MHz) in the TDWR Channel A 

spectrum. ........................................................................................................................................31

Figure 57. Pulse envelope, Point A (f = -100 MHz) in the TDWR Channel B 

spectrum. ........................................................................................................................................31

Figure 58. Pulse envelope, Point B (f = -90 MHz) in the TDWR Channel B 

spectrum. ........................................................................................................................................32

Figure 59. Pulse envelope, Point C (f = -80 MHz) in the TDWR Channel B 

spectrum. ........................................................................................................................................32

Figure 60. Pulse envelope, Point D (f = -70 MHz) in the TDWR Channel B 

spectrum. ........................................................................................................................................33

Figure 61. Pulse envelope, Point E (f = -60 MHz) in the TDWR Channel B 

spectrum. ........................................................................................................................................33

Figure 62. Pulse envelope, Point F (f = -50 MHz) in the TDWR Channel B 

spectrum. ........................................................................................................................................34

Figure 63. Pulse envelope, Point G (f = -40 MHz) in the TDWR Channel B 

spectrum. ........................................................................................................................................34

Figure 64. Pulse envelope, Point H (f = -30 MHz) in the TDWR Channel B 

spectrum. ........................................................................................................................................35

Figure 65. Pulse envelope, Point I (f = -20 MHz) in the TDWR Channel B 

spectrum. ........................................................................................................................................35

Figure 66. Classic rabbit ears pulse envelope, Point J (f = -10 MHz) in the 

TDWR Channel B spectrum. .........................................................................................................36

Figure 67. Pulse envelope, Point K (f0) in the TDWR Channel B spectrum. ................................36

Figure 68. Classic rabbit ears pulse envelope, Point L (f = +10 MHz) in the 

TDWR Channel B spectrum. .........................................................................................................37

Figure 69. Pulse envelope, Point M (f = +20 MHz) in the TDWR Channel B 

spectrum. ........................................................................................................................................37
x

Figure 70. Pulse envelope, Point N (f = +30 MHz) in the TDWR Channel B 

spectrum. ........................................................................................................................................38

Figure 71. Pulse envelope, Point P (f = +40 MHz) in the TDWR Channel B 

spectrum. ........................................................................................................................................38

Figure 72. Pulse envelope, Point Q (f = +50 MHz) in the TDWR Channel B 

spectrum. ........................................................................................................................................39

Figure 73. Pulse envelope, Point R (f = +60 MHz) in the TDWR Channel B 

spectrum. ........................................................................................................................................39

Figure 74. Pulse envelope, Point S (f = +70 MHz) in the TDWR Channel B 

spectrum. ........................................................................................................................................40

Figure 75. Pulse envelope, Point T (f = +80 MHz) in the TDWR Channel B 

spectrum. ........................................................................................................................................40

Figure 76. Pulse envelope, Point U (f = +90 MHz) in the TDWR Channel B 

spectrum. ........................................................................................................................................41

Figure 77. Pulse envelope, Point V (f = +100 MHz) in the TDWR Channel B 

spectrum. ........................................................................................................................................41

Figure 78. SA display of rabbit ears when SA tuned frequency was 10 MHz 

below the center of 20 MHz wide chirped-pulse carrier. SA settings were as 

shown in Table 1, with RBW = 8 MHz. ........................................................................................46

Figure 79. SA display of rabbit ears when SA tuned frequency was 10 MHz above 

the center of the 20 MHz wide chirped-pulse carrier. Pulse duration of 60 s is 

observed between the rabbit ears in both Figures 78 and 79. ........................................................46

Figure 80. SA display when tuned to the center of the chirped-pulse FM range. 

Calculation of pulse duration and FM range is shown in notes in red. ..........................................47
xi

TABLES 

Table 1. SA settings for chirped-pulse parameter measurements. .................................................45
xii

ABBREVIATIONS/ACRONYMS

DFS dynamic frequency selection 

FAA Federal Aviation Administration 

FM frequency modulation 

IF intermediate frequency 

ITS Institute for Telecommunication Sciences (NTIA) 

LFM linear frequency modulation 

NEXRAD Next Generation Weather Radar (WSR-88D) 

NLFM non-linear frequency modulation 

NTIA National Telecommunications and Information Administration 

NWS National Weather Service 

OOB out of band (radio emissions) 

PRI pulse repetition interval 

RBW resolution bandwidth 

RF radio frequency 

SA spectrum analyzer 

TDWR Terminal Doppler Weather Radar (FAA equivalent of the NEXRAD) 

VBW video bandwidth 

VSA vector signal analyzer 

VSG vector signal generator 

WR-100 civilian weather radar equivalent to the WSR-74C 

WSR-74C NWS weather radar now supplanted by the WSR-88D (NEXRAD) 

WSR-88D NEXRAD weather surveillance radar, accepted 1988, Doppler-capable 
THE RABBIT EARS PULSE-ENVELOPE PHENOMENON IN OFF-FUNDAMENTAL 

DETECTION OF PULSED SIGNALS 

Frank H. Sanders1

When radiofrequency pulse envelopes are observed away from their fundamental 

frequency their shapes differ from those at their fundamental frequency. Off-fundamental 

pulse envelopes tend to exhibit spikes at their rising and falling edges with loweramplitude energy between the spikes. This phenomenon, called the rabbit ears effect, is 

described in this NTIA Report. Examples of rabbit ears pulse envelopes are provided in a 

mathematical simulation and from measurements of off-fundamental pulse envelopes of 

two models of 5 GHz weather radars. Morphologies of rabbit ears pulses are examined. A 

method is provided for using this effect to determine the durations and bandwidths of 

chirped pulses in bandwidth-limited measurement systems. Implications for offfundamental detection of pulsed signals for dynamic frequency selection (DFS) 

algorithms are considered. 

Key words: chirped pulses; dynamic frequency selection (DFS); off-fundamental signal 

detection; pulsed radiofrequency (RF) signals; rabbit ears; radar spectrum 

1 MATHEMATICAL DEMONSTRATION OF THE RABBIT EARS EFFECT 

1.1 Time-Domain Envelopes of Spectrum Power Measured Across the Central Spectrum 

Lobe 

When an energy source is pulsed on and off in the time domain, its emitted energy is spread 

across a non-zero portion of the radio frequency (RF) domain. When the pulses are generated at 

a fundamental oscillation frequency, f0, the resulting spectrum is distributed across the spectrum 

at frequencies above and below f0. The Fourier transform and inverse Fourier transform connect 

the time-domain and frequency-domain behaviors of the pulsed energy. An example of a set of 

theoretically generated time-domain oscillatory-energy pulses and their corresponding theoretical 

power spectrum is shown in Figures 1 and 2. 

As shown in Figure 2, the pulsed-energy power spectrum consists of a set of emission lines 

spaced at intervals of (1/PRI), where PRI = time-domain pulse repetition interval. The power 

levels of those individual frequency-domain lines formed as a series of lobes that follow a 

classical (sin(x)/x)2envelope, where x is the separation between any given frequency and f0. The 

null-to-null width of the central lobe is (2/pulse) and all other spectrum lobes have a width of 

(1/pulse), where pulse is the time-domain pulse width (Figure 2).

 

1

 The author is with the Institute for Telecommunication Sciences, National Telecommunications and 

Information Administration, U.S. Department of Commerce, Boulder, Colorado 80305. 
2

Figure 1. Time-domain view of a sequence of simulated RF pulses. In this simulation the 

parameters are f0 = 10 MHz, pulse = 100 s, PRI = 3 ms.

Figure 2. Fourier transform of the pulse sequence of Figure 1. A classic sinc2

power spectrum is 

generated. Power is plotted linearly. 

Suppose that a transmitter generates pulsed radio frequency energy at frequency f0. If a spectrum 

measurement system collects energy from that transmitter through a bandwidth window that 

convolves the lines in the central spectrum lobe around f0, and if the measurement system output 

is fed to a time-domain visualization device such as a crystal detector and an oscilloscope, then 
3

the device’s display will show pulsed envelopes that are a close approximation of the 

transmitter’s pulse-envelope modulation. This effect is demonstrated in the mathematical 

simulation of Figure 3, which is an inverse Fourier transform of the spectrum of Figure 2. 

Figure 3. Inverse Fourier transform of the spectrum of Figure 2, from 0 MHz to 20 MHz. The 

original pulse sequence of Figure 1 is reconstructed in the time domain. 

1.2 Time-Domain Envelopes of Spectrum Power Not Measured Across the Central 

Spectrum Lobe 

If the convolution bandwidth of a measurement system integrates energy across frequencies that 

are exclusive of the lines in the central spectrum lobe, and the collected energy is again fed to a 

time-domain display, then the loss of the energy in the central-lobe lines causes a change in the 

shapes of the detected time-domain pulse envelopes. As shown in the inverse Fourier transforms 

of Figures 4–7, under this circumstance there is a tendency for the centers of the pulses to be 

reduced in power; center-pulse energy goes missing. Relatively high levels of energy remain in 

the rising and falling edges of the pulses. As a result, off-fundamental pulse envelopes tend to be 

pairs of spikes separated by the transmitted pulse width, pulse, with relatively low power levels 

observed in the center of each pulse. This visual signature is called the rabbit ears effect due to 

its putative resemblance to a long-eared lagomorph seen face-on. The pulse-to-pulse PRI is 

unaffected by this phenomenon (as demonstrated in Figures 4–7).
4

Figure 4. Inverse Fourier transform of spectrum 5 MHz off-tuned from f0 showing two pulses in 

the time domain. Convolution bandwidth = 20 MHz. The rabbit ears effect is barely visible. 

Figure 5. Inverse Fourier transform of spectrum 10 MHz off-tuned from f0 showing two pulses in 

the time domain. Convolution bandwidth = 20 MHz. The rabbit ears effect is easily visible. 
5

Figure 6. Inverse Fourier transform of spectrum 15 MHz off-tuned from f0 showing two pulses in 

the time domain. Convolution bandwidth = 20 MHz. The rabbit ears troughs deepen as the offtuned distance from f0 increases. 

Figure 7. Inverse Fourier transform of spectrum 20 MHz off-tuned from f0 showing two pulses in 

the time domain. Convolution bandwidth = 20 MHz. 
6

2 MEASURED RABBIT EARS PULSES IN RADAR EMISSION SPECTRA 

2.1 Real-World Pulse Envelope Characteristics 

Although the rabbit ears effect can be demonstrated mathematically as in Figures 4–7 the 

simulation is a highly simplified representation of the emissions of actual radio transmitters. 

Real-world RF pulse envelopes, as generated for example by radar transmitters, are more 

complexly shaped. They do not have rectangular envelopes, their rising and falling edges have 

non-zero lengths and are not entirely smooth, and they usually exhibit some transient features. 

The transitions between pulse centers and their rising and falling edges are complexly shaped 

and sometimes exhibit transient features. The pulse centers are neither completely smooth nor 

flat. The falling edges of real pulses are usually longer than the rising edges and thus the pulse 

envelopes are unsymmetrical in the time domain. All of these complexities of real pulse 

envelopes result from inherent and unavoidable behaviors of radio transmitters’ baseband pulseforming networks and final-stage RF amplifiers. 

2.2 Expectations for Real-World Rabbit Ears Characteristics 

Complexities in the time-domain envelopes of transmitted pulses translate into complexities in 

the emission spectra of transmitters, causing those spectra to deviate from the ideal sinc2

shape 

shown in Figure 2. If the off-fundamental emission lines of real-world spectra are convolved by 

measurement systems while the center-lobe lines of the spectra are excluded, then rabbit ears 

envelopes will be observed in the time domain outputs of such measurement systems. 

Qualitatively these real-world rabbit ears envelopes will resemble the idealized representations 

of Figures 4–7. But the non-ideal characteristics of real-world pulses and their emission spectra 

mean that real-world rabbit ears can be expected to deviate from idealized rabbit ears in many of 

their details. 

2.3 Rabbit Ears Pulse Envelopes Across a Real-World 5 GHz Radar Spectrum 

The author has measured and recorded rabbit ears pulse envelopes at frequencies between 4 to 

7 GHz across the emission spectrum of a 5 GHz weather radar. The measured emission spectrum 

of this WR-100 radar, a civilian version of the WSR-74C,2

is shown in Figure 8. The spectrum 

was measured in an 8 MHz bandwidth. This bandwidth, wider than recommended in [1], 

exaggerates the amplitudes of the radar’s out-of-band (OOB) and spurious emission levels 

relative to the power at the fundamental. That bandwidth was used to collect time-domain 

measurements of pulse envelopes at each of the frequencies indicated by letters A-Z in Figure 8. 

The spectrum of Figure 8 and the pulse envelopes of Figures 9–33 were all measured via radiated 

energy from the radar. Aside from the wide bandwidth, the spectrum was measured in 

accordance with the guidelines in [1], including the stepped-frequency algorithm described in 

that document. The pulse envelopes were measured with a crystal detector and digital 

oscilloscope connected to the output of the wideband video output of the measurement system. 

The center-frequency pulse shape of the radar is superimposed in red on each of the measured 

rabbit ears envelopes as a visualization aid. 

 

2 WSR-74C radars were once used by the United States National Weather Service but have been 

superseded by WSR-88D NEXRAD radars. 
7

Figure 8. Radiated emission spectrum of a WR-100 (WSR-74C) weather radar, measured in an 

8-MHz bandwidth. The transmitter used a magnetron oscillator as its final-stage output device. 

Pulse envelopes were measured at each of the frequencies labeled A-Z.

Figure 9. Pulse envelope in an 8-MHz bandwidth, Point A (f = -1500 MHz) in the WR-100 

spectrum. The red line shows the radar’s pulse envelope at its fundamental frequency. 
8

Figure 10. Pulse envelope, Point B (f = -1490 MHz) in the WR-100 spectrum. 

Figure 11. Pulse envelope, Point C (f = -1430 MHz) in the WR-100 spectrum. 
9

Figure 12. Pulse envelope, Point D (f = -1360 MHz) in the WR-100 spectrum. 

Figure 13. Pulse envelope, Point E (f = -1330 MHz) in the WR-100 spectrum. 
10

Figure 14. Pulse envelope, Point F (f = -1250 MHz) in the WR-100 spectrum. 

Figure 15. Pulse envelope, Point G (f = -1150 MHz) in the WR-100 spectrum. 
11

Figure 16. Pulse envelope, Point H (f = -1000 MHz) in the WR-100 spectrum. 

Figure 17. Classical rabbit ears, Point I (f = -850 MHz) in the WR-100 spectrum. 
12

Figure 18. Pulse envelope, Point J (f = -670 MHz) in the WR-100 spectrum. 

Figure 19. Pulse envelope, Point K (f = -550 MHz) in the WR-100 spectrum. 
13

Figure 20. Pulse envelope, Point L (f = -490 MHz) in the WR-100 spectrum. 

Figure 21. Pulse envelope, Point M (f = -300 MHz) in the WR-100 spectrum. 
14

Figure 22. Pulse envelope, Point N (f = -160 MHz) in the WR-100 spectrum. 

Figure 23. Pulse envelope, Point P (f = -100 MHz) in the WR-100 spectrum. 
15

Figure 24. Pulse envelope, Point Q (f0) in the WR-100 spectrum. 

Figure 25. Pulse envelope, Point R (f = +150 MHz) in the WR-100 spectrum. 
16

Figure 26. Pulse envelope, Point S (f = +490 MHz) in the WR-100 spectrum.

Figure 27. Pulse envelope, Point T (f = +650 MHz) in the WR-100 spectrum. 
17

Figure 28. Pulse envelope, Point U (f = +730 MHz) in the WR-100 spectrum. 

Figure 29. Pulse envelope, Point V (f = +900 MHz) in the WR-100 spectrum. 
18

Figure 30. Pulse envelope, Point W (f = +1040 MHz) in the WR-100 spectrum. 

Figure 31. Pulse envelope, Point X (f = +1250 MHz) in the WR-100 spectrum. 
19

Figure 32. Pulse envelope, Point Y (f = +1350 MHz) in the WR-100 spectrum. 

Figure 33. Pulse envelope, Point Z (f = +1430 MHz) in the WR-100 spectrum. 
20

2.4 Rabbit Ears Pulse Envelopes Across the Spectrum of Another 5 GHz Radar 

The author has measured and recorded rabbit ears pulse envelopes across the spectrum of a 

5 GHz Federal Aviation Administration (FAA) Terminal Doppler Weather Radar (TDWR). This 

was an engineering test-bed unit that drives a high-power klystron final RF output stage with 

either a conventional pulse-forming network or else a newly designed digital pulse former. The 

two options are called Channels A and B respectively. Their spectra are shown in Figures 34–35. 

Pulse envelopes measured in an 8-MHz bandwidth are shown in Figures 36–77.

Figure 34. TDWR spectrum formed with Channel A (analog) pulse modulator. Off-fundamental 

pulse envelopes were measured every 10 MHz as indicated. 

Figure 35. TDWR spectrum formed with Channel B (digital) pulse modulator. Off-fundamental 

pulse envelopes were measured every 10 MHz as indicated. 
21

Figure 36. Pulse envelope, Point A (f = -100 MHz) in the TDWR Channel A spectrum. The red 

curve is a graphical aid showing the pulse envelope at f0.

Figure 37. Pulse envelope, Point B (f = -90 MHz) in the TDWR Channel A spectrum. 
22

Figure 38. Pulse envelope, Point C (f = -80 MHz) in the TDWR Channel A spectrum. 

Figure 39. Pulse envelope, Point D (f = -70 MHz) in the TDWR Channel A spectrum. 
23

Figure 40. Pulse envelope, Point E (f = -60 MHz) in the TDWR Channel A spectrum. 

Figure 41. Pulse envelope, Point F (f = -50 MHz) in the TDWR Channel A spectrum. 
24

Figure 42. Pulse envelope, Point G (f = -40 MHz) in the TDWR Channel A spectrum. 

Figure 43. Pulse envelope, Point H (f = -30 MHz) in the TDWR Channel A spectrum. 
25

Figure 44. Pulse envelope, Point I (f = -20 MHz) in the TDWR Channel A spectrum. 

Figure 45. Pulse envelope, Point J (f = -10 MHz) in the TDWR Channel A spectrum. 
26

Figure 46. Pulse envelope, Point K (f0) in the TDWR Channel A spectrum. 

Figure 47. Pulse envelope, Point L (f = +10 MHz) in the TDWR Channel A spectrum. 
27

Figure 48. Pulse envelope, Point M (f = +20 MHz) in the TDWR Channel A spectrum. 

Figure 49. Pulse envelope, Point N (f = +30 MHz) in the TDWR Channel A spectrum. 
28

Figure 50. Pulse envelope, Point P (f = +40 MHz) in the TDWR Channel A spectrum. 

Figure 51. Pulse envelope, Point Q (f = +50 MHz) in the TDWR Channel A spectrum. 
29

Figure 52. Pulse envelope, Point R (f = +60 MHz) in the TDWR Channel A spectrum. 

Figure 53. Pulse envelope, Point S (f = +70 MHz) in the TDWR Channel A spectrum. 
30

Figure 54. Pulse envelope, Point T (f = +80 MHz) in the TDWR Channel A spectrum. 

Figure 55. Pulse envelope, Point U (f = +90 MHz) in the TDWR Channel A spectrum. 
31

Figure 56. Pulse envelope, Point V (f = +100 MHz) in the TDWR Channel A spectrum. 

Figure 57. Pulse envelope, Point A (f = -100 MHz) in the TDWR Channel B spectrum. 
32

Figure 58. Pulse envelope, Point B (f = -90 MHz) in the TDWR Channel B spectrum. 

Figure 59. Pulse envelope, Point C (f = -80 MHz) in the TDWR Channel B spectrum. 
33

Figure 60. Pulse envelope, Point D (f = -70 MHz) in the TDWR Channel B spectrum. 

Figure 61. Pulse envelope, Point E (f = -60 MHz) in the TDWR Channel B spectrum. 
34

Figure 62. Pulse envelope, Point F (f = -50 MHz) in the TDWR Channel B spectrum. 

Figure 63. Pulse envelope, Point G (f = -40 MHz) in the TDWR Channel B spectrum. 
35

Figure 64. Pulse envelope, Point H (f = -30 MHz) in the TDWR Channel B spectrum. 

Figure 65. Pulse envelope, Point I (f = -20 MHz) in the TDWR Channel B spectrum. 
36

Figure 66. Classic rabbit ears pulse envelope, Point J (f = -10 MHz) in the TDWR Channel B 

spectrum. 

Figure 67. Pulse envelope, Point K (f0) in the TDWR Channel B spectrum. 
37

Figure 68. Classic rabbit ears pulse envelope, Point L (f = +10 MHz) in the TDWR Channel B 

spectrum. 

Figure 69. Pulse envelope, Point M (f = +20 MHz) in the TDWR Channel B spectrum. 
38

Figure 70. Pulse envelope, Point N (f = +30 MHz) in the TDWR Channel B spectrum. 

Figure 71. Pulse envelope, Point P (f = +40 MHz) in the TDWR Channel B spectrum. 
39

Figure 72. Pulse envelope, Point Q (f = +50 MHz) in the TDWR Channel B spectrum. 

Figure 73. Pulse envelope, Point R (f = +60 MHz) in the TDWR Channel B spectrum. 
40

Figure 74. Pulse envelope, Point S (f = +70 MHz) in the TDWR Channel B spectrum. 

Figure 75. Pulse envelope, Point T (f = +80 MHz) in the TDWR Channel B spectrum. 
41

Figure 76. Pulse envelope, Point U (f = +90 MHz) in the TDWR Channel B spectrum. 

Figure 77. Pulse envelope, Point V (f = +100 MHz) in the TDWR Channel B spectrum. 
42

3 OBSERVATIONS OF RABBIT EARS MORPHOLOGIES AND IMPLICATIONS FOR 

OFF-FUNDAMENTAL DETECTION OF PULSED SIGNALS 

3.1 Observed Rabbit Ears Morphologies 

Rabbit ears morphologies observed in the off-fundamental pulses measured from the WR-100 

and TDWR 5 GHz weather radars include: pairs of high-amplitude rising and falling edges with 

lower-amplitude noise in-between (“classic rabbit ears”); single rising edges followed by loweramplitude noise; single falling edges preceded by lower-amplitude noise; and noise-like pulses 

with no prominent rising or falling edges. These morphologies vary continuously across the 

measured emission spectra. They vary as a function of location in the spectrum and of 

measurement bandwidth. No method is known for predicting a priori the morphology that will 

be observed on any given frequency in any given measurement bandwidth. 

3.2 Implications for Off-Fundamental Detection of Pulsed Signals 

If off-fundamental detection of pulsed signals needs to be performed (as for example in dynamic 

frequency selection (DFS) systems), then designers of such systems should be aware of the 

rabbit ears phenomenon. Off-fundamental detection techniques that depend on sensing of pulse 

widths, for example, may not be as robust as desired if the short rabbit ears spikes (and pairs of 

short rabbit ears spikes) are detected in a receiver’s bandwidth instead of the expected centerfrequency pulse width of the transmitter. Designers should also aware that the observed widths 

of the spikes will vary as a function of their receivers’ detection bandwidths. 

Conversely, if DFS or other spectrum-sharing algorithms do not depend on sensing of pulse 

widths, but instead rely on (for example) the PRI patterns from pulse to pulse, then the rabbit 

ears effect may not necessarily be a concern. This is because the rabbit ears effect only applies to

individual pulses but does not affect the PRIs between pulses. 

3.3 Multipath Effects 

Although not the subject of this study, it is noted that some multipath effects are observed in 

some of the measured data. Multipath effects are a function of the geometries between 

measurement systems and DFS receivers. As with the rabbit ears effect, multipath effects would 

tend to complicate DFS sensing that is dependent on pulse widths but might pose little or no 

problem for DFS techniques that rely on PRIs. 
43

4 PRACTICAL APPLICATION: USING THE RABBIT EARS EFFECT TO MEASURE 

PARAMETERS OF CHIRPED-FREQUENCY PULSES 

4.1 Frequency-Modulated (Chirped) Pulses 

Many radar systems use frequency-modulated (FM) pulses as a method of pulse compression. 

Chirping, as this technique is called, allows radar receivers to sustain a high degree of range 

resolution (via compression techniques in radar receivers) even though the transmitted pulse 

widths are lengthened to compensate for the limited peak power generated by solid-state 

transmitters. In modern solid-state radars it is not uncommon to encounter chirped pulse 

durations that are 100 s long and chirped bandwidths that are 100 MHz wide. This is in contrast 

to conventional radar pulse widths that might be about 1 s long and are not chirped. 

4.2 Measurement of Chirped Pulse Parameters through Use of the Rabbit Ears Effect 

Vector signal analyzers (VSAs) can be used to measure the characteristics of chirped pulses up 

to a point. VSAs sample at high speed (typically at about 95 megasamples (Msa, 106) per 

second) in the time domain and then perform Fourier transforms to show pulse duration and 

chirp bandwidth information. VSAs are powerful measurement devices but they do not 

necessarily have enough bandwidth to directly measure the wider chirp bandwidths of modern 

radars. A 95 Msa/sec VSA digitizer, for example, will only provide up to about 36 MHz of 

measurement bandwidth, just a fraction of the bandwidth of some types of chirped radar 

transmitters. 

The rabbit ears effect can be used as a practical tool to measure both the duration and bandwidth 

of arbitrarily long-duration and wide-bandwidth chirped pulses. Furthermore, the effect can be 

used with both spectrum analyzers (SAs) and VSAs. The author has used this effect to measure 

the chirped-pulse characteristics of highly sophisticated, state-of-the-art radars in places and 

times where he either did not have a VSA or else the pulse characteristics exceeded even the 

capabilities of the best available VSAs. In the author’s experience, the rabbit ears effect can be 

used to achieve chirped-pulse characteristics measurements with greater speed and ease than 

direct measurements of these characteristics. (Related information on engineering aspects of the 

rabbit ears effect in bandwidth-limited receivers is provided in [2].) 

This measurement technique is founded on the recognition that, if the measurement system is 

bandwidth-limited compared to the chirp width of the pulses to be measured, then the carrier 

frequencies of each of the chirped pulses will spend some time outside the convolution 

bandwidth of the measurement system. Therefore chirped pulses are measured off their 

fundamental frequencies by bandwidth-limited measurement systems during part of each of their 

cycles, even if the measurement system is center-tuned to their center frequencies. The 

conclusion drawn from this observation is that bandwidth-limited measurement of widebandwidth chirped pulses is equivalent to the rabbit ears measurement of non-chirped pulses. 

Imagine that a measurement system has a smaller convolution bandwidth than the bandwidth of 

the chirped pulses that are to be measured, and that it is center-tuned to the same frequency as 

the center of the intentional FM chirp range of the pulses. Then the chirped pulses’ carriers begin 

and end on frequencies outside the measurement system bandwidth, and the pulses sweep 
44

through the measurement system bandwidth between the two spikes. The measurement system 

will show a rabbit ear spike in the time domain at the beginning and ending of each pulse; the 

time between those rabbit ears must be the pulse duration. 

As the chirp carrier tunes (frequency modulates) through the measurement system bandwidth, a 

bump will be formed in the time domain. This bump will be centered between the two rabbit ear 

spikes if the measurement is tuned to the center frequency of the chirped pulses. If the 

measurement system’s tuned frequency is offset slightly below or above that fundamental 

frequency then the bump will move slightly left or right, respectively, in the time domain 

between the rabbit ears. 

The ratio of the width of the time-domain bump to the time between the rabbit ears is equal to 

the ratio of the measurement bandwidth to the bandwidth of the chirped pulses. If the chirp 

bandwidth is Bc, the measurement bandwidth is Bmeas, the pulse width determined from the rabbit 

ears spike spacing is T, and the duration of the bump between the rabbit ears is tdelta, then the 

proportionality relationship between these parameters is3

: 

 

 

 

 

 

(1a)

and therefore 

 (

 

 

) (1b)

4.3 Example Chirp-Parameter Measurement Using the Rabbit Ears Effect 

The author has used this technique for many years in measurements of chirped pulses transmitted 

by operational and developmental radar systems; he has undertaken a simple demonstration of 

the technique for this NTIA Report. For this demonstration a vector signal generator (VSG) 

output is fed to an SA. The VSG is programmed to produce RF pulses at 5500 MHz that are 

60 s long with a chirp bandwidth of 20 MHz, the chirp direction running in time from lower 

frequency to higher frequency. Although the 20 MHz chirp bandwidth is within the bandwidth 

capabilities of existing VSAs, it exceeds the maximum 8 MHz bandwidth of the SA used for this 

demonstration. In any event, the purpose of the exercise is simply to provide an example of this 

measurement technique. 

The rabbit ears measurement of chirped-pulse bandwidth is performed with the SA parameters 

shown in Table 1. Figures 78–80 show the results from the SA display. 

 

3 This relationship is strictly true only if the FM of the chirped pulses is linear. However, non-linear FM 

(NLFM) in chirped pulses of existing radars is so close to linear FM (LFM) in existing radar designs that 

for practical purposes the linear ratio may be used as a good approximation even for NLFM chirped 

pulses. 
45

Table 1. SA settings for chirped-pulse parameter measurements. 

SA Parameter SA Parameter Setting

frequency span zero hertz (time domain)

detection positive peak

trace mode clear-write

resolution (IF) bandwidth (RBW) widest available in SA being used

video bandwidth (VBW) at least as wide as RBW

sweep time longer than the likely chirped-pulse duration

RF attenuation as much as needed to prevent IF-stage overload

reference Level highest possible with the selected RF attenuation

Figures 78 and 79 show the rabbit ears that are observed when the spectrum analyzer is tuned 

10 MHz below and above the center frequency of 20 MHz chirped pulses, respectively. Each of 

these two figures shows classic pairs of time domain spikes that can be used to determine chirped 

pulse duration, T, by measuring the time interval between them. In these two figures, however, 

the chirp bandwidth cannot be determined because of the large amount of off-tuning of the 

measurement system from the center frequency of the chirp range. 

Figure 80 shows the SA display when the measurement system is tuned to the center frequency 

of the chirp range. The data in this figure may be used to determine both the pulse duration, T, 

and the FM range of the chirped pulses, Bc. In this figure the pulse carrier successively enters the 

SA convolution bandwidth (its intermediate frequency (IF), or resolution bandwidth (RBW)) of 

8 MHz, passes through the SA’s tuned frequency (coincident with the center frequency of the 

chirping), and finally passes out of the SA’s convolution bandwidth. The bump between the 

rabbit ears is actually the convolution of the SA RBW with the chirp-pulse carrier; the chirppulse carrier sweeps out the shape of the SA RBW as it passes through the tuned frequency of 

the SA. Thus the 3 dB bandwidth of the bump is the 3 dB bandwidth of the SA RBW. 

In addition to the measurement of T from the interval between the rabbit ears, another interval, 

tdelta, is measured at the 3 dB points of the bump between the rabbit ears, as shown in Figure 80. 

The chirp bandwidth, Bc, is then computed by multiplying the value of the SA RBW, Bmeas, by 

the pulse duration, T, and dividing by tdelta. These computations are shown in red in Figure 80. 

In Figure 80 the convolution of the SA RBW is centered between the pulses because the SA is 

center-tuned to the chirped-pulse FM range. If the SA were off-tuned slightly below or above the 

center of the chirped-pulse FM range then the bump would be moved slightly to the left or right, 

respectively, but it would still be possible to measure the critically important value of tdelta and 

therefore still compute Bc. If the amount of off-tuning were to become too large, however, then 

the bump would be moved so far to the left or right that the result would become what is shown 

in Figures 78 and 79. In that case it becomes impossible to measure tdelta and then Bc cannot be 

computed. It is reiterated that this measurement technique for parameters of chirped pulses can 

be used with any bandwidth-limited measurement system, including both SAs and VSAs. 
46

Figure 78. SA display of rabbit ears when SA tuned frequency was 10 MHz below the center of 

20 MHz wide chirped-pulse carrier. SA settings were as shown in Table 1, with RBW = 8 MHz. 

Figure 79. SA display of rabbit ears when SA tuned frequency was 10 MHz above the center of 

the 20 MHz wide chirped-pulse carrier. Pulse duration of 60 s is observed between the rabbit 

ears in both Figures 78 and 79.
47

Figure 80. SA display when tuned to the center of the chirped-pulse FM range. Calculation of 

pulse duration and FM range is shown in notes in red. 
48

5 SUMMARY 

When RF pulse envelopes are observed away from their fundamental frequency their shapes 

differ from those at their fundamental frequency. Off-fundamental pulse envelopes tend to 

exhibit spikes at their rising and falling edges with lower-amplitude energy between the spikes, a 

phenomenon called the rabbit ears effect. This time-domain phenomenon results from the 

convolution by a bandwidth-limited measurement system or radio receiver of a limited number 

of off-fundamental Fourier lines in the emission spectra of pulsed signals. Examples of rabbit 

ears pulse envelopes have been provided in this report in a mathematical simulation and in 

measurements of off-fundamental pulse envelopes of two models of 5 GHz weather radars. 

The morphologies of rabbit ears pulses show a wide range of variation. The shapes of rabbit ears 

on any given frequency and in any given receiver bandwidth are not predictable a priori on any 

available theoretical basis—they can only be observed on an ad hoc basis on each frequency in 

any particular receiver or measurement bandwidth. 

The characteristics of rabbit ears would complicate DFS techniques that depend on off-frequency 

sensing of pulse widths or pulse shapes. Conversely, on-frequency DFS sensing would not be 

affected. The rabbit ears effect has no effect on transmitted PRI sequences. Therefore DFS 

techniques that use PRI information to avoid radar frequencies would not be impacted by this 

effect. 

The rabbit ears phenomenon has a useful, practical application. This is the measurement of pulse 

durations and FM bandwidths of chirped radar pulses with measurement systems that are 

themselves bandwidth-limited. This is useful and important because state-of-the-art solid-state 

radars that transmit chirped pulses often generate chirped FM ranges that exceed the directmeasurement capabilities of even the most capable VSAs. Furthermore, the rabbit ears method of 

measuring chirped-pulse parameters works with any bandwidth-limited measurement system, 

including both SAs and VSAs. 
49

6 ACKNOWLEDGEMENTS 

Mr. Geoff Sanders of ITS is thanked for generating the mathematical creations of the rabbit ears 

effect in Figures 1–7 using the software product MATLAB©. Mr. John Carroll of ITS and Mr. 

Bob Sole of NTIA’s Office of Spectrum Measurement (OSM) were instrumental in the 

measurement of TDWR rabbit ears pulse envelopes. The staff of the TDWR Program Support 

Facility at the Federal Aviation Administration’s Mike Monroney Aeronautical Center in 

Oklahoma City, Oklahoma are thanked for making the TDWR available for the pulse-shape 

measurements. Ms. Kristen Davis of ITS is thanked for her preparation and formatting of this 

Technical Report. 
50

7 REFERENCES 

[1] Sanders, F. H., R. L. Hinkle, and B. J. Ramsey, “Measurement Procedures for the Radar 

Spectrum Engineering Criteria (RSEC)”, NTIA Report TR-05-420, NTIA, U.S. 

Department of Commerce, March 2005. http://www.its.bldrdoc.gov/publications/2450.aspx

[2] Sanders, F. H., and R. Sole, “Test results and simulations illustrating the effective duty 

cycle of frequency modulated pulsed radiolocation and EESS system waveforms in marine 

radionavigation receivers”, ITU-R Report M.2128, International Telecommunications 

Union Radiocommunication Sector (ITU-R), 2008. http://www.itu.int/pub/R-REP-M.2128
FORM NTIA-29 U.S. DEPARTMENT OF COMMERCE 

(4-80) NAT’L. TELECOMMUNICATIONS AND INFORMATION ADMINISTRATION

BIBLIOGRAPHIC DATA SHEET

1. PUBLICATION NO. 2. Government Accession No. 3. Recipient’s Accession No.

4. TITLE AND SUBTITLE

The Rabbit Ears Pulse-Envelope Phenomenon in Off-Fundamental Detection of Pulsed 

Signals

5. Publication Date

July 2012

6. Performing Organization Code

NTIA/ITS.T

7. AUTHOR(S)

Frank H. Sanders

9. Project/Task/Work Unit No.

8. PERFORMING ORGANIZATION NAME AND ADDRESS 6462000-200

Institute for Telecommunication Sciences

National Telecommunications & Information Administration

U.S. Department of Commerce

325 Broadway

Boulder, CO 80305

10. Contract/Grant Number.

11. Sponsoring Organization Name and Address

National Telecommunications & Information Administration

Herbert C. Hoover Building

14th & Constitution Ave., NW

Washington, DC 20230

12. Type of Report and Period Covered

14. SUPPLEMENTARY NOTES

15. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant bibliography or 

literature survey, mention it here.)

When radiofrequency pulse envelopes are observed away from their fundamental frequency their shapes differ from those at 

their fundamental frequency. Off-fundamental pulse envelopes tend to exhibit spikes at their rising and falling edges with 

lower-amplitude energy between the spikes. This phenomenon, called the rabbit ears effect, is described in this NTIA Report. 

Examples of rabbit ears pulse envelopes are provided in a mathematical simulation and from measurements of offfundamental pulse envelopes of two models of 5 GHz weather radars. Morphologies of rabbit ears pulses are examined. A 

method is provided for using this effect to determine the durations and bandwidths of chirped pulses in bandwidth-limited 

measurement systems. Implications for off-fundamental detection of pulsed signals for dynamic frequency selection (DFS) 

algorithms are considered.

16. Key Words (Alphabetical order, separated by semicolons)

chirped pulses; dynamic frequency selection (DFS); off-fundamental signal detection; pulsed radiofrequency (RF) signals; 

rabbit ears; radar spectrum

17. AVAILABILITY STATEMENT

UNLIMITED.

FOR OFFICIAL DISTRIBUTION.

18. Security Class. (This report)

Unclassified

20. Number of pages

66

19. Security Class. (This page)

Unclassified

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