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

report series

NTIA Technical Report TR-12-488

Relationships Between Measured Power 

and Measurement Bandwidth for 

Frequency-Modulated (Chirped) Pulses

Frank H. Sanders

Roger A. Dalke

U.S. DEPARTMENT OF COMMERCE

August 2012

NTIA Technical Report TR-12-488

Relationships Between Measured Power 

and Measurement Bandwidth for 

Frequency-Modulated (Chirped) Pulses

Frank H. Sanders

Roger A. Dalke

DISCLAIMER 

This report identifies certain commercial equipment and materials to adequately specify 

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

recommendation or endorsement by the National Telecommunications and Information 

Administration (NTIA), nor does it imply that the material or equipment identified is the best 

available for this purpose. 

iii
iv
v

CONTENTS

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

Abbreviations/Acronyms.............................................................................................................. vii

1 Heuristic Derivation of Frequency-Modulated (Chirped) Pulse Emission Bandwidth ................1

1.1 Introduction.............................................................................................................................1

1.2 Heuristic Approach to Understanding Power-Bandwidth Relationships in Chirped 

Pulses......................................................................................................................................2

1.3 Measurement Bandwidth Required to Obtain Full Peak Power of Chirped Pulses................3

1.4 Power Per Spectrum Emission Line Relative to Total Power ................................................5

1.4.1 Power Per Spectrum Emission Line Relative to Total Peak Power...................................5

1.4.2 Power Per Spectrum Emission Line Relative to Total Average Power .............................6

1.5 Summary of Relationships Among Peak Power, Average Power and Bmeas ..........................7

2 Rigorous Derivation of Chirped Pulse Emission Bandwidth .......................................................9

2.1 Introduction.............................................................................................................................9

2.2 Chirped Pulsed FMCW Radar Signal and Fourier Transform................................................9

2.3 Minimum Bandwidth Required to Obtain Peak Voltage of the Chirp..................................10

2.4 Miscellaneous Relations for Peak and Average Power When 0  B  .............................12

3 Power-Measurement Data Supporting Theoretical Results........................................................14

3.1 Introduction...........................................................................................................................14

3.2 Measurement Setup...............................................................................................................14

3.3 Measurement Results............................................................................................................14

3.4 Comparison of Theoretical and Measured Results...............................................................14

3.5 Summary...............................................................................................................................18

4 Acknowledgments.......................................................................................................................19

5 References...................................................................................................................................20
vi

FIGURES

Figure 1. Measured frequency modulation versus time behavior for linear and 

non-linear chirped radar pulses. Many modern radars use non-linear chirped 

pulses................................................................................................................................................2

Figure 2. Diagram of proportional geometric relationships among Bmeas, Bchirp, 

tmeas and chirp. The relationship is exact for LFM pulses and approximate for 

NLFM pulses. ..................................................................................................................................4

Figure 3. Relationships among Bmeas, chirp pulse parameters, and measured peak 

and average power. ..........................................................................................................................8

Figure 4. Measured time-domain envelopes of a series of chirped pulses. The full 

pulse width is observed in this graph because the measurement bandwidth of 8 

MHz exceeded the chirp bandwidth of 1 MHz..............................................................................15

Figure 5. Measured linear chirp (1 MHz in 100 s) of one pulse from the 

sequence of Figure 4. .....................................................................................................................16

Figure 6. Measured line spectrum detail for the chirped-pulse sequence of Figures 

4 and 5............................................................................................................................................16

Figure 7. Measured average-power chirped-pulse spectra as a function of Bmeas..........................17

Figure 8. Measured peak-power chirped-pulse spectra as a function of Bmeas...............................17

Figure 9. Measured peak and average power for example chirped pulses as a 

function of measurement bandwidth. Measured power levels are from Figures 7 

and 8. Comparison of this figure to Figure 3 shows strong agreement between 

theoretical predictions and measured results. ................................................................................18
vii

ABBREVIATIONS/ACRONYMS

FM frequency modulation

FMCW frequency modulated carrier wave

ITS Institute for Telecommunication Sciences (NTIA)

LFM linear frequency modulation (linear chirp)

NLFM non-linear frequency modulation (non-linear chirp)

NTIA National Telecommunications and Information Administration

P0N amplitude-only pulse modulation (pronounced “p-on”)

pri pulse repetition interval

RBW resolution bandwidth

RF radio frequency

SA spectrum analyzer

VBW video bandwidth

VSA vector signal analyzer

VSG vector signal generator
viii
RELATIONSHIPS BETWEEN MEASURED POWER AND MEASUREMENT 

BANDWIDTH FOR FREQUENCY-MODULATED (CHIRPED) PULSES

Frank H. Sanders and Roger A. Dalke1

Measured power levels for radio frequency (RF) pulses that are frequency modulated 

(chirped) vary as a function of the bandwidth in which the measurement is performed; if 

chirped pulses cause RF interference, the power levels of the pulses in victim receivers 

will likewise vary as a function of receiver bandwidth. This report provides both heuristic 

and rigorous derivations of the relationships among chirped pulse parameters and the 

measured peak and average power levels of chirped pulses as a function of measurement 

bandwidth. These relationships may be best understood via a single graph (Figure 3)

presented in this report. This report supplements NTIA Technical Reports TR-05-420, 

TR-10-465 and TR-10-466, in which the formula for minimum bandwidth needed for 

measurement of full peak power in chirped pulses is presented but not derived.

Key words: chirped pulses; electromagnetic compatibility; frequency-modulated carrier wave

(FMCW chirped) pulses; pulse power measurements; radar power measurement; 

radar spectrum measurement; radio frequency (RF) interference; RF measurement

1 HEURISTIC DERIVATION OF FREQUENCY-MODULATED (CHIRPED) PULSE 

EMISSION BANDWIDTH

1.1 Introduction

Most radar signals and some radio jamming signals consist of radio frequency (RF) carriers that 

are formed into repetitive pulse sequences. Other than being switched on and off (amplitude 

modulated) to form trapezoidal time-domain envelopes, pulse carriers are not necessarily 

additionally modulated; such are said to have P0N (pronounced “P-on”) modulation [1]. Some 

types of radar and jamming pulses are, however, additionally modulated in either frequency or 

phase. Frequency modulated carrier wave (FMCW) pulses, called chirped because of the sound 

they make when they are demodulated at audio frequencies, show variation in their measured 

power levels as a function of the bandwidth used for the power measurement.

Meaningful power measurements of chirped pulses depend on accurate knowledge of the 

variation of measured power as a function of measurement bandwidth. This knowledge depends 

in turn on understanding the emission bandwidths of chirped pulses as a function of the pertinent 

parameters (frequency range and time interval) of the chirp. In this report we derive the 

relationships between chirped-pulse parameters and measured power levels as a function of 

measurement bandwidth. We provide the derivations twice, in heuristic and rigorous 

mathematical forms, and compare the results to measurements of actual chirped pulses. This 

 

1 The authors are with the Institute for Telecommunication Sciences, National Telecommunications and

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

report supplements NTIA Technical Reports [2], [3] and [4] in which a formula for the minimum 

bandwidth for measurement of full peak power in chirped pulses is presented but not derived.

2

1.2 Heuristic Approach to Understanding Power-Bandwidth Relationships in Chirped 

Pulses

While a rigorous mathematical derivation is required to definitively demonstrate the 

relationships between power and bandwidth for chirped pulses, we provide this heuristic 

presentation first because it may be more accessible by a wider audience than the mathematical 

derivation that follows in Section 2 of this report.

Consider the frequency-in-time modulation of chirped pulses of two operational radars, as

measured with a vector signal analyzer (VSA) and shown in Figure 1. The frequency may be 

ramped upward or downward in time (up-chirped or down-chirped) during each pulse, and the 

frequency modulation may be either linear or non-linear (LFM or NLFM, respectively) in time.

Figure 1. Measured frequency modulation versus time behavior for linear and non-linear chirped 

radar pulses. Many modern radars use non-linear chirped pulses.

 

2 Results will vary slightly depending on the exact shapes of measurement or receiver bandwidths. We 

use a rectangular, brick-wall shape for theoretical analysis as it is more mathematically tractable. 

Corroborating measurement data in this report, however, were taken with Gaussian filter shapes in a 

spectrum analyzer. As shown in Section 3, the brick-wall theoretical assumption agrees closely with the 

Gaussian implementation results. This indicates that the results are insensitive to details of filter shape.
3

In this analysis we assume that pulses of RF energy are transmitted in a continuous sequence 

with a pulse repetition interval (pri) of Tpri. The pulse envelopes in the time domain are assumed 

to be trapezoidal with a center-pulse width of chirp; their rise and fall times are short compared to 

chirp. The bandwidth of the chirped-frequency modulation is Bchirp, 

 | 

|, (1) 

where f2 and f1 are the extreme frequencies of the chirp modulation. 

As for any periodic waveform, the pulse sequence produces emission lines in the frequency 

domain that are spaced (1/Tpri) apart. Just as for simple P0N pulses, the amplitude (voltage) 

envelope of these lines follows a sin(x)/x or sinc curve, and the frequency domain envelope of 

the lines’ intensity (power) spectrum is a sinc2(“sinc squared”) function. (If the pulse repetition 

interval is irregular (staggered) instead of constant, then the average pulse interval is substituted 

for Tpri and the sinc2envelope still holds.) 

The measurement bandwidth Bmeas (or victim receiver bandwidth in interference studies) is 

centered on the central lobe of the emission spectrum. This bandwidth may be varied; as it 

expands or contracts, the number of convolved emission spectrum lines varies proportionately. 

The minimum value of Bmeas is (1/Tpri), so that it always contains at least one emission line. If the 

measurement bandwidth is less than the half-power points of the spectrum central lobe, the 

amplitude and power levels of the convolved emission spectrum lines are approximately constant 

within the measurement bandwidth. 

1.3 Measurement Bandwidth Required to Obtain Full Peak Power of Chirped Pulses 

From general Fourier transform relationships between the length of a pulse in the time domain 

and the spread of the same pulse in the frequency domain, 

, (2) 

where  is the pulse width and f is the approximate spectral width of the majority of the pulsed 

energy. Therefore to measure the total power of simple P0N pulsed energy, the measurement 

bandwidth Bmeas_total_power must be at least as wide as the inverse of the pulse widths: 

. (3) 

This is the condition for measurement of total peak power in a pulse or equivalently for the 

convolution of the full power of a pulse to be coupled into a radio receiver’s circuitry. (The 

condition for measurement of the total average power in the pulse sequence is addressed below.) 

To obtain the minimum bandwidth required to observe the full peak power of a chirped pulse, 

consider Figure 2, in which Bmeas is drawn as an arbitrary bandwidth relative to the chirp 

bandwidth, Bchirp, of a pulse. 

!f " 1

!

Bmeas _ total _ power ! 1

!
4

Figure 2. Diagram of proportional geometric relationships among Bmeas, Bchirp, tmeas and chirp. 

The relationship is exact for LFM pulses and approximate for NLFM pulses. 

In this figure, the pulse width chirp is the time interval of the chirping from the initial frequency 

f1 to the final frequency f2:  = chirp. Since Bchirp = f2 – f1 and the chirp bandwidth is swept 

linearly (or nearly so, even for NLFM pulses) during the time interval chirp, the following 

geometric proportionality3holds: 

, (4) 

where tmeas is the interval of time during which the chirped pulse is observed in the measurement 

bandwidth Bmeas. If Bchirp  Bmeas then of course chirp  tmeas. Rearranging (4), the proportionality 

relationship among these parameters is 

. (5a) 

From the fundamental frequency-time relationship of (2) the minimum bandwidth in which a 

full-power measurement may be obtained must be 

 

3 This proportionality is only approximately true for NLFM pulses.

tmeas = Bmeas

Bchirp

!" chirp

tmeas

Bmeas

= ! chirp

Bchirp
5

. (5b) 

Combining (5a) and (5b): 

(5c) 

and therefore the full power can only be measured if 

. (6a) 

Introducing a shorthand variable, , for the ratio ⁄ ), the bandwidth condition for 

measurement of total chirped-pulse peak power is: 

. (6b) 

Equation (6b) is used in NTIA Reports [2], [3] and [4], but we have not published its derivation 

until now. The equation is only approximately true for NLFM pulses because the proportionality 

relationship of (4) is only approximately true under that condition.4

When using (6b) it is often convenient to render bandwidths in units of megahertz and chirp 

intervals in units of microseconds; likewise units of kilohertz and milliseconds may be used 

together. Any units that are reciprocals of each other may legitimately be used together in the 

equation. The relationship of (6b) is also true for the total power convolved in the circuitry of 

radio receivers that might experience interference from chirped pulses: only victim receivers that 

have an effective bandwidth equaling or exceeding the value given in (6b) will couple the full 

power of the chirped pulses into their circuitry. 

1.4 Power Per Spectrum Emission Line Relative to Total Power 

1.4.1 Power Per Spectrum Emission Line Relative to Total Peak Power

The number of emission lines contributing to the peak power within (treating this as a brickwall bandwidth) is the ratio of that bandwidth to the line spacing of (1/Tpri): 

 

4 As a practical matter NLFM pulses in modern radar systems are close to being linear through most of 

the chirp cycle. This near-linear characteristic is seen in the NLFM curves of Figures 1 and 2, which 

were both measured from operational radars.

Bmeas !

1

tmeas

Bmeas !

" chirp

Bchirp

#

$

%

&

'

( = t( meas ) )

1

Bmeas

#

$

%

&

'

(

Bmeas

2 !

Bchirp

" chirp

Bmeas ! !

!
6

, (7) 

where dc, the ratio of the pulse width to the pulse repetition interval, is the duty cycle of the pulse 

sequence. 

Power goes as the square of voltage. For an emission spectrum line with voltage 

Vline

the power 

measured in the line, 

pline

, goes as 

Vline

2

. (Peak and average power values are identical for single 

emission lines.) The peak-detected power, 

ppeak _detected

, measured across all of the spectrum lines, 

nlines

, occurring within the measurement bandwidth (or receiver bandwidth) goes as the square of 

the number of lines within the measurement bandwidth: 

. (8) 

Since 

nlines

is proportional to Bmeas, 

(9a) 

or, in decibel terms, 

(9b) 

where Ppeak_detected = 10log(ppeak_detected) and Pline = 10log(pline).

Combining (7) and (8), the relationship between the power in a spectrum line and total peakdetected power is: 

 (10a) 

or, in decibel terms, 

. (10b) 

1.4.2 Power Per Spectrum Emission Line Relative to Total Average Power 

As already noted, the power measured in a line is identical for peak and average detection. But 

the average power within Bmeas goes in direct proportion to Bmeas. Since 

nlines

is proportional to 

Bmeas, 

(11a) 

Npeak _lines = !

1

Tpri

!

"

#

#

$

%

&

& = Tpri

" chirp

Bchirp" chirp = Bchirp" chirp

dc

ppeak _det ected ! nlines ( "Vline )

2

= n

2

lines " pline

ppeak _det ected ! B2

meas " pline

Ppeak _det ected ! Pline + 20logBmeas

ptotal _ peak = Tpri

! chirp

Bchirp! chirp

!

"

#

$

%

&

2

' pline = BchirpTpri

2

! chirp

' pline = "Tpri

2 ' pline

Ptotal _ peak =10log(Bchirp ) !10log !( chirp ) + 20log(Tpri) + Pline

paverage ! Bmeas " pline
7

or, in decibel terms, 

. (11b) 

The total average power is distributed across Bchirp. The number of lines within this bandwidth is

. (12) 

The total average power is the total peak power times the duty cycle: 

, (13) 

so the total average power is 

(14a) 

or, in decibel terms, 

Ptotal _ average= Pline+10log(BchirpTpri). (14b) 

1.5 Summary of Relationships Among Peak Power, Average Power and Bmeas

. (15a) 

In decibel terms, 

. (15b) 

The peak-detected power measured in bandwidths varies at a rate of 

20log(Bmeas).

The power per line normalized to the average power is the peak power times the duty cycle, or 

naverage_lines = Bchirp

1

Tpri

!

"

#

#

$

%

&

& = BchirpTpri

ptotal _ average = dc ! ppeak _ total = ! chirp

Tpri

! ptotal _ peak

ptotal _ average = BchirpTpri ! pline

pline_ normalized _ to_ total _ peak = ! chirp

BchirpTpri

2

Pline_ normalized _ to_ total _ peak =10 log !( chirp ) !10log(Bchirp ) ! 20log(Tpri)

(1 Tpri) < Bmeas ! !

Total peak power occurs in bandwidths of Bmeas ! ! , while total average power occurs in 

bandwidths of Bmeas ! Bchirp . If the total peak power is normalized to unity (zero decibels), then 

the power (peak-detected equaling average) per line in the central lobe of the spectrum is, from 

(10a):

Paverage ! Pline +10 logBmeas
8

chirp pri

line normalized to average B T

p

1

_ _ _  . (16a) 

In decibel terms, 

    Pline normalized to average Bchirp Tpri _ _ _  10log 10log . (16b) 

The average power measured in bandwidths varies at a rate of 10log(Bmeas). 

Peak and average power are related to each other by the duty cycle: 

(17a) 

or, in decibel terms, 

. (17b) 

The relationships among chirp pulse parameters, Bmeas, and the measured peak and average 

power levels of chirped pulses are illustrated graphically in the log-log plot of Figure 3. 

Figure 3. Relationships among Bmeas, chirp pulse parameters, and measured peak and average 

power. 

(1 Tpri) < Bmeas ! !

paverage = dc ! ppeak

Paverage = Ppeak + DC = Ppeak +10 log !( chirp ) !10log(Tpri)
9

2 RIGOROUS DERIVATION OF CHIRPED PULSE EMISSION BANDWIDTH 

2.1 Introduction 

In this section we rigorously derive (6b) for the minimum measurement bandwidth required to 

obtain the peak envelope voltage of a periodic chirped pulsed radar signal. First we define the 

signal. Then we obtain a mathematical expression for its Fourier transform. The Fourier 

transforms of the chirped pulse and the measurement filter are used to derive an expression for 

the peak measured voltage as a function of bandwidth and hence (6b). The results given here 

assume an ideal brick-wall filter and use the large argument approximation for Fresnel integrals. 

Hence, (6b) applies to situations that are reasonably characterized by these assumptions. 

2.2 Chirped Pulsed FMCW Radar Signal and Fourier Transform 

The chirped pulsed FMCW radar signal is defined as 

(18) 

where p(t) defines the envelope of the chirp, fc is the center frequency and Tpri is the repetition 

period. The parameter  is the ratio of the range of the frequency sweep (fc −f)  f 

(fc + f) to the duration of the sweep 2T or  = (f /T) = (Bchirp /chirp) and we assume Tpri  2T. 

We can derive (18) by starting with the baseband expression for a single chirp, 

. (19) 

To obtain (18), the chirp is convolved with an infinite series of Dirac delta functions regularly 

spaced at time intervals Tpri and up-converted to fc. The FMCW signal of (18) is just the real part. 

Since convolution with an infinite series of delta functions in the time domain is equivalent to 

sampling in the frequency domain, the baseband FMCW spectrum is obtained from the spectrum 

for a single chirp by sampling at the repetition frequency 1/Tpri. 

We consider the case of a rectangular pulse, p(t),

p(t) = 1 where (–T  t  T), and p(t) = 0 everywhere else (20) 

and a linear frequency sweep ( is a constant). The baseband spectrum of a single chirp is 

given by 

. (21) 

Equation (21) reminds us of Fresnel integrals that are usually defined as follows [5]: 

s(t) = p(t ! nTpri) n=!"

"

# cos 2! fct + !" (t ! nTpri)

2

( )

c(t) = p(t)e

i!"t

2

C( f ) = e

i!"t

2+i2! ft

dt

!T

T

"
10

. (22) 

Note the symmetry property F(−z) = −F(z) which will be used in the sequel. Completing the 

square of the integrand in (21) and a change in variables yields 

(23) 

where - = (f − f)/ and + = (f + f)/. Hence, the Fourier transform can be written in terms 

of Fresnel integrals as follows: 

. (24) 

We will use both the small and large argument approximations for the Fresnel integrals [5]. For 

small arguments 

(25) 

and for large arguments where |arg z| < /2

. (26) 

In the derivation that follows, we will use only the first term of this series. If we keep only the 

first term and z  5, the maximum error is about 12 per cent and if z  10, the maximum error is 

about 6 per cent. 

2.3 Minimum Bandwidth Required to Obtain Peak Voltage of the Chirp 

We assume that the measurement system simply filters the periodic chirped pulsed FMCW 

signal and the measured voltage is given by 

(27) 

where H(f) is the measurement system transfer function. To obtain the minimum measurement 

bandwidth required to obtain the peak envelope voltage of the chirp, we assume an ideal 

rectangular brick-wall filter, i.e., 

F(z) = e

i!t

2

2

0

z

! dt

C( f ) = e

i! f 2

"

"

e

i!t

2

dt

# !

# +

"

C( f ) = e

i! f 2

!

2!

F( 2! + ) ! F( 2! ! ) "

# $%

F(z) = z + O z

7

( )

F(z) ! 1+ i

2 " e

i!z

2

2

!z

i +

1

!z

2 + ("1)

m 1#3###(4m "1)

!z

2 ( )

2m i +

4m(4m +1)

!z

2

$

%

& '

(

) m=0

*

+ $

%

&

&

'

(

)

)

!(t) = 1

Tpri

C n

Tpri

!

"

#

$

%

& H n

Tpri

!

"

#

$

%

& e

i 2!nt

Tpri

n='(

(

)
11

H(f) = 1 where (–B/2  f  B/2), and H(f) = 0 everywhere else (28) 

and B is the bandwidth of the filter. For this case, the measured voltage can be written as 

(29) 

where I = [−TpriB/2, TpriB/2]. 

Returning to (24), the arguments of the Fresnel integrals can be written as 

. (30) 

If we restrict ourselves to 

f

, and chirp parameters so that is sufficiently large, we 

can use the first term of (26) as an approximation and obtain the following result for the Fourier 

transform of a chirp: 

. (31) 

Furthermore, when is large, we can reasonably approximate the sum in (29) by an 

integral 

(32) 

which can also be written in terms of Fresnel integrals. For convenience we define 

, (33) 

which can also be written in terms of Fresnel integrals. For convenience we define 

. (34) 

Now if we define 

 

so that |F( 

 

)| = max |F( 

 

)|, then |F ( 

 +  

) + F ( 

 

- 

 

)|  2|F( 

 

)|

and equality is achieved only when 

 = 0. Furthermore, it can be shown (e.g., using numerical 

methods) that max | 

 

)| = | )|for 

 

and 

 . Hence, the peak measured 

voltage is given by the following approximation: 

!(t) = 1

Tpri

C n

Tpri

!

"

#

$

%

& e

i 2"nt

Tpri

n'I

(

2! ± = 2 f

"

!

"

# $

%

& ± 'fT

< ! !fT

C( f ) ! (1+ i)e

"i! f 2

"

2"

!fT0

!(t) = 1

!

e

i" f 2

!+i2" ft

!B

2

B

2

" df

!(t') ! 1

2

F(B'+ t') + F(B'" t')

B' = B

2!

, t' = 2!t
12

(35) 

and for 

 √ , we can use the small argument approximation 

. (36) 

We now have the desired rule for the minimum bandwidth required to measure the peak voltage 

of the chirp. For measurement bandwidths in the range 

, (37) 

i.e., the peak measured voltage is proportional to the measurement bandwidth and when , 

the peak measured voltage is approximately equal to the peak envelope voltage of the chirp (i.e. 

max |c(t)| given in (19)). 

Since  = (f /T) = (Bchirp/chirp), the condition ) for the peak 

measured voltage to equal the peak envelope voltage of the chirp in the rigorously derived (37) is 

identical to the requirement for the measurement bandwidth to capture the full power of a 

chirped pulse, as derived heuristically as (6b). In the identity of (6b) and (37) we have met our 

goal of independently deriving this relationship both ways. 

2.4 Miscellaneous Relations for Peak and Average Power When 0  B  √ 

Having derived (37), we now briefly extend that work to provide some miscellaneous additional 

equations that parallel several results in the heuristic development of (7) through (17). 

In Sections 2.2 and 2.3, the amplitude of the chirped pulse was taken to be unity. If we let the 

amplitude be denoted as A, the previous results scale accordingly, e.g., (31) and (37) are scaled 

as follows: 

. (38) 

As noted previously, all results are approximate. Recognizing this, we use equal signs in what 

follows. Using these results, the peak power in the measurement bandwidth is 

. (39) 

F(B') ! B'

!(0) ! B

"

B ! ! (i.e, B ! Bchirp ! chirp

!(0) ! 2 F(B') for 0 " B' " B'

m and #fT >>1

!(0) ! A B

"

and C( f ) ! A(1+ i)e

"i# f "

2"

Ppeak = !(0)

2

= A2

B2

"

! A2as B ! "

0 ! B ! !

B ! !
13

We need to examine periodic signals in terms of their spectral lines and average power. The 

number of lines in the full measurement bandwidth 

Bmeas

is 

nlines

= 

BmeasTpri

 (the floor function, 

or integer value of 

BmeasTpri

) or . Since the power per line is 

(40) 

the average power is 

(41) 

and

ppeak= B

2

Tpri

2

pline= nlines

2

pline

. (42) 

The ratio of peak to average power, 

pr

, is 

pr=

ppeak

pavg

= nlines

(43) 

and

pline

ppeak

=

1

nlines

2

. (44) 

Bmeas ! nlines Tpri for BmeasTpri !1

pline = C(0)

2

Tpri

2 = A2

!Tpri

2

pavg = nlines ! pline = A2

B

!Tpri
14

3 POWER-MEASUREMENT DATA SUPPORTING THEORETICAL RESULTS 

3.1 Introduction 

To illustrate the relationships among peak power, average power and Bmeas with a physical 

chirped pulsed FMCW signal, we set up RF-generation hardware to produce chirped pulses in a 

repeating sequence. We measured the RF emissions with peak and average detection in 

bandwidths ranging from (1/Tpri) to somewhat wider than Bchirp; we provide the results of those 

measurements in this section. 

3.2 Measurement Setup 

We configured a vector signal generator (VSG) to generate linearly chirped pulses. We adjusted 

the chirp bandwidth to1 MHz, set the pulse width at 100 s, and set the pulse repetition rate at 

2500/sec = 2.5 kHz. We calculated the quantity √ ⁄ to be 100 kHz. We 

connected a spectrum analyzer to the VSG output and measured the spectrum in bandwidths of 

1 kHz, 3 kHz, 10 kHz, 30 kHz, 100 kHz, 300 kHz, 1 MHz and 3 MHz with average detection, 

and again with peak detection in all of the same bandwidths except 3 MHz. We used an Agilent 

E4440A spectrum analyzer with maximum-hold trace mode, 50 msec sweep time and 10 dB 

input attenuation. The measured pulse repetition sequence is documented in the time domain 

(from the E4440A, operated in zero-hertz span) in Figure 4. The measured linear chirp is 

documented in the modulation domain in Figure 5; we acquired the Figure 5 data with an Agilent 

E9600 Series VSA. 

3.3 Measurement Results 

Measurement results are shown in Figures 6–9. Figure 6 shows a detailed portion of the emission 

line spectrum, with the lines spaced 2.5 kHz apart at a constant amplitude from one to the next. 

Figures 7 and 8 show emission spectra measured with average and peak detection, respectively, 

in a variety of bandwidths Bmeas. Using the data in these two graphs, we graphed the power levels 

observed in each bandwidth and detection mode as a function of Bmeas in Figure 9. 

3.4 Comparison of Theoretical and Measured Results 

The theoretical predictions graphed in Figure 3 can be compared directly to the measurement 

values shown in Figure 9. Both figures show identical power-measurement results at bandwidths 

that resolve individual emission lines. The line power relative to the maximum peak power is

 ( )

 

)) = -32 dB,

as predicted by (15), and relative to the maximum average power the line power is

 ( )) = -26 dB,

as predicted by (16). The peak power maximizes at a normalized level of 0 dB when

 √ ⁄  = 0.1 MHz = 100 kHz,
15

as predicted by (6b) and (37), and average power maximizes when Bmeas = Bchirp = 1 MHz, also 

as predicted. The measured difference between maximum peak and average power is -6 dB, as 

predicted by (17) from the time domain duty cycle of

(100 μs/400 μs) = 1/4.

As bandwidths increase from the line spacing, the average power levels go as 10logBmeas and 

peak power levels go as 20logBmeas, as predicted by (9) and (11) and the parallel (41) and (42).

Figure 4. Measured time-domain envelopes of a series of chirped pulses. The full pulse width is 

observed in this graph because the measurement bandwidth of 8 MHz exceeded the chirp 

bandwidth of 1 MHz.
16

Figure 5. Measured linear chirp (1 MHz in 100 s) of one pulse from the sequence of Figure 4.

Figure 6. Measured line spectrum detail for the chirped-pulse sequence of Figures 4 and 5.
17

Figure 7. Measured average-power chirped-pulse spectra as a function of Bmeas.

Figure 8. Measured peak-power chirped-pulse spectra as a function of Bmeas.
18

Figure 9. Measured peak and average power for example chirped pulses as a function of 

measurement bandwidth. Measured power levels are from Figures 7 and 8. Comparison of this 

figure to Figure 3 shows strong agreement between theoretical predictions and measured results. 

3.5 Summary 

Unlike the case for unmodulated P0N pulses, the bandwidths at which measured peak and 

average power are maximized for chirped pulses are not necessarily identical, although peak and 

average power measurements always yield identical power levels for individual emission 

spectrum lines. As is the case for P0N pulses, peak power measurements of chirped pulses vary 

as 20log of the measurement bandwidth, while average power measurements of these pulses vary 

as 10log of the measurement bandwidth. 

The relationships derived in this report may be used for electromagnetic compatibility studies of 

interference from chirped-pulsed emitters to victim radio receivers. In such studies, victim 

receiver bandwidth should be substituted for the variable Bmeas in the equations in this report. 

We have derived the relationships among peak power, average power and measurement 

bandwidth for chirped FMCW pulses both heuristically and rigorously in this report. These 

relationships may be best understood and utilized via the graph of Figure 3. We see strong 

agreement between theoretical predictions and measurements of actual chirped pulsed emissions 

in comparison of our prediction graph (Figure 3) with measurement results (Figure 9). 
19

4 ACKNOWLEDGMENTS

Discussions with Dr. Michael Spencer of the Jet Propulsion Laboratory (JPL) in Pasadena, 

California prompted our creation of this report; Dr. Spencer identified the need to have this 

material published in a referenceable document. We thank Geoff Sanders and John Carroll of the 

NTIA/ITS laboratory for setting up the VSG that created the chirped RF pulses for the 

measurements we have presented in this report.
20

5 REFERENCES

[1] Manual of Regulations and Procedures for Federal Radio Frequency Management, 

National Telecommunications and Information Administration, U.S. Department of 

Commerce, May 2010 Revision of the 2008 Edition, Section 9.8.2, pp. 9–21.

[2] Sanders, F. H., R. L. Hinkle and B. J. Ramsey, “Measurement procedures for the radar 

spectrum engineering criteria (RSEC),” NTIA Report TR-05-420, U.S. Dept. of Commerce, 

Mar. 2005.

[3] Sanders, F. H., R. T. Johnk, M. A. McFarland and J. R. Hoffman, “Emission measurement 

results for a cellular and PCS signal-jamming transmitter,” NTIA Report TR-10-465, U. S. 

Dept. of Commerce, Feb. 2010.

[4] Sanders, F. H. and R. T. Johnk, “Emission measurements of a cellular and PCS jammer at a 

prison facility,” NTIA Report TR-10-466, U. S. Dept. of Commerce, May 2010.

[5] Abramowitz, M. and L. A. Stegun, Handbook of Mathematical Functions, National Bureau 

of Standards, Applied Mathematics Series, Publication 55, U.S. Department of Commerce, 

1964.
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

Relationships Between Measured Power and Measurement Bandwidth for FrequencyModulated (Chirped) Pulses

5. Publication Date

August 2012

6. Performing Organization Code

ITS.T

7. AUTHOR(S)

Frank H. Sanders and Roger A. Dalke

9. Project/Task/Work Unit No.

8. PERFORMING ORGANIZATION NAME AND ADDRESS 6454000-200

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

Technical Report, FY12

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.)

Measured power levels for radio frequency (RF) pulses that are frequency modulated (chirped) vary as a function of the 

bandwidth in which the measurement is performed; if chirped pulses cause RF interference, the power levels of the pulses in 

victim receivers will likewise vary as a function of receiver bandwidth. This report provides both heuristic and rigorous 

derivations of the relationships among chirped pulse parameters and the measured peak and average power levels of chirped 

pulses as a function of measurement bandwidth. These relationships may be best understood via a single graph (Figure 3) 

presented in this report. This report supplements NTIA Technical Reports TR-05-420, TR-10-465 and TR-10-466, in which 

the formula for minimum bandwidth needed for measurement of full peak power in chirped pulses is presented but not 

derived.

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

chirped pulses; electromagnetic compatibility; frequency-modulated carrier wave (FMCW chirped) pulses; pulse power 

measurements; radar power measurement; radar spectrum measurement; radio frequency (RF) interference; RF measurement

17. AVAILABILITY STATEMENT

UNLIMITED.

FOR OFFICIAL DISTRIBUTION.

18. Security Class. (This report)

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20. Number of pages

 20

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TR-12-488
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