IEEE TRANSACTIONS ON ELECTROMAGNETIC
COMPAT\I11ILITY
VOL.
EMC-9,
[6] L. Linderer et al., "Mileasurements oln two Nb superconductive
RF cavities," Phys. Letters, vol. 2, pp. 119-120, September 1962.
[7] C. R. Haden, J. M. Victor, and W. H. Hartwig, "R. F. residual
losses in superconductors," presented at 1966 Southwestern IEEE
Conf., Dallas, Tex., April 20-22, 1966.
[8] W. H. Hartwig and D. Grissom, "Dielectric dissipation measturements below 7.2°K," in Low Teooperature Physics. New York:
Plenum Press, 1965.
[9] R. J. Allen and N. S. Nahman, "Analysis and performanice of
superconductive coaxial transmission lines," Proc. IEEE, vol. 52,
pp. 1147-1154, October 1964.
[10] T. A. Buchold, "About the nature of the surface losses in
superconductors at low frequencies," Cryogenics, vol. 3, pp. 141-149,
September 1963.
NO.
3
DECEMBER
[11] C. P. Bean, "M\Iagnetization of high field superconductors,"
Rev. Mod. Phys., vol. 36, pp. 31-39, January 1964.
[12] C. P. Bean et al., "A research investigation on the factors that
affect the superconducting properties of materials," General Electric
Research Laboratory, Schenectady, N. Y., Tech Rept. AFAIL-TR65-431, March 1966.
[13] S. Becker et al., "Interference analysis of new compon-ents anid
circu'its," Airbourne Instruments Laboratory, Melville, N. Y., Tech.
Rept. RADC-TDR-64-161, 1964; see also "A rule of thumb for
prediction of third-order intermodulation," IEEE Spectru7n, vol. 1,
p. 5, May 1964.
[14] K. Siegel, R. Domehick, and F. Arams, "Superconducting
High-Q radio-frequiency eircuit usin g niobium stannoide above
4.20K," Proc. IEEE, vol. 55, pp. 457-458, March 1967.
Frequencies
Spectrum Analysis
The Significance of Negative
ROBERT B. MAARCUS,
Abstract-In EMC measurements spectra generated by pulsed
signals are often measured. To calculate the spectra theoretically,
Fourier methods are usually used. The Fourier transform yields a
spectrum which contains positive and negative frequencies. The
exponential form of the Fourier series also yields a spectrum which
contains positive and negative frequencies. However, the trigonometric form of the Fourier series yields a spectrum containing only
positive frequencies. Since there seems to be some doubt about the
physical interpretation of negative frequencies, the relationship between the double sided spectrum of the exponential Fourier series
and the single sided spectrum of the trigonometric Fourier series is
shown. The reasoning is then extended to the Fourier transform to
show the relationship between the double and single sided spectrum
obtained by the Fourier transform. A method of obtaining spectral
levels for positive frequencies only using the Fourier transform is
shown.
IN E1\JC WORK it is ofteu necessary to measure pulsed
spectra. In theoretical studies associated with the
measurements, Fourier analysis has been used for many
years to predict the spectra theoretically.
In Fourier analysis one canl use Fourier tranisfoims or
Fourier series. The Fourier series gives a spectrum which
cointaiins discrete spectrunm linies. It is usually cumbersome
to perform mathematical operations OIl each spectrum
line to determine the spectral characteristics of a pulsed
signal or to determinie the output of circuits whose input
consists of pulsed signals. It has been found much more
convenient to use the spectrum of a single isolated pulse to
w
Mlanuscript received May 13, 1967; revised August 11, 1967. This
ork was supported uinder Air Force Contract 19(628)-5049.
The aujthor is with the IIT Research Institute, Anniiapolis, Md.
123"
1967
in
SENIOR MEMBER, IEEE
determine the output waveshape or other signal parameters of a circuit whose input is a pulsed signal. The
Fourier transform gives the spectral level of a single
isolated pulse. If the actual input to the circuit is a repetitively pulsed signal, the output signial due to a single pulse
of the pulse train can be founid by the Fourier tranisform method provided that the circuit has no residual
currenit or voltage from the previous pulse.
If any standard textbook on Fourier trainsforms were
investigated, it could be seeni that the spectra found by
means of Fourier transforms conitain negative anid
positive frequencies. The question arises, what are
negative frequencies and what siginificanrce do they have
in spectrum analysis? In RF measurements negative
frequenicies cannot be defined. Are the negative frequenicies
merely discarded and looked upon as mathematical
curiosities?
The negative frequeiicies cannot be discarded mathematically. The following expositioni will help shed some
light on the significance and the definitionr of niegative
frequencies in spectrum analysis.
Let us first examine the Fourier selies in exponienitial
form. It will be shown that the spectrum described by the
Fourier series has negative frequeiicy componieiits in the
same sense as the spectrum described by the Fourier
transform. The relationship betweeni the niegative frequeincies in the two spectrums will be showni later.
The Fourier series in exponential form is
f c
f = (t)n=-coaE ico
n9t(
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124
IEEE
TRANSACTIONS
where
f(t)
=
WT
=
=
E
T
DECEMBER
1967
If a time function f(t) is real, then
f
atn
ON ELECTROMAGNETIC COMPATIBILITY
any
27rfr
frequency of periodic function
E
=-
=
=
a°-n
periodic function of time
T/2
,f(t)
inw
'trdt
=
(3)
an
where
(2)
maximum amplitude of periodic function of time
period of periodic function.
a,
O
conjugate of
an
T/2
=
TJf(t) e-inw
rdt.
If an were to be put in trigonometric form, we would obAn examination of (1) will show that a periodic function tain
of time consists of an infinite sum of discrete spectral lines;
E fT/2
the frequency of each line is nfT. Since n has all integer
ff(t) (cos ncw,t - j sin ncw,t) dt.
atn = T
(4)
JT/2
values from minus infinity to plus infinity, the spectrum
mathematically exists in the negative frequency region as
The real part of an will be called Rn and the imaginary
well as the positive frequency region. However, when n
of a,, will be called Xn; therefore
part
equals zero, the frequency is zero which indicates the
spectrum has a dc component. Some of the spectral lines
inieluding the dc component may have zero amplitude and
Rn=T
f(i't) cos nwo,t dt
(5)
;T2
thus vanish. If the nature of each spectral line is examined
it can be seen that each line is a time function whose
E T/2
Xn = -sin
dt
amplitude is ao. The time varying part of the function
(6)
T-T12 f(t) rnw,t
is En T. The time function is not a sinusoidal time function
but it is a rotating vector whose amplitude is an and whose
an - Rn + jXn
(7)
angular frequency is nw,. Equation (1) shows that n takes
(8)
on all integer values from minus infinity to plus infinity.
In the negative region the angular frequency of the
rotating vector is negative. The negative frequency
can be interpreted as clockwise rotation of the vector
X
--X-n
while positive frequency represents counter-clockwise
When n equals zero a new quantity ao will be defined as
rotation of the vector. There is no other significance to
negative frequencies. It should be apparent now that the
E JT/2
ao -o=f(t)dt.
exponential form of the series yields spectral components
(9)
T JT/2
which are not sinusoids but rotating vectors of various
Equation (9) defines the amplitude of the dc component
amplitudes and frequencies. Since the amplitude of each
vector is constant it is not possible to show any direct of the spectrum.
The series can now be written as
physical interpretation of the spectral lines. The doublesided spectrum containing negative frequencies is only a
-1
mathematical and not a physical entity.
E anEjnwrt + E an jncrt
(10)
f(t) == ao +n=-co
n=1
Most of us are familiar with the Fourier series in trigoSubstituting (3) in (10) yields
nometric form. The trigonometric form contains no
negative frequencies. Further, the trigonometric form
shows spectral lines to be of sinusoidal form. Sinusoids
(11)
f(t) = ao + E a*n 6 -n- rt + E an jnc rt
n=1
n=1
have physical as well as mathematical meaning. Within
Substituting (7) and (8) in (11) yields
mathematical restrictions, which will be shown, the sinusoidal form of the series is exactly equivalent to the
exponential form. The transition from the exponential f(t) = ao + E (RAn jX-n) e
n=1
form to the trigonometric form will be shown shortly.
To be perfectly general mathematically, a function of
+ E (Rn + jXn) Ejnlrwt
time can be real or complex. While a complex function of
n=1
time has no physical interpretation, it has important
mathematical uses in Fourier analyses which will not be
Putting the exponential in trigonometric form yields
covered here. The foregoing statements on going from
the double-sided spectrum to the single-sided spectrum
ao +
(Rn jXn) (cos nw,t j sin nw,t)
apply only to real time functions. Therefore, the restriction f(t)
n=1
on going to the one-sided or positive frequency only
spectrum from the two-sided spectrum is that the time
+ n=1 (Rn + jXn)(cos nfwTt + j sin nfwrt)
function has to be real for the transition to be valid.
*
-n
°n=
iX7,
=
at-
co
00
00
00
=
-
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-
MARLCUS: NEGXITIVE
FREQUENCIES IN
SPIECTP'lUM A.N ALYSIS
125
where
coo
f(t)
=
ao
+
Z
ii= 1
[Rn
- j(RI, Sill ncOt +
cOs
Xn
-n Sill lCOrt
co
f(t) - ao + E
n
1
[2R,,
nw,rt
COS
Xnsi5
-
nw,)rt
nflrt)] + E
n
+ j(RI
cos nWrt
-
=
1
Sill ncort +
E
[R,
X,Z
2Xn Sill nflrt].
COS
COS
J(t)
e-
T 1T/2
nllOrt
nwcrt)
T12
an = -
c.idt.
(16)
If f(t) is not periodic but a single isolated pulse, the
I time function f(t) can be written as
(12)
f(t)
=
F(27f) e i2wrftd
(e17)
(t)ej2ft.
(18)
0
Equation (12) is the familiar trigonometric form of the
Fourier series. It is exactly equivalent to the exponenitial
form providing the time function is real. Equationi (12)
containis no negative frequencies. What has happened to
the uegative frequencies? The negative frequency came
about by considering the time varying trigoniometric
fuuctioins to be made up of two vectors rotating in opposite
directions. There is no physical meaniing to the negative
frequencies.
The Fourier transform will lnow be conisidered. If onie
has a single isolated pulse, the spectrum resulting therefrom can be described by means of the Fourier transform.
The spectrum has uo discrete lines but is coiltilluous. The
Fourier transforms describe a spectral level in volts per
Hertz at every frequency.
The direct Fourier transform is
F(w)
f(t) -wdt.
=
(13)
Equation (13) gives the speetral level of the spectrum at
any angular frequenicy co.
If the spectrum is knowni the time funietion canl be
founid from the iniverse tranisform which is
where
F(2wrf)
f
=
f-l
OD
Equations (17) and (18) are analogous to (15) and (16).
It is easier to visualize the functioni of time in (17) as
being the integral of a continuous spectrum whose amplitudes are described by (18) and whose frequency cornponents are described by the exponential part of (17).
The variable in (17) and (18) can be changed from f to
o, thus
27r J
co=
(19)
(20)
dw = 2wr df.
Multiplying and dividing (17) by 27r yields
f(t) =22 7r
F(27rf) E2' 7f2wdj.
_co
(21)
Substituting (19) and (20) in (21) yields
wtdw.
(14)
MNIathematicians coinsider (13) aild (14) as a tranisformii
F(co) =
f(t) -ij-tdt
pair; (13) gives an amplitude spectrum from a time
fuinctioni anid (14) gives a time functioni from ain amplitude
It cail IlOW be said that
spectrum.
However, an amplitude spectrum has ino physical
I
f(t) 2r _ coF(w)Uw't dco
meaning to the enigineer. Furthermore, it seems very
strange that one can obtaini a time fuiletioin by iintegratinig
an amplitude funetion. In (14) the amplitude function wheire
F(w) is multiplied by a time funietioni EjXt; therefore, the
illtegral yields a time funietioni. The poillt is that the
F(co)
f(t) -i@tdt
mathematicians have separated the amplitudes of the
spectral componients from the time funetioni and have and
pgotteii a transform pair. Enginieers would like to coinsider
F(co) = R(w) + jX(w)
spectral compoinents as time funictionis with associated
amplitudes and not just as disassociated amplitudes.
R (c) =
f(t) cos wt dt
Let us reconsider the expoinential series
(13)
(t)
JI F(w)ejictdw.
7=
22l r -0
(14)
f(t)
=
2r
f
_co F()We
Substituting (19) in (18) yields
_C
=
F
E
=
(14)
(13)
_co
f(t)
(23)
00
=
n
E a1nE
(1)
= -co
If 27rf, is substituted for Cor, then
co
f(t)
(22)
=
n =-o
a,, e j27rfrt
(15)
X(co)
-
f f(t) sin cot dl.
(24)
_co
The spectral components are F(CO) E i'. These componeints are not discrete lines but are continuous. The
spectral components are time functions and mathemati-
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126
IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY
cally exist at negative as well as positive frequencies. However, just as in the exponential series, the time functions
are rotating vectors which do not change amplitude with
time. Again if only real time functions are considered, then
F(co) = F(-c,,)
R(w) =R(-)
X(W) = -X(-Cw).
Substituting (22) in (14) yields
1I
co
-
f(t)
=
f(t)
=
I-
2 7r
_co
I
2
oo
r
[R(w) +
j'X(G)](cos wt
{R(w)cos
wt
(25)
+ j sin wt)dw
-X(w) sin cut
+ j[LXo)
wt + R(w) sin wt]}dw
cos
since
X(W) =
w)
anid
(-cw)
R(w)
it follows that
[X(w)
cos
cot + R(w) sin wt]dw O.
=
=
I
[2R(w) cos wt - 2X(o) sin tl]dc. (27)
Equation (27) is the Fourier transform for the singlesided spectrum. It contains no negative frequencies.
It can be seen that the amplitudes of the spectral components of the one-sided spectrum are exactly double
those of the two-sided spectrum. It must still be remembered that the foregoing applies only to real time functions.
Negative frequencies are associated with time functions
representing clockwise rotating vectors and positive
frequencies are associated with time functions representing
counter-clockwise rotating vectors. These are mathematical representations of the frequency components of
the spectrum and do not have direct physical interpretation. The mathematical combination of the two oppositely
rotating vectors produces sinewaves. The sinewave
components of the spectrum have physical meaning and
it is these sinewaves that constitute the actual physical
spectrum. It has been shown how to combine the rotating
vectors mathematically to produce a spectrum of sinewaves which contain no negative frequencies. Therefore
negative frequencies are mathematical tools which have
no direct physical interpretation.
REFERENCES
Therefore
f(t)
1967
Equation (26) is the Fourier transform for the spectrum
in trigonometric form. The frequency limits of the spectrum range from minus infinity to plus infinity and therefore the spectrum is double-sided. But since R(w) =
R(-w) and X(co) = -X(-w), then
f(t)
[R (X) + jX (w)]Ie Jtdw.
f(t) =2r
co
Putting f-Xt in trigonometric form yields
DECEMBER
= 2
2
J
r
co0
[R(w)
cos
wt
-
X(w) sinl wt]dw.
(26)
[1] A. Papoulis, The Fourier Integral and Its Applications. New
York: McGraw-Hill, 1962.
[2] M. Javid and E. Brenner, Analysis, Transmission, and Filtering
of Sign,als. New York: McGraw-Hill, 1963.
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