PROCEEDINGS OF THE I-R-E

On the Filter Problem of the Power-SpectrumAnalyzer*

S. S. L. CHANGt, SENIOR MEMBER, IRIF

Summary-The function of a power-spectrum analyzer is to pre-dict from a single finite length of wave record, which has the statisti-cal properties of filtered random noise (e.g. ocean wave, turbulenceetc.), the average power per unit bandwidth of an ensemble of suchrecords at various frequencies. The signal derived from repeatingthe record is heterodyned with another signal whose frequency isscanned at a uniform rate across the entire spectrum and the result-ant wave is passed through a narrow filter and then detected by asquare-law detector. Two problems arise:

1. Due to its finite bandwidth, the filter performs a necessaryweighted average on the power spectrum. What is the best filterresponse to minimize the intrinsic error associated with the predic-tion of an average characteristic of an ensemble from a singlerecord? What practical filter is closest to the ideally best?

2. How fast can the frequency be scanned without appreciablydeviating the filter response?

Definite solutions are given to the above problems. Eqs. (26) and(27) together with Table I give the lowest probable error for filterswith various shapes of response curve. Eq. (32) defines the idealfliter which minimizes this error. The ideal filter can be very closelyapproximated by cascading a single resonant circuit to a pair ofcritically coupled resonant circuits with a Q-value \/2 times that ofthe former. The filter response to varying frequency would not alterappreciably from its response to steady sinusoidal wave if the rate offrequency change is smaller than the square of the half bandwidth.

INTRODUCTIONT HE PHYSICAL ACTION of repeating a wave

record of finite duration indefinitely breaks acontinuous power-density spectrum into discrete

components. To approximate the continuous power-density spectrum by performing a weighted average ofa finite number, m, of neighboring components intro-duces two essential sources of error, namely:

1. the statistical error, which is approximately equalto 1/Vm`, and

2. the blurring error, which is due to the averagingprocesses and increases as m is increased.

It is apparent that for the weighted average powerto have any significance at all, the power densityE(.,,) must not undergo appreciable change in the fre-quency spacing between two neighboring components.This can be accomplished physically by making theduration of wave record sufficiently long.

In order to minimize the statistical error, a largenumber of components are generally included in theaveraging process. However, in so doing, the power den-sity E(w,7) undergoes appreciable change in the rangeof frequencies being so included and the blurring errorbecomes large. To determine the optimum bandwidthand attenuation curve of the filter both componentsmust be taken into account.

* Decimal classification: R143.2. Original manuscript receivedby the IRE, November 9, 1953; revised manuscript received, March5, 1954. This work was performed under Contract No. DA-49-055-Eng. 32 of Beach Erosion Board, Corps of Engineers.

t Dept. of Electrical Eng., New York University, N. Y., N. Y.

A mathematical expression is obtained for the over-allprobable error including both of the above mentionedfactors. While other sources of error are also present,they are of similar smaller order of magnitude andneglected in the analysis:

1. In (3), the difference between the ensemble aver-age power En of a discrete spectral line and the cor-responding area -qE(w-,) of the continuous power-densityspectrum is neglected. This is justifiable as the primaryassumption for the averaging process is such that E(co),)and E(wco+1) do not differ appreciably.

2. In (4), the area of the pass band of the filter is usedas the normalizing quotient instead of the sum of filterresponses at discrete lines. It is not in accord with com-mon practice employed in manual averaging computa-tions, but represents the actual situation in the elec-tronic analyzer briefly described in the summary. In anycase, the difference between the two methods of normal-ization is a small fraction of a line in the total sum whilethe statistical error referring to the same basis for com-parison would be V/mi lines.

Using pass band areas as the normalizing factor, meas-ured power density P(co) has the same dimensions asspectral density E(co). It reduces to E(co) in the idealcase of 'm being infinite while Mi7 being infinitesimal.Once the mathematical expression of the probable

error is obtained, variational methods are used to de-termine the minimum probable error and the associatedoptimum bandwidth as well as optimum shape of at-tenuation curve.

Error coefficients KA are determined for varioustypes of filters as a measure of their desirability. If theshape of the attenuation curve remains unaltered whilethe bandwidth of the filter varies, so that the over-allprobable error is always at its minimum value, thisminimum error is proportional to K2N. However, as thebandwidth is fixed in actual equipments, an alternativebasis for comparison is to select different bandwidthsfor various filters so that they give the same statisticalerrors, and to compare the respective blurring errors.The blurring errors are proportional to KA5. Physically,with the same equipment and everything else, the blur-ring error introduced by a double-tuned circuit would be85 per cent more than that introduced by the approxi-mate ideal filter.The value Ka= o for the single-tuned circuit points

to the fact that its cut off is too gradual to resolve thepower spectrum sufficiently. Mathematically, as itseffective pass band is not clearly defined, there are al-ways some points on the power-density spectrum forwhich d2E(o,)/dw2 and the resulting expression for theprobable error is indeterminate.

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Chang: On the Filter Problem of the Power-Spectrum Analyzer

The result of the second part of the analysis about theallowable rate of scanning is what one would expectfrom purely physical considerations. The time for aresonant signal to build up in a narrow pass-band filteris approximately the reciprocal of the half bandwidth. Ifeach spectral line is allowed to build up approximatelyto its full strength, it should stay in the pass band for afew times that duration. This is the same as sayingthat the rate of frequency change should be smallerthan the square of the half bandwidth. The analysisalso gives the difference in response between the tran-sient case and the steady-state case so that one maymake design estimations.

CRITERION OF THE OPTIMUM FILTER

In repeating a wave record of T seconds, the signalderived can be expressed as a Fourier series,

2irnt . r27rnt.cAncos + bn s Tn

According to Rice,' the probability that An has a valuewhich lies between An and An+dAn is:

1 2e.e-An /EndAn

where En is the mean square error. The coefficient bn isindependent of An and has the same probability dis-tribution. Let P. denote the power of the nth spectralline, that is:

P. = An2 + bn2.

analyzer is to recover E(X) from the finite wave recordwith the least probable error.From (2), we see that the power contained in each

Fourier component has a probable value from zero to afew times its ensemble mean. To record each componentseparately would cause a probable error of 100 per cent.The statistical error can be reduced by averaging thepower contained in a number of components in theimmediate neighborhood.2 This weighted averagingprocess is automatically accomplished by the electricalwave filter in the power-spectrum analyzer.

While the statistical error is being reduced by aver-aging over a large number of components, anothersource of error is introduced by the averaging process.Evidently, the averaged quantity would fail to followclosely the variations in the power spectrum (or theensemble mean), especially when the variations arerapid. An optimum filter should minimize the combinederror due to both the statistical effect and the blurringeffect.

PROBABLE ERRORLet the function e 2a of a filter be represented by

U2(Q/A), where Q=w-wo, wo being the center fre-quency, and A is some sort of bandwidth such thatU2(Q/A) is small for values of Q| larger than A. Thenormalized measured power at the output of the filter is:

(4)

(1)

0,,P(Wo) =

lU2 -dQJ 0'aA

The probability that the power Pn has a value be-tween Pn and Pn+dPrw is:

pndP_ = - e-P-1E-dPnEn

(2)

where pn is the probability density. Eq. (2) follows di-rectly from the probability distributions of the coeffi-cients An to bn.

Evidently, the ensemble average power for the nthharmonic is E.. It is related to the continuous power-density spectrum by the following equation; approxi-mately:

En= yE(COn) (3)

where = 27r/T, and con= n-q. The purpose of the wave

P(Do)- E(wo) =

(Pn- En)U2(- ) - 4tn=l iU

U2{ dQJ-O \A

(5)

where

V = E(co) U2(y) d - E EnU2(-) (6)

Vt is the error due to blurring. Note that it is inde-pendent of P,The mean square error is the weighted average of the

square of P(coo) -E(wo). From (2) and (5), it is:

00o 0°o - oo O'nX - 2 oof. .f[ E(Pn-EEn)U2)-V] IIPdPn

:- n=l AUn=l.~ ~ ~ ~ ~

[, 22Qd ]-oo0 A

(7)

2 J. W. Tukey, "The Sampling Theory of Power Spectrum Es-timates. Symposium on Applications of Autocorrelation Analysis to

I S. O. Rice, "Mathematical analysis of random noise." Bell Sys. Physical Problems," Woods Hole, Mass., Office of Naval Research,Tech. Jour., vol. 23, 1944, vol. 24; 1945. Washington, D. C.; 1949.

The error is

1954 1279

EZ2

PROCEEDINGS OF THE I-R-E

(8)

(9)

From (2), it follows:r0-(P En)ptdPn 0

-(Pn-En)2pndPn= E.,

f =PndPn = 1

Using (8), (9), and (10), (7) can be readily integrated.It is:

12 = u()+ 2}

where I is the integral fGo.U2(Q/1A)dQ.Eq. (11) shows that the over-all probable error e

can be expressed as

=l 2 + Eb2 (12)

where 6p is the statistical error,

P 2 =-E En2U4E (13)12 ¼!

and 6lb iS the error due to blurring.

Elb - I*(14)

The statistical error can be simplified by noting that, tothe first order of approximation, En?E(wo), asU(f2n/A) has appreciable magnitude only when Qn issmall. To the same order of approximation the summa-tion can be replaced by an integral. Therefore:

2~ fnE2(Wo) f U4( )d (15)

Next, the function i1 is to be evaluated. Since the sta-tistical properties of the ensemble is similar to that offiltered random noise, E(w) is same as the summationof the functions e-2 of a number of physical networks.As this function of a passive network with dissipativeelements is analytic and bounded for real values of fre-quency9'4 the mean-value theorem states that:

/dE\E(co) = E(ooo) + k--) (w-o)

dx 0

From (15), E. can be expressed as:

/dE\En=- E(wn) =-E(wo) +717 ) .On

Qn2+ 7iV(co, ')o)

2(18)

(10) Substituting (17) into (6), and replacing sumnmationsby integrals it becomes:

-dE -wQ4- = I U2(- IQd

1 Fd2E1 r Q2 dW2J too -0JX A,)

(11)

In deriving (18), V(cw, w0) is replaced by V(wo, wo), asU2(Q/IA) has appreciable value only in the immediateneighborhood of w =o.As the values of dE/dw and d2E/d W2 at wo are inde-

pendent quantities, to minimize the aggregated average41 with respect to U for all possible values of the deriva-tives is mathematically equivalent to minimizing theabsolute values of the two integrals of (18). The firstintegral vanishes if U2(Q/A) is symmetrical with re-spect to U. While this condition is not the only possi-bility, a direct calculation without it by minimizationof the second integral and (15) while holding the firstintegral equal to zero leads to the symmetrical filter of(32). As the calculation is straightforward, it will beomitted here. Instead, a slightly more complicatedmethod which not only derives the ideal filter but alsomakes it possible to determine the relative merits ofvarious practical filters will be presented.

EFFECT OF BANDWIDTHWith the assumption of symmetry of U2(j2/A), (12)

becomes:

K 1 J2E2 = nE2(cc0) -+- E(2)2(Coo) -

J2 4 12(19)

where E(2) (WO) is the value of the second derivative ofE(w) at w=co0, and the functions J, K, and I denote:

I= U2) dQJ 00 A

(18a)

1+ - V(co, WO5) (( - W,O)2

2(16)

w-here V(w, coo) is equal to the second derivative d2E/dco2at some value of co between (A and coo. It follows that:

Vc2, )

V(woo Coo) = K 2dw o(17)

3 W. J. Pierson, Jr., "A Unified Mathematical Theory for theAnalysis Propagation and Refraction of Storm Generated OceanSurface Waves-Parts I and II," Research Div., College of Eng.,New York University; 1952.

4 W. J. Pierson, Jr., and W. Marks, "The Power Spectrum Analy-sis of Ocean Wave Records." Trans. Amer. Geophys. Union, vol. 33;Dec., 1952.

= -2 d =A)I

J U2 (Q2dQ

100 /Q / 2 X= A3 U2

2 d A=3J1

K=f U4(-dQ

= Af U4Q-)d() = AK,.

(20)

(21)

(22)

1280 A ugust

Chang: On the Filter Problem of the Power-Spectrum Analyzer

The definite integrals I1, J1, and K1 are independent of A.Eq. (19) becomes:

62 = E2(o) K11I 1 J12El2n 2(O)- - + - E(2)2(WO) - M4.I[12 a 4 112

A()=O, for Q | > A

U2(Z = A {1 -()}(32)

for I Q| < A.(23)

tEq. (23) states that while the statistical error is in-versely proportional to the square root of bandwidth,error due to blurring is proportional to the fourth powerof bandwidth. There exists a bandwidth for which theover-all error is a minimum. Setting d62/dA = 0, it is:

A6 E2(o) K1 (24)

E(2)2(wo) J12

Substitute (24) into (23); it becomes

62 = - 7l8El 6(o)E(2) 4(Co) (25)4I2

or

(26)I = KeE8(wo) [n2E(2)(wo)] 2

\5 K, 4J1 2K 2=-

2 I1

The error coefficient Ke, as defined by (27), gives aquantitative evaluation of the minimum error associatedwith filters of various shapes.

THE OPTIMUM FILTER AND ITSPRACTICAL APPROXIMATIONS

To determine the shape of the filter which minimizesKe, its first-order variation with respect to U is set tozero:

(5ln Ke = 0. (28)

(27)

Evidently the exact performance cannot be obtainedwith an actual filter. Eq. (32) can be written as:

/Q\ AU2 1+ ()2 ()4+ ()6 . (33)

Eq. (33) is exactly equivalent to (32) for all values ofU. A good enough approximation can be obtained bystopping at the sixth power:

A

1 + ()+ ()+( )

A

[1 +()2[i +

Eq. (34) can be realized by isolated cascading of asingle tuned circuit having a Q factor of wo/2A and apair of coupled tuned circuits having Q factors ofwo/x/2A respectively.The constant K£ is calculated for different types of

filters and the results are given in Table I.

TABLE I

Filter Type KgSingle Tuned Circuit GoDouble Tuned Circuit (isolated) .77Triple Tuned Circuit (isolated) .71Tukey's Rectangular Response Filter2'6 .68Filter of Equation 34 .68Optimum Filter .66

* (34)

Eq. (28) and the fact that (5U is entirely arbitrary forvarious values of Q/A leads to the following equation:

1.6U3At

+L

K, f1 I'0. (29)

This gives:

u( )=O (30)

or

U2 ( 0 ) = 5 K1 1 K /(-2 (31)\A/ 4 I 4 J1 \A/

Thus the response curve of the optimum filter at variousvalues of Q must be in accord with either (30) or (31).A simple but tedious calculation shows that among thevarious possibilities, the one which gives the lowestvalue of Kg is:

EFFECT OF VARYING FREQUENCYLet the response of the filter to unit pulse at time r

be denoted by F(t-r). The relation between the in-stantaneous input signal es and output signal eo is:

eo(t) = f ei(r)F(t - r)dr. (35)-00

For filters with a bandwidth A much smaller than themedium frequency coo, the function F(t-r) is a wavetrain of frequency c0 starting at t=r and lasts for aninterval of the order of 1/A. Hence the history ofei(r) which contributes to eo(t) in (35) is,limited to theinterval from t-0(1/A) to t, the notation 0(1/A) mean-ing a time interval of the order of 1/A. Generally theperiod of scanning is many times 0(1/A), and marginsof safety are left at both ends of the entire spectrum.

6 J. W. Tukey and R. W. Hamming, "Measuring Noise Color,"Bell Telephone Lab. Notes; 1949.

-t-~~~~~~~~~

19514 1281

0U2

9 Q 2 9AU 2U

A A A

PROCEEDINGS OF THE I-R-E

Thus the frequency of a single component of the inputsignal is:

CO = CO1 + qt (36)

eo(t) =\/ Ei

{e1- (wi+COo)+(t-71)j2/(4&12+q2/A2)cj)s41(t)

2 (l 14+-)

where q is the rate of change of frequency, and w, is aconstant. The phase angle is:

iColt + qt2 +± o, (37)

Oo is the constant of integration. The input signal canbe written as:

ei(t) = Es sin (wit + 14qt2 + P0). (38)

As the essential interest of this analysis lies in thegeneral relationship between the filter response to theallowable value of q which does not alter it appreciablyrather than the exact value of the output signal for cer-tain particular filters, some convenient approximateform of F(t) may be used. For the narrow-band filters,the envelope of its unit pulse-response function canbe fairly well approximated by the function e-Ai2(-TTj)2where ri is the time interval between the incident-pulseand peak-output response, and 1/A1 is the approximateduration of the wave train. The approximate expressionfor F(t) is:

F(t) = e-A12 ( t_Tl)2 sin cot. (39)

Substitute (38) and (39) into (35), it becomes

rteo(t) = Ei sin (coi7 + 1 qr2 + (o)

-00

e-A12(t-Tri) sin woo(t - -r)dr. (40)

As the exponential factor has negligible magnitude fornegative values of (t-r), to the same order of ap-proximation as (39), the upper limit of the aboveintegral can be extended to oo. The sine terms can bechanged into the difference of two cosine terms. Eq.(40) becomes:

eo(t) = e-A2(tTro)2 _0oCOS [(i1 + COO)7 + 2qr2 + o- Cot]dr

i e-Al 2(t_r_rl)22 _0

Cos [(WI - WO)7 + 'qT2 + (0 + coot]dr. (41)

By writing the cosine factor as the real part of an ex-ponential function, (41) can be readily integrated. Theresult is:

+e---{ (wl-wo)+q(t-7j)}2/(4Ai +q /Al2) COS +2(t). (42)In (42) q5l(t) and 'P2(t) are complicated expressions of tand will be omitted here as only the envelope of theoutput signal is of interest.The first term is entirely negligible since 0>>Aj.

Noting that co =ci+qt, the second term may be writtenas:

eo(t) -\/7r Ei e_(w-coo-qri) I(4A 2+q2/A12) COS02(t). (43)

2q(24~1142 Al4+4/

The sinusoidal response of the filter is obtained by tak-ing the limit of (43) with q approaching zero. Its halfbandwidth taken at the e-i point is 2Ai.With finite rate of change of frequency, (43) states:1. The peak response is reduced by a factor of

*1+ 4q2 ,-1/4+(h.b.w.)4

2. The bandwidth is increased by a factor of

{l+ 4q2 }1/2+(h.b.w.)4

3. The center frequency is shifted approximatelyby qrj;:q/Al=2q/h.b.w., since the time required toreach peak response is approximately 1/Al.The criterion for not altering appreciably the filter

performance is, therefore:

(h.b.w.)2q <<

2 (44)

In the inequality (44), the half bandwidth and the rateof change of frequency are expressed in radians persecond and radians per second per second respectively.

In the above derivations, q is treated as constant.However, the results are equally valid for varying ratesof frequency change as long as the variation is smallwithin a time interval of the order of the reciprocal ofhalf bandwidth.

ACKNOWLEDGMENTThe writer is indebted to Joseph Caldwell and W. J.

Pierson for their valuable discussions and help.

1282 Z. -1; o,v,