1960 IRE TRANSACTIONS ON SPACE ELECTRONICS AND TELEMETRY 85

The Effect of Different Types of Video Filters onPDM\-FM and PCM-FM Radio Telemetry*

M. H. NICHOLSt AND A. T. BUBLITZt MEMBER, IRE

Summary-The effects of two limiting types of video filters of an FM demodulator,2 the portion of the noise power(namely the ideal low-pass vertical cutoff herein called the ILPF spectrum intercepted by the filter will be proportionaland the Gaussian) on the performance of PDM-FM are compared. to the cube of the bandwidth of the filter; i.e., the rmsIt is found that minimum allowable pulse height controls the mini-mum allowed bandwidth of the Gaussian filter and if crosstalk video noise voltage at the output of the filter will besuppression of 50 to 60 db is required, crosstalk controls the mini- proportional to the three-halves power of the filter band-mum allowed bandwidth of the ILPF. In PCM-FM using the width. For any low-pass filter other than the ILPF, asampling video detector, the critical effect is the overlapping of definition of ban-dwidth F, has to be chosen-such aspulses at sampling time; using the integrating detector, the criticaleffect is the difference in area under the pulse with adjacent pulses the 3-db poit of the amplitude response curve whichnot present and the area with no pulse present but the two adjacent is used in this paper. The rms output signal-to-noise ratiopulses present. In general, the results of the theory agree well with of a PDM-FM system using a particular low-pass filterexperimental results of Aeronutronic [8] and aid in their interpreta- is proportional to the ratio of the slope of the step re-tion. sponse of the video filter at the slicing level to the rms

PART I-PDM-FM noise voltage at the output of the filter (which includesthe second and third of the above three criteria for low-

I. INTRODUCTION pass filters).UXGLOW [1] has compared analytical results of The response of an ideal vertical cutoff low-pass filter

PDM-FM telemetry based on two different video has an initial overshoot of 9 per cent to a unit step inputfilter characteristics. The results compared by with the oscillations dying off approximately inversely

Uglow were taken from Feldman and Bennett's analysis [2] proportional to time [3] and a slope of 2.0 F, at the slicingwith a Gaussian video filter (the model used by Feldman level (slicing level taken equal to half height) where F,and Bennett for PPM-FM applies equally well to PDM- is the cutoff frequency and the equivalent noise band-FM) and from Nichols and Rauch [3] with an ideal width is FC. A Gaussian filter has no overshoot in itssharp cutoff low-pass filter hereafter abbreviated as ILPF. response to a unit step input; it has an effective noisePart I of this paper corrects the method of optimization bandwidth of 1.34 F, with an FM video noise spectrum,used in Chapter 11 of Nichols and Rauch, and extends where F, is the 3-db frequency of the Gaussian filterthe analysis of Feldman and Bennett. and the slope of the step response is 3.0 F, at the slicing

level.3 The ratio of the maximum slope of the step re-II. COMPARISON OF FILTERS sponse to the three-halves power of the equivalent noise

Both the Gaussian and the ILPF characteristics are bandwidth of an ILPF (used in a PDM-FM\ system)impossible to realize. In practice, the ILPF is usually is 2.0 Fc12, and for the Gaussian filter it is 1.96 Fc12 _approximated by a maximally flat gain or Butterworth 2.0 Fc 1J2.3 As shown later in a numerical illustration,type of filter characteristic; the Gaussian low-pass filter the use of the Gaussian filter in practice eliminates theis most often approximated by a maximally flat phase problem of crosstalk in PDM-FM.or Thomson type of filter characteristic.! From the pointof view of using such a filter at the output of the de- OUTPUT SIGNAL T ERmodulator in a PDM-FM system, three characteristics OF A PDM-FM SYSTEMof the filter performance are of interest: 1) the rate of The rms output signal-to-noise ratio, S/N, of a PDM-decay of the transient in the response to a step input, FM system when well above threshold is approximately:4which determines the amount of crosstalk betweeinadjacelnt pulses; 2) the rate of rise (slope) of the response 2 In this paper, we have used the well-known triangular videoto a step input at the slicing level; and 3) the equivalent noise voltage spectrum for FM; for example, see Nichols and Rauch[3], p. 57, Fig. 4.6.1. The calculations of Stumpers and Middletonnoise bandwidth of the filter (using as basis of comparison have indicated that, with sinusoidal modulation and heavv limiting,an ILPF). When a low-pass filter is used at the output this is a good approximation for IF signal-to-noise ratio of 10 db

or better; see, for example, (33) of Stumpers [5] and Fig. 11 ofMiddleton [9]. In Middleton [9], a Gaussian IF filter characteristic

* Received by the PGSET, June 3, 1960. This work< was supported is used which causes some departure from the linear characteristicby USAF Contract No. AF 33(616)-5796, Telemetry and Range which results when an ideal vertical cutoff IF characteristic isSafety Section, Control Equipment Branch, Wright Air Dev. assumed. Aeronutronic [8] has reported a significant departure fromDivision, ARDC. the triangular noise spectrum w-hen complex modulation is applied;

t College of Engineering, University of Michigan, Ann Arbor, see, for example, Fig. I-A-13 [8].MiCh. 3See Appendix I.l'See, for example, references listed in Henderson and Kautz [4]. 4See, for example, Nichols and Rauch, reference 3, p. 60.

86 IRE TRANSACTIONS ON SPACE ELECTRONICS AND TELEMETRY JuneS (rms full modulation time) (pulse slope at slicing level) Fp 112 St = bkIB'12 (2b)N rms fluctuation noise voltage in video 2fm 1) where S, is the rms value of the carrier at threshold.where The constant p can be estimated from the calculationsF = number of samples per channel per second of Salinger [6] on the response of an ideal vertical cutofff-cutoff frequency of the low-pass output filter in band-pass filter to a step change in frequency. In a

each channel PDM-FM system, the carrier frequency f, is modulatedby shifting it from fo - fD to f, + fD which, accordingpulse slope at slicing leveld 4 V3k5SFJtD to the results of Salinger, gives a pulse at the output

rms fluctuation noise voltage in video kjB3N/2 of the discriminator approximately of the form Si(rrB0t)where plus a damped ringing oscillation equal in frequency toS = rms value of signal in IF in volts (referred to 2fD, The apparent damping as well as the magnitude

of this ringing is dependent on the value of the ratiooutput)1 r . Box 2fo3 = p; and as the value of p approaches 1.0, thek.5= a constant defined so that the slope of the unit . = . . a

step responise of the filter is 2 kF at the slicing magnitude of the ringing approaches a maximum and thelevel 5 a damping approaches zero. For deviation ratios, fD F, = D,level3-db point of filter in cps greater than 0.5, this ringing oscillation will be attenuated

F., = poinof fieinecps to an amount depending on the actual characteristics.rms ct anoise votaerperrooun band of the IF band-pass and of the video low-pass filter.w=equidthentnoiFchannl( edt tovideoufiltpu forFM Curves giving the time response of the carrier frequency

RN= equivalent noise bcnandwidth ofvidefilterfo to a sudden shift are given in Salinger's paper and also= k6F, where k6 iS a constant

tD = peak frequency swing of FM link when fully in Appendix 13 of Nichols and Rauch [3]. From thesemodulated curves, it appears as though p = 1.4 would be a con-

1 \ servative choice. If the deviation ratio is greater thanrms full modulation time6 = 3sr y - 0.5 and the action of the video low-pass filter is con-

where sidered, then the value of p can be allowed to approachcloser to i.O.'

n = number of channelsT = guard time = spacing between pulses to reduce IV. OUTPUT SIGNAL-TO-NOISE RATIO

crosstalk plus minimum pulse length. OF A PDM-FM SYSTEM WITHThe factor (F/2fm)"2 in (1) takes into account the A GAUSSIAN VIDEO FILTER

frequency components of the noise lying between F12 The guard time r depends on the characteristic of theand fm taken out by the individual channel low-pass video filter and must be chosen to satisfy crosstalkoutput filter which is assumed to cut off sharply at tf. criteria and the minimum allowable pulse height. TheBy substitution into (1), the rms individual channel minimum allowable pulse height determines the mini-

output signal-to-noise ration- at carrier threshold (SIN), mum pulse length. Since the step response of a Gaussianis obtained filter has no overshoot,9 it turns out that r is controlled

19 3/2 112 by the minimum pulse height requirement. Fig. 1 illus--= -Fc($ -tF p (2a) trates schematically the pulse height effect for a parti-

where r2rBo = IF channel noise bandwidth h/2p = a constant = Boj/2ftD

7^773/2k7 = 5/k5k6 Fig. 1-Schematic showing reduction of pulse height.The constant 6 is the FMI carrier threshold factor given

by7 cular spacing with a Gaussian filter where X r1 + T2.If T1 = -2, then h, _ h2. The values of h1 and h2 determine

5 Slicing level assumed to occur at half steady-state pulse height. the leeway for noise discrimination at minimum modula-6 The factor 1/23/2 appears since this derivation assumes

sinusoidal modulation; thus the rms value is 1/23'2 of the peak-to- tion and full modulation, respectively. If the instantaneouspeak swing. video noise voltage exceeds h, the pulse plus noise will7 This assumes for purposes of analysis that a well-defined vthreshold exists which is constant for all deviation ratios. Some exceed the slicing level for the long pulse and if theidea of the validity of this assumption can be obtained by reference iLLstantaneous video noise voltage is less than -h,, theto the results published by Stumpers [51 and Middleton [10]. Ac-cording to Stumpers (Figs. 3 and 8), the assumption is approximately short pulse plus noise will nlot reach the slicing level. Thusjustified for IF bandwidth to video bandwtidth ratios of 4 or more h2 and h, determine the video threshold for minimumfor the ideal vertical cutoff IF characteristic and for two or morefor a Gaussian IF characteristic. The latter agrees within about2 db with Middleton [101, Fig. 8. Stumper's Fig. 8 indicates a a of 8N=ichols and Rauch [31, p. 65.about 10 db and Middleton's Fig. 8 indicates a S varying from about 9In practice, it is possible to approximate very closely the6 db for very low deviation ratios to about 8 db for the higher Gaussian low-pass filter. For example, a tenth-order low-pass Besseldeviation ratios. In this paper, we utse a a of 9 db for numerical filter has a step function response very close to that of a Gaussianillulstrations. filter with only 0.12 per cent overshoot [41.

1960 Nichols and Bublitz: The Effect of Video Filters on PDM-FM and PCM-FM Radio Telemetry 87and maximum modulation.10 Also, Fig. I shows that illustration, the maximum relative cross talk is approxi-the slopes at the slicing levels are reduced, which in- mately 106, which is certainly adequate! It is also neces-creases the output noise at full and minimum modulation sary to check to see that the video threshold is satisfiedwhen above threshold.1" These considerations determine when the carrier is above threshold, i.e., for (6b) to bethe minimum permissible values of T1 and r,. The quanti- valid, St > S, where S, is the rms value of the carrierties T1 and 1-2 so determined can be written as required for carrier threshold, and S. is the rms value

k a k of the carrier required for video threshold. Using theTj = F, and T2 = Fe conditions for video threshold from Nichols and Rauch [3],

page 66, and the expression for carrier threshold fromwhere k, and k4 are constants. Eq. (2a) then becomes for (2b) anid substituting for Bo from (6a) and F, from (4),this case, using the relation 2PfD = B,, the ratio of the two threshold

s 3 F, Bo 3/2~~~ 1/2 values of S becomes:k7 k4 (32

__ __

(N): di2nF )(F,) p (2f,l ( S, 11.3k5(k3 + k4)Following the procedure of Feldman and Bennett [2], St _2f,,_12 S)(S/N), can be maximized by varying F,, holding the F NItmaxother variables constant; i.e., for constant IF ban-dwidth Using the above numerical illustratioIn, 7, = 1.5 andB, and constant nF. This gives: k, + k, = 0.61; also (2fm F)1/2 is the order of unity and

F we choose (S/N)tmax = 100. With these values,c = 3(k3 k+) (4)nFS,

Substitution of (4) inlto (3) gives: St 10/Sh t2nF 1/2 ,6 F 1/2 Thus for reasonable values of the variables, the video( = k )/2 F7/N tmax 3F p \2f (5) threshold is satisfied for a lower signal strength thanl

the carrier threshold and the results are valid in thisThe required IF bandwidth Bo to give (S/N) Imax at respect.carrier threshold is obtained from (5): Since Fig. 1 and the calculations relative to minimum

Bo _ 1.22 (p\2f_\1J2/F \1/2/5\ 12/3 allowable pulse height and crosstalk were based on thenF L k7 t ~AJFm) ].nF} Vl\TJtma (6a) step response of the Gaussian filter, it is necessary to

check to see that the assumption of step-shaped pulsesior purposes of illustration, we will assume Tr f2, inito the video filter is justified. Using the relationswvhich makes h1 _ h2, and that the threshold require- F, = 1.8 nF and k7 _ 1.0, (6a) can be writtenment is that h, - h2 = h/4. The total guard time isT = r, + Tr. Writing the transfer function of the Gaussian Bo _ 1 22Ipu{2fn l(1/2(iS 12/3 (61c)filter as exp (-c2f2 F2) where F, = the frequency for F, 1.8 L \6I\ F / kNltmax j3 db attenuation and c; = 0.35, the guard time r for To evaluate this, we choose p = 1.4, 6 = 2 V2, 21m- F,this case is computed by use of error function tables a .T g B m1to be 0.61/FCseconlds. From (4), F,/InF =1.8, and ad(/fta 0.Tl le oF1 spfrom Se0tion I,sco 1.02 so (5) becomes viously noted, the response of the discriminator is ap-fro Section'IIc- - 1.0, SO (5) becomes proximately Si(rBot). Thus the spectrum into the Gaus-

s AO 3/2 ; F 1/2 sian video filter is essentially flat out to the frequency(N/ tmax - \nF/ p (2fm Bo/2 = 5F, where it cuts off sharply. At 5 F,, the attenua-tion of the Gaussian filter is approximately iO-4 or -80 db.It iS now necessary to check to see that crosstalk iS at tio ofteGusa.fle sapoiaey1-r-8 bIt is now ne rThis justifies the assumption of step-shaped (rectangular)a tolerable level under the modulatLion condition illus- t G v f 3

trated in Fig. 1. This is done by dividing the value ofthe "tail-off" of the long pulse at the time the trailing V. OUTPUT SIGNAL-To-NOISE RATIO OF Aedge of the short pulse crossed the slicing level by the PDM-FM SYSTEM WITH AN ILPFslope of the short pulse at that time. To obtain thecrosstalk relative to full modulation, we divide by the In the case of the Gaussian filter, the consideration offull modulation time, that is, by (1/nF - T). For this minimum allowable pulse height determines the guard

time r required. Since the envelope of the step function"0Assuming that the leading edges of the pulses are fixed, the response of an ILPF dies off approximately inversely

modulation sensitivity near full mo9dulation will decrease because with the time, it turns out that crosstalk reduction ratherof the presence of the leading edge of the next pulse. This effect thnmnmmplehih eemnstegadtmcan be calibrated out.thnmnmmplehihdee ieshegadie

"1It is conceivable that in some cases, the reduction of the slopeat the slicing level is of more concern than the reduction of thepulse height, so that a minimum allowable slope becomes thecriterion for minimum allowable pulse width. The general procedure 13 Since 2pfD = Bo, this gives a deviation ratio of about 3.6.is the same in either case. Fig. 8 of Middleton [10)] and Fig. 8 of Stumpers [5] both indicate a12 See Appendixes I and II. sharp carrier threshold for this case with a a of 8-10 db.

88 IRE TRANSACTIONS ON SPACE ELECTRONICS AND TELEMETRY Juner required. Following the procedure in Nichols and Rauch V,V/DEO THRESHOLD[3], page 245, the maximum relative crosstalk y is given by: St

0.023nF CARRIER THRESHOLD= 2 *l (7) \Fer(l - nFr) (

Solving for r and choosing the root which gives T -* 0as F. ---- O :_ _ __ _ _ _ __ _ _ _

[(:i 2 00231/22nF - L(_ ) - 0.02F21j (8) Fig. 2.

Following the same procedure as in the previous section, calculated above, the output signal-to-noise ratio for thethis value of r is substituted into (2a) and (SIN), maxi- Gaussian type of video filter is approximately twice thatmized against F,, holding nF and Bo constant, giving:14 for the ILPF.

e 2 .041 (9) VI. COMPARISON WITH EXPERIMENTnF ly7