Full report

Original page 1

., DEPARTMENT OF ELECTRICAL ENGINEERING SCHOOL OF ENGINEERING AND SCIENCE ' _ NEW YORK UNIVERSITY New York 53, New York

Original page 2

i ! TECHNICAL M_2_ORANDUM NO. 41 WIDE 3A_D PHASE DISTORTION EQUALIZATION • C. Capetanopoulos July 1964 Contract No. NAS 9-3908 i Prepared !. by i NN YORK UNIVNITY SCHOOL OF ENGINE_ING ANS SCIENCE DEPARTM_T OF ELECTRICAL ENGINEERING Laboratory for Electrosc!ence Research ! University Hei@hts I New York, New York 10453 i for Ii NATIONAL AERONAUTICS AND SPACE ADMINISTRATION GODDARD SPACE FLIGHT C_TER GRE_{BELT, MARYLAND ! ,i !

Original page 3

m ii ABSTRACT This report gives the design criteria for wide band phase realization, The design of passive lattice phase equalizers is introduced, These are all-pass passive networks that can correct the phase response of a particular system without affecting its amplitude response; they have the advantage of requiring no p_rer and have been proven to give reliable performance in many systems where phase equalization is required, The T_in part of this report consists of the design of a particula_ lattice equalizer _hose phase vs, frequency characteristic has the form of an '-S- curve, This particular phase characteristic can be used for phase correction in a wide variety of systems, One particularly effective method for almost any phase correction required is to cascade a.number of "theselattice networks designed for a socalled "staggered" arrangement,

Original page 4

TABLE OF CONTE_IS INTRODUCTION DESIGN PROCEDURE TYPICAL DESIGN CONCLUSIONS ill P_ge _ 2 7 8

Original page 5

! WIDE BAND PHASE DISTORTION EQUALIZATION INTRODUCTION The departure of the phase vs. frequency curve from a straight line for a tape recorder accounts for the distortion in a signal t4at is due to the different time delays of the different frequency components of the signal. d@ Since the time delay Td is defined as Xd = d_ ' it is evident that unless @ = k_ + c there will be time delay distortion. It is experimentally observed that the time delay is minimum for a certain band of frequencies and increases for frequencie_ outside this band. Thus it is typical to represent the time delay vs. frequency curve by a parabola centered at a frequency a_. Then the phase vs. frequency curve has the form of y = AXs + B with its point of inflection displaced to the right by _ and upwards so that a tangent through this point will cross the ordinate at the zero or 2n_ point, as shown in Figure 1. To compensate for the distortion of the signal arising from such a phase characteristic, a network should be designed which would have complementary phase vs. frequency characteristics (i.e., and-Scurve) that is, it would introduce little delay at low and high frequencies sad considerable delay in the center ud of frequencies. A" typical phase vs. frequency curve is shown in Figure l the minimum delay is assumed to occur at lO0 kc. and the phase variation is over _, radlans to assure appreciable time delay. [

Original page 6

DESIGN PROCEDURE Tae basic network of the phase distortion equalizer will be a lattice net_rork. This network has the advantage over all other passive t_o-ports of offering th_ designer a greater versatility in his specifications. Consider the followi.ug lattice network: ZA o --'I za ; o _ The design equations for the symmetrical lattice are the following: The open and short circuited impedances are z_zB i_ L. The characteristic (or image) impedance is The propagation constant is given by e7 = e_ej_= 1 +_ZA/ZB - ZA i I ZA/ ; ,:2t=- --- ..................................... , ,.,,,,,i , i iiii iii ii i ii I Lt h L . ii i i l ii ii iii i i

Original page 7

It is desired that the lattice network be at.all-pass structure so that the amplitude characteristic viii not need any further compensation. In order that the symmetrical lattice be an all-pass structure it is necessary that ZA and ZB be pure reactances of opposite sign at all frequencies. Under these circumstances eaeJ_ = i + JX. and thus _ = 0 i - jx Then the characteristic impedance is purely real The simplest all-pass lattice l_s the form ZA:L ZB--C For this network it can be shown that the phase angle varies wlth frequency as ¢ = 2 t_n"_ o The slope of the phase vs. frequency curv which represents the time is a monotonically der:reasing delay (3 d - de d'J function of the freque_cy, thus such a network cannot improve the low frequency phase response. Consider the following symmetrical lattice network. P

Original page 8

° [ IY CI ZA= J_C_ = l % ! J_ i - L% c zA = l-L- =. _c, ,io.rm. Tm order that the lattice st_ct_e be all-pass it is nece.ary that the ratio ZA/ZB be negative for all ,. i'nlsis possible _.tL denominator of the expression is a perfect square. _hus L:Ca = I%Ca = b Condition for All-Pass

Original page 9

Then the phase angle ¢ = 2 tan-I j Ba__L_C becomes ' = _ tan-' [I _ L_C "] If we let a _ Iuc ¢ = 2 tan-Z 1 s In order to improve the low frequency phase response it is desired that the phase vs. frequency response of the equalizing network have a region where it is concave upwards, this in turn implies that the curge must have a point of inflection. So a relationship between a end b must be obtained in order that the curve have a point of inflection. The slope of the phase curve is given in The point of infle=tion is given by the condition as* - 'bSe+ bS The solution + as - 3b = 0 (i) ,_ = -b.41 - 4b(aS-,_b) s

Original page 10

6 In order that o be 8 real frequency it is necessary that -b If b >> 1 the condition reduces aS< "b - b(a-3b>)0 to 4 UDder this condition there will be one and only one point of inflection in the phase curve. Equation (i) ms be solved for b for two different values of a. hus: : b=--- (for a_ = b) b - Z.I (for a"_ << b) • he design equations for the network are as fo]lows: 1 [where _ is the desired frequency at whic the point of inflection Occurs ] (b) ZC= 0 =./ (c) L.c, < -- ) It can be seen that the designer has two degrees of freedom, ° He can arbitrarily specify the character£stlc resistance and the frequency a which the point of inflection will occur. Values of the _ =,=

Original page 11

? different elements must be chosen such that condition c is satisfied. The phase a_le is given by = 2 tan-_ TYPICAL DESIGN -L_C, 2 Let it be required to design a compensating nebwork having the characteristic :of Figure 2 (II), with a characteristic resistance of i00 _ aa a center (inflection) frequency of i00 kc., and a phase varlation over 4 raalans, L | Two identical stages will be used to btain the phase variation of 4, radians. The conditions to be satisfied are: 1 C,, = i0(2.) s X l ° 2 z,,cl (2.)" x z ° The elements are _ _ lO 4 C,, 2 (z.)a x zo_° 1 x 10-3K7 ; O_: 1 ,,,x 10-7 Fa ; h : Jzo (2.) Jlo (2._ :-xzO "'m ; _: Jzo xzo"s 2. Z.

Original page 12

h 0 " - CONCLUSIONS 8 /. 0 i In the previous example a single phase equslizl_ network was designed to compensate the distortion arising from a parabolic time k dealy vs. frequency curve. However, it is often colnon that the time delay vs. frequency curve instead of being a parabola is essentially flat over a range of frequencies. Undr- these conditions _" single network as the one described will not be atisfactory. A considerable improvement should i result if 3 networks having phase characteristics of Figure 2 (Ill) I g with identical characteristic impedances but center (inflection) frequencies properly chosen to give a considerable and approximately constant time delay over the center band of frequencies as shown. If r it is desired to have a continuous variation over the shap of the h frequency curve, this can be done by varying the inductance _, but the other elements must be properly chosen and mechanically linked to '.' this inductance. J f

Original page 13

Phase , I ; , I I r q_e lattice network designed can be used as a i_sic structure to equalize a wide variety of time delay vs. frequency curves. Consider for example that the phase vs. frequency recorder has the form shown below. / / •/ 3./- i / / # / t , / i/ __,.... i curve for a particular tape / _ .....j.;._L i i I T

Original page 14

I L -F This phase variation mlgh_ be explained in terms of simple RC network p equivalents representing the record and playback systems. This phase curve indicates that there is a considerable time delay over the center of the frequency range and very little delay outside the band. Phase equalization can be rovidea by colmtructing two lattice networks in cascade ha'ring identical image impedances, with their inflection frequencies chosen such as to provide a considerable time delay over the low and high frequency band. The inflection frequency for the first lattice shoul be chosen to be a little less than half the .nflection frequency of the phase curve. For the seco lattice its inflection ifrequency should be chosen to be approximately twice that of the phase curve. The slopes could then be adjusted for best equalization. I f l m w M u

Original page 15

.i tn 0u_ 8 0u3 0 u3 • , \ \ 0 0 , , , \ " " t 0. • \ _ /L\ , \ o W -- \ o I..- W / °" / ,, xx _ _ _ _ o o

Original page 16