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a = [(1 + a. cos cor)2 + (a sin cor)2]* (8) « 1 + a cos cor (9)
+ tan'1
a cos cor
«* coT" -\ a sin cor
(10)
(ID
The approximate equations (9) and (11) indicate that the description of the echo in the overall transmission characteristics consists of a ripple or scallop which is cosinusoidal for the amplitude and sinusoidal for the phase. While the exact equations (8) and (10) do not indicate precisely the same shape of ripple, its general features are little different. The approximate equations indicate that the ripple amplitude, in nepers or radians, is equal to the relative echo amplitude a.
The wavelength of the ripples, along the radian frequency scale, is the change in co which causes a change of 2ir in the argument cor. Hence
or
ACO = 27T/T
A/ = l/i
(12)
(13)
In particular call A/ = /„ = co0/27r when it extends from zero frequency to the effective upper video cutoff. One half-cycle of frequency /0, or re = l/(2/0) = TT/WO, then defines one picture element. Any echo delay r can be measured in picture elements re, as :
T/re = r/ (ir/coo) = co0r/7r (14)
The envelope delay, d<p/dw, from equation (11) is:
d<p/du
ar cos COQT
(15)
The amplitude of the ripples in the envelope delay is ar. This is proportional to the echo amplitude a, and inversely proportional to the ripple wavelength A/ or 1/T.
The phase delay c^/co, is :
If to the signal of equation (5) there is added a second echo of the same amplitude as the first, but leading the original signal by time r, that equation becomes:
v = cos co (/ T) + a cos co (/ T r) + a cos co (/ — r -| T)
(17)
= (1 + 2a cos cor) cos co (/ — T)
(18)
= T + (a/co) sin cor
(16)
Thus the pair of echoes is described by transmission characteristics having a ripple only in the amplitude, and no distortion whatever in the phase.
If the second echo is given an amplitude which is the negative of the first, equation (5) becomes:
v = cos co (t — T) + a cos co (/ — T — T) — a cos co (/ — T -\ r)
(19)
= cos co(/ — T) + 2a sin cor sin co(/ — T)
(20)
With this pair of echoes the ripples in the transmission characteristics add in the tangential direction, and cancel in the radial direction (in the complex plot). Thus first order ripples appear in the phase, but cancel in the amplitude. When the echoes are small enough, therefore, they are described by a ripple in the phase and only infinitesimal distortion in the amplitude. When the echoes are large there is a second-order ripple in the amplitude which is not cancelled. It requires more than a pair of echoes to correspond to a characteristic which shows no distortion whatever in the amplitude.
References
1. D. K. Gannett and E. I. Green, "Wire transmission system for television," Trans. AIEE, 46: 946-953, June 1927.
2. F. W. Reynolds, "A new telephotograph system," Bell Sys. Tech. 7., 15: 549-574, Oct. 1936.
3. H. A. Wheeler, "The interpretation of amplitude and phase distortion in terms of paired echoes," Proc. IRE, 27: 359-385, June 1939. This reference
Pierre Mertz: Television Transmission Echoes
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