A proposed digital carrier-signal-tracking loop in a radio transponder could be programmed to operate in either a perfect-integration or an imperfect-integration mode. Although originally intended for use in a transponder aboard a spacecraft at a great distance from the Earth, the proposed loop might also be advantageously incorporated into terrestrial communication systems in which it is necessary to track the phases of received carrier signals.

Either imperfect or perfect integration can be advantageous or disadvantageous, depending on state of signal reception. Among specialists in the design of carrier-signal-tracking loops, it is well known that as long as a carrier signal is present, the tracking performance of a loop that contains a perfect integrator is better than that of a loop that contains an imperfect integrator. For example, when the frequency of the received carrier signal is offset from the best-lock frequency by an amount d*f*, a loop that contains a perfect integrator exhibits zero phase error, whereas a loop that contains an imperfect integrator exhibits a phase error equal asdf to 2π d

*Δ*/a

*K*, where a

*K*is the loop gain. On the other hand, when a loop idles (that is, when the input to the loop consists solely of noise), then for a given loop bandwidth, the best-lock frequency of the loop drifts less if the integration is imperfect than it does if the integration is perfect.

The proposed loop design would make it possible to choose whichever integration mode — perfect or imperfect — is currently more advantageous. The figure is a block diagram of a loop filter that can implement either mode. The transfer function of the loop can be given by *A*_{1}*z*^{-1} + *A*_{2}/(*z* – *A*_{3}), where *A*_{1}, *A*_{2}, and *A*_{3} are arbitrary parameters and *z*is the argument of the *z* transform (*z* = *e ^{Ts}*, where

*T*is the sample period of the digital circuitry and

*s*is the complex-frequency variable of the Laplace transform). The transfer function is that of a perfect or imperfect integrator, depending on the choice of

*A*

_{1},

*A*

_{2}, and

*A*

_{3}.

To obtain a perfect integrator, one must choose

*A*_{1} = *K*_{1},

*A*_{2} = *K*_{2}*T*_{U}, and

*A*_{3} = 1,

where *K*_{1} and *K*_{2} are parameters that determine the loop performance and TU is the sample period at the output of the loop error accumulator.

To obtain an imperfect integrator, one must choose

*A*_{1} *K*(*T*_{U} – τ _{2})/(*T*_{U} – τ_{1}),

*A*_{2} = *K*{(τ_{2}/τ_{1})– [(*T*_{U} – τ_{2})/(*T*_{U} – τ_{1})]}, and

*A*_{3} = 1– (*T*_{U}/τ_{1}),

where *K* is the strong-signal loop gain and τ_{1} and τ_{2} are the loop time-constant parameters.

It is not necessary to select the parameters *A*_{1} and *A*_{2} with high precision: it suffices to set these parameters within about 1 percent of the values given in the equations above. However, the performance of the loop is quite sensitive to the value of *A*_{3}: For an imperfect integrator, *A*_{3} must be set at a value that is less than 1 by a small, precise amount.

*This work was done by Jeff Berner, James M. Layland, and Peter Kinman of Caltech for*NASA's Jet Propulsion Laboratory*. For further information, access the Technical Support Package (TSP)*free on-line at www.nasatech.com/tsp under the Electronics & Computers category.

*NPO-20845*