An algorithm estimates the Doppler frequency shift in a received radio signal with square-wave-pulse binary differential-phase-shift-keyed (DPSK) modulation. [With little modification, the algorithm is also extensible to *M*-ary (where *M* is an integer > 2) DPSK modulation with pulse shapes other than square.] The algorithm was developed especially for use in processing a received K- or K_{a}-band signal in a ground mobile receiver in a ground-mobile/satellite communication system. There is a need for estimation of, and compensation for, large Doppler frequency shifts in such a receiver. The algorithm implements a feedforward (open-loop) estimation scheme, providing the needed Doppler estimate prior to detection and subsequent processing (see figure) for extraction of data symbols from the modulation.

Older closed-loop schemes for estimating and compensating for Doppler shifts do not work at Doppler shifts greater than fractions of symbol rates, nor do they work in the presence of the deep fading that often occurs in mobile operation. An older open-loop scheme is also limited to small Doppler shifts and depends on symbol synchronization. The present algorithm does not depend on symbol synchronization and performs well even at Doppler shifts well beyond the symbol rate. Furthermore, because it is a feedforward rather than a feedback algorithm, it offers at least the potential for better performance in the presence of fading.

Prior to processing via this algorithm, the input signal is mixed to baseband, then passed through a low-pass filter of cutoff frequency *B*/2 to reduce out-of-band noise. The in-phase (I) and quadrature (Q) components of the resulting input signal *r*(*t*) can be represented in complex form as

*r*(*t*) = *s*(*t*)exp(*-* *jω*_{0}*t*) + *n*(*t*),

where *t* is time; *s*(*t*) is the square-wave-pulse-shaped binary data-modulation signal with symbol period *T*; *n*(*t*) is the low-pass-filtered version of the additive white Gaussian noise, with independent I and Q components, that is received along with the signal; and *ω*_{0}denotes the Doppler-frequency shift, which one seeks to estimate, and which is assumed to be constant during the time needed to make the estimate.

In this algorithm, *r*(*t*) is split into two paths, on one of which it is delayed by an interval *τ*, where 0 __<__ *τ *< *T*. The complex conjugate of the delayed signal *r*(*t**-* *τ*) is mixed with the undelayed signal *r*(*t*) to obtain *z*_{0}(*t*). The part of the algorithm described thus far can be characterized as implementing a *τ*-delay differential detector. It can be shown that provided the noise is negligible, one can estimate *ω*_{0}*τ* as minus the arctangent of the ratio between the dc (zero-frequency) Q and I components of the output *z*_{0}(*t*) of this detector. To isolate these zero-frequency I and Q components, it is necessary to filter out frequency components at the fundamental and harmonics of the symbol frequency 1/*T*; this is accomplished by low-pass-filtering *z*_{0}(*t*) to obtain the complex dc signal *z*(*t*). The estimate *ω̂ _{0}* of the Doppler shift is then computed from

*ω̂ _{0} = *(-1/τ) arctan [ z

_{Q}(

*t*)/z

_{I}(

*t*) ],

where *z*_{Q}(*t*) and *z*_{I}(*t*) denote the Q and I components of *z*(*t*). The maximum size of the Doppler shift that can be tracked without incurring the integer-multiple-of-2*π *phase ambiguity is given by

*|ω̂ _{0}|max = π/τ*

*This work was done by Thomas C. Jedrey, Edgar H. Satorius, and Martin J. Agan of Caltech for *NASA's Jet Propulsion Laboratory*. For further information, access the Technical Support Package (TSP) **free on-line at www.techbriefs.com** under the Electronic Systems category,*or circle no. 150* on the TSP Order card in this issue to receive a copy by mail ($5 charge).*

*NPO-19812*

##### NASA Tech Briefs Magazine

This article first appeared in the February, 1998 issue of *NASA Tech Briefs* Magazine.

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