BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING
FIG. 1 is a block diagram view of a typical receiver for processing an RF signal for display.
FIG. 2 is a block diagram view of a prior art interpolation technique used in the typical receiver of FIG. 1.
FIG. 3 is a graphic view of a pulsed continuous wave (CW) signal indicating a problem with the interpolation technique of FIG. 2.
FIG. 4 is a block diagram view of another prior art interpolation technique used in the typical receiver of FIG. 1.
FIG. 5 is a graphic view of a pulsed CW signal indicating a problem with the interpolation technique of FIG. 3.
FIG. 6 is a graphic view illustrating the cause of ripple shown in FIG. 5.
FIG. 7 is a block diagram view of an improved interpolation technique according to the present invention.
FIG. 8 is a graphic view illustrating a pulsed CW signal produced according to the present invention that does not have the problems of the prior art.
DETAILED DESCRIPTION OF THE INVENTION
Referring now to FIG. 7 the improved interpolation technique of the present invention is shown, based upon the separate interpolation of the I and Q components of a sampled complex signal. The IQ baseband signal samples produced by a quadrature signal generator from an input sampled signal, as shown in FIG. 1, are input to a compensation module 12 and to a carrier phase estimation module 14. Also input to the carrier phase estimation module 14 is known information about the signal being analyzed in the form of signal model parameters, as described below. The carrier phase estimation module 14 estimates a carrier phase from the IQ baseband signal at each sample time using the signal model parameters. The estimated carrier phase is input to a correction module 16 for generating a correction equation for the IQ baseband signal. Output from the compensation module 12 are corrected I and Q sampled components which are then separately interpolated by respective interpolators 18, 20. The interpolated results are then input to a magnitude computation module 22 to produce the interpolated magnitude samples. The output from the compensation module 12 is corrected so that the sampled signal has no frequency variations. The resulting reconstructed signal is shown in FIG. 8 without any negative magnitudes and with virtually no ripple except at the leading and trailing edges of the CW pulse.
For a signal having a constant envelope, A, such as a CW pulse signal, the complex baseband representation of the signal s(t) may be expressed as:
s(t)=Aejφ(t)
The value of φ(t) is estimated from the sampled signal, s(k), where k is an integer, t=kts and ts is the sampling period. Let φ(k) be the estimated phase, so the signal phase may be corrected by a sample wise product
s(k)e−φ(k)
The corrected phase should be almost zero, and the magnitude is exactly the same as the original signal, A. Therefore interpolation independently in I and Q may be safely performed without ripple.
In a CW signal the phase, φ(t), may be modeled as
φ(k)=ak+b
where a is the frequency offset and b is the initial phase offset at k=0. Since for this application b is not significant, the frequency offset, a, may be estimated. A common method is the Least Square Estimation (LSE).
For a radar pulse Linear FM (LFM) is commonly used (“chirp” signal), where the phase, φ(t), may be modeled as
φ(k)=ak2+bk+c
and 2a is the frequency modulation.
The carrier phase estimation module 14 may use a least squares estimation (LSE), which only requires a signal model, uses simple computation and still provides a reasonable estimation. The principle is to find an estimator, θ̂, that minimizes the sum of error squares, i.e.,
d/dθ̂
i{Σ(error)2}=0
where θ1=a, θ2=b and θ3=c.
The signal model is a linear combination of parameters to be estimated that may be reduced in matrix form to
y=Hθ+e
where y represents the measured phase, e represents the error and Hθ represents the ideal signal φ. The estimator, θ̂, becomes
θ̂=(HTH)−1HTy
At each sample time of the IQ baseband signal a phase sample is determined (see FIG. 6). The LSE is applied to the phase samples to produce the phase estimator which is input to the correction module 16. For a constant frequency signal this is a fit-to-phase measurement—φ=at+b+e—resulting in a linear fit. A fit-to-frequency measurement takes the derivative of the phase over time—d/dt{φ}=a+e′—resulting in a constant value. For the LFM signal (varying frequency signal) the two models produce either a parabolic fit—φ=at2+bt+c+e—or a linear fit—f=d/dt{φ}=2at+b+e′.
Other estimator algorithms may be used in lieu of the LSE algorithm. For example another estimator is a Minimum Variance Unbiased Estimator (MVUE). Other classical estimators include Minimum Likelihood Estimator (MLE) among others, and Bayesian estimators include Minimum Mean Squared Error (MMSE), Maximum a Posteriori (MAP), etc.
Therefore the system model is provided to the estimation module 14 in the form of signal model information. If possible the Cramer-Rao Lower Bound (CRLB) is found to access accuracy limits. An estimation algorithm is used on as many raw data points as possible. Other estimators may be used, if possible, and the results for the one having the lowest variance is used.
Thus the present invention provides an improved interpolation for reconstructing sampled complex signals by estimating a carrier phase for the baseband signal from the sampled complex signal and known signal model parameters, and by using the estimated carrier phase to compensate for frequency variation in the sampled complex signal prior to interpolating the complex signal components separately.
Patent Application
20080049879
