The present invention relates to the display of electrical signals, and more particularly to an improved interpolation of sampled complex signals for increasing the accuracy of the displayed signal.
There are many demands for observing the envelope magnitude of a radio frequency (RF) band-limited signal in the time domain. Modern measurement instruments, such as spectrum analyzers as represented in
There are two common methods for computing the magnitude of interpolated complex signals using traditional techniques. However both methods have some difficulties. One method is to take the magnitude of the original sampled signal and then apply interpolation, as shown in
What is desired is an improved interpolation for complex signals that does not produce negative magnitudes and does not introduce excessive ripple in order to provide a more accurate representation of the input signal.
Accordingly the present invention provides an improved interpolation scheme for reconstructing a complex signal from sampled data by estimating a carrier phase for each sampled data time. The carrier phase is then used to compensate for frequency variation in the complex signal. The complex components of the sampled data are then interpolated separately with the interpolated results being used to produce an interpolated magnitude that is the reconstructed complex signal.
The objects, advantages and other novel features of the present invention are apparent from the following detailed description when read in conjunction with the appended claims and attached drawing.
Referring now to
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, θ^, that minimizes the sum of error squares, i.e.,
d/dθ^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, θ^, becomes
θ^=(HTH)−1HTy
At each sample time of the IQ baseband signal a phase sample is determined (see
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.
Number | Name | Date | Kind |
---|---|---|---|
6628707 | Rafie et al. | Sep 2003 | B2 |
7010062 | Joshi et al. | Mar 2006 | B2 |
7342980 | Nair | Mar 2008 | B2 |
20010046847 | Wakamatsu et al. | Nov 2001 | A1 |
20040120304 | Kloos et al. | Jun 2004 | A1 |
Number | Date | Country | |
---|---|---|---|
20080049879 A1 | Feb 2008 | US |