Accurate Spectral Estimation Based on Measurements with a Distorted-timebase Digitizer

The timebase distortion present in an equivalent-time sampling oscilloscope introduces errors in the estimation of the values of the spectral components of a microwave signal when a classical discrete Fourier transform is used. In this article a method is developed to avoid these errors. The method is tested both in practice and with simulations. Two parts can be distinguished. At first, the timebase distortion is measured. This is done by digitizing a sinusoidal signal applied at the oscilloscope’s input, and by calculating the phase of the analytical signal of the digitized waveform. Other possible methods to measure the timebase distortion are discussed, and it is shown why the method used is the most appropriate for our specific application. The knowledge of the timebase distortion is then used to build a least-squares-error estimator for the values of the spectral components of a digitized microwave signal. An experimental verification is done, from which is concluded that the method effectively removes the spectral estimation errors due to a timebase distortion. Introduction If one wants to measure both amplitude and phase of the spectral components of a microwave signal, a spectrum analyzer can no longer be used. An equivalent-time broadband sampling oscilloscope with an appropriate triggering is a viable solution. A classical approach is to digitize the signal and to apply a discrete Fourier transform (DFT). Unfortunately the results of this method can be incorrect because of timebase deviations from ideality, which are typically present in equivalent-time sampling oscilloscopes. Two kinds of timebase errors can be distinguished. A first kind of error is the timing jitter. Several articles are available in literature which cover this subject [1] [2]. Much less available however are articles concerning the effects of the presence of a systematic timebase distortion in equivalent-time sampling oscilloscopes. In this paper a method is described to estimate the values of the spectral components of a signal with good accuracy, even with a significant timebase error present in the digitizer. Two parts can be distinguished. At first the timebase deviation from ideality is measured by digitizing a sinusoidal signal. The method is similar to the method mentioned in [3], and is based on the calculation of the instantaneous phase of the analytic signal [4] of the digitized waveform, which in fact corresponds to performing a digital phase demodulation. Since the digitized sinusoidal signal appears to be slightly phase modulated by the timebase distortion, the demodulated phase is a good measure for this distortion. The effect of additive noise is studied, and it is explained how windowing techniques [5] can increase accuracy. Three other methods to measure the timebase error are also discussed, together with a short description of their main characteristics. It is explained why the one actually used is suited for our specific application. The use of the method is illustrated by a simulation and an experiment. Once the timebase distortion is characterized, this information can be used to estimate the values of the spectral components in a microwave signal with good accuracy. A leastsquares-error approach is used to actually perform the estimation. The results of an experimental verification are reported in the last section. These results prove that the novel method proposed effectively removes the erroneous effects which a digitizer timebase distortion has on the estimation of the values of spectral components. Mathematical notations xyz : vector or matrix named xyz xyz[k] : component with index k of vector xyz (first index is 0) xyz[i,j] : component of matrix xyz on row i and column j xyzt : transpose of xyz : element by element multiplication applied to a and b f * g : circular convolution of f and g : discrete Fourier transform (DFT) of f : complex conjugate of f : vector with i-indexed value 1 and the other elements 0 : argument of complex number a (in rad) Timebase distortion measurement Mathematical theory of the “Analytical signal”-method In this section will theoretically be explained how an a priori unknown timebase distortion can be calculated out of a digitized sinusoidal signal. Because the method is based on the calculation of the analytical signal [4] associated with a digitized signal, we will call it the “Analytical signal”-method. Suppose that there are N sampling points and that the ideal sampling period, this is the sampling period of the digitizer if it would have no timebase error, equals Ts. We then define ωbase by Eq.1. (Eq. 1) If t denotes the vector of the actual sampling instants, then it is possible to define the a priori unknown timebase distortion, called dis, by the following equation: . (Eq. 2) Let ωcal be the angular frequency of the cosine waveform, called calibration signal, that will be used to measure the timebase distortion. ωcal will be chosen such that it equals an integer, called L, times ωbase: a b • F f ( ) f∗ δi φ a ( ) ωbase 2π TsN --------= dis k [ ] t k [ ] kTs – = . (Eq. 3) We then define the complex vector w and d by Eq.4 and Eq.5. (Eq. 4) (Eq. 5) For ease of notation we will assume that the calibration signal has an amplitude of 2. It can however easily be verified that the algorithm is insensitive to this amplitude. When this signal is applied at the input of the digitizer, the vector of sampled values, called s, will be given by Eq.6. (Eq. 6) If Eq.2, Eq.4 and Eq.5 are substituted in Eq.6 the following results: . (Eq. 7) Next the DFT of s is calculated, called f, this results in Eq.8. (Eq. 8) If we define i as being (L modulo N) and using Eq.1, Eq.3 and Eq.4, we can write Eq.8 as: . (Eq. 9) In practice, with L not exceedingly large, will have about all of his energy concentrated around DC. This can be explained by approximating d, defined by Eq.5, by the first two terms of its Taylor series: . (Eq. 10) The approximation of d[k] as a linear function of dis[k] will be valid if is much smaller than 1. In practice dis[k] represents the residual error of the timebase of an instrument. Looking at Eq.3, this means that Eq.10 will hold for all values of L smaller then a certain value LLIN. For these values of L, can be approximated as follows: ωcal Lωbase = w k [ ] e jωcalkTs = d k [ ] e jωcaldis k [ ] = s k [ ] e jωcalt k [ ] e jωcalt k [ ] – + =