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The work presented in this paper builds on previous research done by the authors in detailing a novel procedure for obtaining a very fast measurement of the integral nonlinearity of an analog-to-digital converter (ADC). The core of the method is the parametric spectral estimation of the ADC output; the static characteristic is subsequently reconstructed as a sum of Chebyshev polynomials, in accordance with a previously developed procedure. The method allows one to test an ADC with sinusoids of any reasonable amplitude (even a slight overdrive is allowed), frequency (no synchronization is needed), and phase (which is digitally compensated). This approach is less accurate than the histogram test but incomparably faster (about 8000 samples are sufficient regardless of the ADC resolution).