1. An apparatus comprising: an analog-to-digital converter (ADC) that is configured to generate a first digital signal from an analog signal; and a controller that is coupled to the ADC so as to provide a sample signal to the ADC and to receive the first digital signal from the ADC, wherein the frequency of the sample signal is less than a Nyquist frequency for the analog signal, and wherein the controller generates a second digital signal from the first digital signal using a dictionary that compensates for mismatch from a sparsifying basis, and wherein the dictionary is constructed by iteratively isolating each of a plurality of spectral terms and determining its offset, and wherein the second digital signal is approximately equal to an analog-to-digital conversion of the analog signal at the Nyquist frequency.