  A better understanding of the noise causing qubit decoherence is crucial for
improving qubit performance. The noise spectrum affecting the qubit may be
extracted by measuring dephasing under the application of pulse sequences but
requires accurate qubit control and sufficiently long relaxation times, which
are not always available. Here, we describe an alternative method to extract
the spectrum from correlations of single-shot measurement outcomes of
successive free induction decays. This method only requires qubit
initialization and readout with a moderate fidelity and also allows independent
tuning of both the overall sensitivity and the frequency region over which it
is sensitive. Thus, it is possible to maintain a good detection contrast over a
very wide frequency range. We discuss using our method for measuring both $1/f$
noise and the fluctuation spectrum of the nuclear bath of GaAs spin qubits.
