It's supposed that computing a spectrogram for a signal is faster using wavelets. However, wavelets need to be applied to the signal for every time step and frequency, giving the implementation a time complexity of $O(t*f)$; however, using the STFT process, we only need to apply the fourier transform over each time step (generating all frequencies composing the windowed signal) - $O(t)$.
Is wavelet convolution over a signal for a specific frequency significantly faster than a fast fourier transform of a time step (window) in the signal?