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I'm trying to detect the presence of a sinusoid in 1/f noise conditions using the STFT and the DWT with Haar wavelets. I find this interesting phenomenon that I'm not able to explain (see plot at 1)

My sampling rate is 1200 Hz and the graphs show the one sided response (0-600 Hz). I am using 2 scales for the Wavelet Transform.

My question is, why in the FFT method can I clearly see the blips in about the correct frequency as my simulated data (180 Hz), while in the WT method, the increase in energy is spread visible across all three bands. Shouldn't I see a bright line only in the highpass segment of the second scale?

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The FFT is much better at detecting a sinusoid than a DWT. The FFT is approximating a periodic signal with a series of periodic signals. The coefficients of the FFT will be maximum in the frequency bins (could be a single bin with sufficient resolution) where the component of the FFT series best matches the periodic signal to be detected. The noise will distribute roughly evenly in all bins, so the signal to be detected will show up readily.

The DWT is representing the signal to be detected (in this case a periodic signal) with a set of non-periodic functions, so this requires lots of components (bands as you refer to them – but DWT really works with scale not frequency) to follow the periodic nature of the signal to be detected. The DWT is better suited to detect / analyze non-periodic signals.

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