This is especially true because the scales "visited" by the DWT are separated by factors of two, and are much less dense than the coverage you can get in the time-frequency plane with the FFT. So using the DWT to examine the time-scale plane isn't going to get you very far. This makes the DWT faster than the FFT, $O(N)$, but also destroys the translation-invariance. This occurs because at every stage of the DWT, the signal is decimated by two. And the comment that made above that the DWT is not translation-invariant is correct. The DWT, or discrete wavelet transform, computes only discrete scales, just as the FFT computes only discrete frequencies. If your signal $x(t)$ is a sine wave, the two transformations are the same. The same transformation in the time-frequency plane is $x(t) \rightarrow x(t-\Delta t) e^$, where $\Delta \omega$ is a shift in frequency. Signals at different places in the time scale plane are related by $x(t) \rightarrow x(\Delta s(t-\Delta t))$, where $\Delta s$ moves you up (or down) in scale and $\Delta t$ shifts you in time. The time-scale plane is not the same as the time-frequency plane, although it might be useful as well. This is very late, but maybe it's worth it anyway. The main point is that computation time is at least roughly similar for both, so I don't think you should worry about it when deciding which to use. (Technically it's a time-scale plane, not time-frequency, but I think they're the same for the complex Morlet wavelet?) FWT is a "critical sampling" of the plane, and produces the same amount of data as FFT, so it seems fair to compare them. I still don't understand how the FWT tiles the time-frequency plane, or if producing n outputs is sufficient to get the same kind of similarity measurement as an n-point circular cross-correlation using FFTs.Meyer wavelet takes 6 times as long as Daubechies to produce the same amount of data. The time varies depending on the wavelet used.They use the abbreviation DWT in their documentation. I don't even know if wavedec() would be considered a Fast Wavelet Transform. It's Python, and the two implementations could be very differently efficient. So the wavelet method might actually require less computation, depending on whether you can use the information you get out of it.Įmpirically, producing n outputs from n real inputs, the multi-level wavelet transform in PyWavelets becomes faster than NumPy's FFT when n is greater than about 4096. The Fast Fourier Transform takes $O(n\log n)$ operations, while the Fast Wavelet Transform takes $O(n)$. Second, I have only a minimal understanding of wavelets, but your assumption that wavelets require more computation might be wrong. Correlation vs coherence vs wavelet-based correlation are all different things, so this question is kind of like asking "Which is better? Screwdrivers or hammers?" It depends on what you're trying to do, and whether you care about similarity in time, frequency spectra, or both. If the delay spread D over a particular cellular communication path in an urban environment is 1.9 µs, then using equation above, the coherence bandwidth is approximately 0.53 MHz, which results in frequency selective fading over the IS-95 bandwidth.First off, you should use whichever tool is appropriate for the job. This is the bandwidth over which the channel transfer function remains virtually constant. The portion of the signal bandwidth over which fading does not occur typically contains enough signal power to sustain reliable communications. Therefore, when fading occurs it occurs only over a relatively small fraction of the total CDMA signal bandwidth. One reason for designing the CDMA IS-95 waveform with a bandwidth of approximately 1.25 MHz is because in many urban signaling environments the coherence bandwidth B c is significantly less than 1.25 MHz. The coherence bandwidth varies over cellular or PCS communications paths because the multipath spread D varies from path to path.įrequencies within a coherence bandwidth of one another tend to all fade in a similar or correlated fashion. It can be reasonably assumed that the channel is flat if the coherence bandwidth is greater than the data signal bandwidth. If the multipath time delay spread equals D seconds, then the coherence bandwidth W c
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