I am little confused about why we need analytic signals so bad in time-frequency analysis. What might happen if I use non-analytic signals to do time-frequency analysis?
Assuming time-frequency aims a providing a separation (at least visual) between signal components, the main reasons could be:
- for quadratic distributions, which tend to yield interference between components, "cancelling" at negative frequencies reduce the quantity of components that can interfere.
- for linear distributions, the filter bank formalism, and especially the down-sampling operators, is simplified, reducing the impact of aliasing errors.
For real signals, the Hermitian symmetry yield that "no information" is lost in the analytic form, a complex-valued function that has no negative frequency components, and it is easy to go back to real.
However, this is not so simple in practice.
- If the analytic signal constructed from a wide-sense stationary (WSS) real signal is always proper, this may not be the case for non-stationary signals, as underlined in Stochastic time-frequency analysis using the analytic signal: why the complementary distribution matters.
- Moreover, computing the analytic signal on discrete, finite-length data is often only approximate, especially for real-time applications. Hence the energy in the negative frequencies is rarely zero.
- Extending the concept of analyticity beyond 1D is not evident, and several design still exist.