Complex Windows

Question

Typically windows are symmetrical and real. Are there any applications where complex windows have been used (other than applying the same real window to the real and imaginary components of a complex number).

A full complex window would have the sum and cross products between the real and imaginary terms as follows if applied as a complex conjugate product:

$$y_C[n] = (w_R+jw_I)(x_R+jx_I)^* = w_R x_R + w_I x_I + j(w_I x_R - w_R x_I)$$

What could be a possible benefit with this and is this actually used anywhere? Are there any types of signals where the typical considerations for windows (such as dynamic range versus resolution bandwidth) would be different for the real and imaginary components of the signal? Or thinking in the frequency domain, would there be any advantage or application to having an asymmetric Kernel?

Answer 1

I can see one use case, where you'd absorb a frequency shift, i.e. a multiplication with an $e^{jfn}$ sequence, into the window. That would only be advantageous if the shift remains constant, and no phase continuity between separately windowed "frames" (or whatever you window) is needed.
Maybe some bursty FSK modem? Q-Tone modulations? OFDM with a differential PSK inside¹?

Other than that: There's GFDM systems, where the ambiguity function (Time/frequency plane of a single pulse in a multi-symbol multi-carrier frame, generalizing the OFDM frame) of a pulse only has nulls at every two subcarrier spacings, and at every two symbol durations. Through alternatingly only using real and imaginary parts of only half the subcarriers, but at a denser raster than the sinc shape of an equivalent OFDM system would allow, the orthogonality of symbols is preserved:

^{From Koslowski, Sebastian: Synchronisation und Entzerrung in Filterbank-Multicarrier-Empfängern, dissertation , Karlsruher Institut für Technologie (KIT), 2018}

Now, if I wanted to build something to detect symbol statistics at different subcarrier frequencies through a DFT of an undersampling of this waveform, I might want to build a window that "follows" the real and imaginary parts. But, honestly, this is as constructed as it gets.

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¹ DAB(+) is an OFDM system which employs a 2048-FFT in a streaming system (not short bursty transmissions), but doesn't want to spend complexity on phase recovery, so it does differential QPSK on each subcarrier; that system complexity tradeoff never ceases to surprise me!