Title: In OQAM modulation, why does the up-sampling process be seperated into two parts?
Tags: signal-analysis
Question: The OQAM modulation is implemented by up-sampling the symbols by two and delaying the imaginary part of each sub-channel by half a symbol. The signals are up-sampled by M/2 and convolved with the impulse response of the pulse shaping filter u[n].Why should this process being separated into two parts?

Comment: I'm really curious: What system uses OQAM? For QPSK, I see the ease of implementation that explains why you'd want to trade bandwidth for limited phase jumps, but for QAMs >4 (essentially, 9QAM and up, but I've never seen anything like 9QAM, so 16QAM up), wouldn't you just code the symbols to achieve that if it was of any concern?

Comment: I mean, I've seen Offset modulations in multicarrier schemes (specifically, things that are OFDM or generalizations of OFDM), but you say "pulse shaping filter"; is this an input to an OFDM-like system?

Comment: @MarcusMüller OQAM in combination with pulse-shaping can well be used for multicarrier systems. When using a pulse shaping filter, each carrier is shaped separately, and it can be considered a generalization of the conventional CP-OFDM (which essentially uses a pulse shaping filter which is a rect of length $T+T_{CP}$ and the subcarriers are $1/T$ apart (apparently, $T$ is the symbol duration). When using a different pulse shaping filter, you likely run into the restriction of the Balian-Low Theorem (which states there is no pulse shaping filter that is well localized ...

Comment: ...  in time and frequency which can achieve orthogonality between carriers with QAM modulation (when subcarrier distance is $1/T$). Here, using OQAM can give you at least quasi-orthogonality (real-domain orthogonality only) between the carriers. Have a look for OFDM-OQAM and FBMC-OQAM for more information.

Comment: I agree with most of your comment. However, I have some question about the classification. I think there are two kinds of technology to avoid ISI: OFDM & FBMC. One way to realize FBMC is OQAM/OFDM.

Comment: Well, I think this is just a terminology thing. Some people use OFDM/OQAM and FBMC/OQAM interchangably. CP-OFDM is a completely different technique (as used in LTE). To my understanding FBMC/OQAM does not require perfect reconstruction (i.e. exact real-domain orthogonality, but only "orthogonal enough"), whereas OFDM/OQAM requires perfect reconstruction (according to the name O-FDM). But, this is just terminology and it depends on you, what you use and how you define it.

Here is the accepted answer: In the end, it counts that the transmit signal of a multicarrier-OQAM system equals

$$x(t)=\sum_k\sum_m j^{k+m}d_{km} g(t-k\frac{T}{2})\exp(j2\pi\frac{mt}{T})$$

where $d_{km}$ is the real-valued data, $k$ is the symbol index and $m$ is the subcarrier index. 

However you achieve this transmit signal structure, your implementation is fine. Though, it is mostly done as you describe for convenience reasons. For sure, you can perform the upsampling by $M$ points and delay one part of the QAM symbols by $M/2$ instead of first upsampling by two and then delaying one part by one sample. However, for implementation reasons, e.g. delaying by $M/2$ samples requires more memory than delaying by only one sample. 

So, the answer is that it's not an algorithmic requirement, but more a convenient way of generating the OQAM multicarrier signal.



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