If we had perfect synchronization in the receiver in time, frequency and phase, and if we had no concerns with implementing a matched filter in the receiver for optimum reception under low signal conditions - then we could sample the waveform at one sample per symbol. In fact, the whole job of the receiver is to remove all those offsets (and equalize for channel distortion, and optimally filter). As I will explain further below, some of these operations can be done at one sample per symbol andsuch that ultimately the conditioned waveform will be at one sample per symbol for the final symbol decision (or soft-decision). As I will explain further below, some of these operations in the receiver can be done at one sample per symbol depending on algorithm used except for the matched filter specifically.
We can understand why that is the case by first understanding what the waveform looks like as received (and as transmitted), where at the transmitter we have additional reasons to create the signal with multiple samples per symbol. Let me cover that first and then come back to the recieverreceiver:
Luckily and for spectrum efficiency and regulatory reasons we would rarely transmit such a signal but instead make use of pulse-shaping which the graphic above demonstrates. By transitioning slowly from one symbol to the next, we end up with a spectrum that is much more contained, as demonstrated by the red trace and spectrum in the plot. If our pulse shaping was unrealizably perfect, the resulting total RF bandwidth would be the symbol rate itself. With realizable pulse shaping it is typically between 8% and 25% wider or more, depending on how complicated we want to make the implementation (and what specifications we need to make). The graphic above shows a whopping 30% increase, so with the 1 MSample/sec the RF bandwidth is 1.3 MHz. The absolute minimum sampling rate with complex I and Q sampling at baseband would be 1.3 MHz, but additional margin is needed for realizable transition bands in subsequent filtering (either in digital upconversionup-conversion, or the reconstruction filter after the DAC). For those considerations it is convenient to use a sampling rate of 2 or 4 samples per symbol (or higher if we really want to simplify subsequent filtering and are not concerned about the increase in digital processing- this is a system trade that should involve someone looking at the whole PHY layer digital + analog architecture).
The pulse shaping implemented at the transmitter is done in such a way to minimize the transmit spectrum as explained above, but also to allow for the implementation of a "matched filter" in the receiver, and such that the cascade of both filters does not introduce inter-symbol-interference (ISI). The matched filter allows for optimum reception (in the presence of white noise), an important feature given sensitivity is often an important performance goal. A "Zero-ISI" pulse shaping filter results in the eye diagram shown below on the right hand side (here for the case of a 25% bandwidth increase) for the case of a "raised cosine Filter"filter" which is a very common pulse shape used. (For more details on eye-diagrams and what they show, see DSP.SE# 64613). The raised cosine filter is split into two filters such that the transmit filter and receiver matched filter are each "root raised cosine filters". This results in the waveform at the transmitter appearing on the left where we clearly do see ISI. The same reason we required multiple samples per symbol to implement the pulse shaping filter in the transmitter is one reason why we need multiple samples per symbol to implement the pulse shaping filter in the receiver. The two filters are matched, but that does not mean they need to be sampled at the same rate; the minimum requirement will be to meet Nyquist and for this reason (and convenience of implementing the timing recovery and decimating to the final one sample per symbol) a two-sample per symbol rate is often more than sufficient.
The above plots also apply to considerations for the other operations in the receiver required for removing distortion (IQ imbalance if the topology introduces it such as Zero-IF receivers) and phase, frequency and time offsets which are removed by carrier and timing recovery loops. However, the synchronization requirement is not a one size catch all requirement that multiple samples per symbol are absolutely required in all cases. There are algorithms for timing recovery that work with multiple samples per symbol such as the Gardner Loop, and then others that work with one sample per symbol such as the Mueller and Mueller Synchronizer (I compare the two at this post,DSP.SE# 75202 and there are several other posts here that detail these timing recovery synchronizers). Similarly there are carrier recovery algorithms that work with one sample per symbol such as the decision-directed carrier recovery loop that I detail at DSP.SE# 17297. Such receivers will have multiple stages of sampling where the operations requiring or benefiting from multiple samples per symbol are processed first and then followed by decimation stages for subsequent operations that operate at one sample per symbol. Out of these, the one operation we can't avoid having multiple samples/symbol is the matched filter, and depending on the level of channel distortion, the equalizer. There are several trades in the overall design (that go well beyond what can be detailed here) that can drive the choices as to which algorithm and when it would make sense to use an approach that operates on multiple samples per symbol.
