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), such 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, frequency and time offsets are not the primary reason for multiple samples per symbol, and algorithms exists that do operate best with one sample per symbol for those specific operations. The universal reason for having multiple samples per symbol in the receiver is to implement the matched filter and for channel equalization. That said, there are effective algorithms for carrier and timing recovery that do require multiple samples per symbol in their implementation but that is a design decision as part of the trade space in the overall receiver design. Further down I provide links to other posts here on Stack Exchange that detail common implementations for timing and carrier recovery and exactly when multiple samples per symbol are required for those implementations specifically.
We can understand when multiple samples per symbol are required 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 receiver:
Below shows two possible spectrums for a 16 QAM waveform with a symbol rate of 1 MSymbol/sec. Also shown is the real component of the time domain waveform at baseband (where with the blue trace we clearly see 4 levels, we will also see 4 levels on the quadrature component leading to the 16 possible symbols for 16 QAM). If we transmitted such rectangular pulses, that instantly transition from one level to the next, we would get the Sinc shaped spectrum in blue (consistent with the Fourier Transform of a rectangular pulse-- yes! For the case of equiprobable data, the Fourier Transform of whatever base pulse we transmit will be the envelope for the resulting spectrum).
An understanding of the sampling theory is required and may need review prior to continuing to read below (there are many other posts here that discuss sampling and aliasing etc, so I am going to assume that is well understood). If we were to transmit this very broad spectrum, we would need to have a very high sampling rate in order to not have to large side-lobes alias back into the primary bandwidth of the signal. This of course results in many samples per symbol.
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 up-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).
Below is an interesting graphic I have that helps demonstrate this:
The first graphic shows the pulse shaping filter implementation if the waveform was sampled at 2 samples per symbol- Notice how we clearly see here the single-sided bandwidth of the same resulting RF signal I showed above:
The seconds graphic shows the pulse shaping we could do with the same filter complexity (same number of filter coefficients) but sampled at 10 samples per symbol:
In both cases we have more than enough of a sampling rate to meet the minimum Nyquist Criterion of 1.3 MHz. But there are some additional very interesting details revealed in these plots. When we sample 2 samples / symbol we are able to get a much better sideband rejection, but will require much more complexity in subsequent filtering to reject digital images which occur at every multiple of the sampling rate (an understanding of the sampling process and the spectrum we would expect to see out of the D/A Converter for both cases is helpful to understand this). This applies if we are connecting directly to a D/A or if we are going through subsequent digital resampling to a higher rate to alleviate the analog filtering: in that case it would be the digital filtering in the resampling process where we would be making that trade. Often the end result is 2 or 4 samples is more than enough for reasonable filter implementations in the digital, but we would often resample to a higher rate if possible ahead of the D/A converter (depending on the D/A rate and its performance.
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" 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.
As first introduced, if the receiver is not perfectly synchronized- meaning if there are small offsets in frequency, phase or time-- or if there are other imbalances such as quadrature or amplitude error, or channel distortions (multi-path etc or asymmetric passband errors in the RF/analog path in the receiver) then it is also necessary to have more than one sample per symbol to correct for these errors. A straight-forward demonstration of this is channel equalization. Distortion within the channel (from multi-path for example) can appear as the equivalent of a filter that affects portions of the spectrum, and the job of the equalizer is to compensate for that distortion. The graphic below demonstrates why a fractionally spaced (multiple samples per symbol) would out-perform a "baud-rate" (one sample/symbol) equalizer, again showing the spectral images of that same waveform we evaluated above with regards to its occupied bandwidth and the sample rate. Both equalizers can be used but this shows the consideration and importance of using multiple samples per symbol in the receiver:
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 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.
Like what you see? These details and graphics are all from an online course I teach (a recent session just started and the recordings are available through Nov 1 so there is still an opportunity to jump in!) The course is "DSP for Software Radio" and details all of these concerns with timing recovery, carrier recovery, AGC, equalization etc specific to single-carrier radio implementations such as QAM, and provides an excellent foundation for further study in multi-carrier and multi-space implementations. After taking this 15 hour / 5 week course, you will completely understand all the implications in deciding how many samples per symbol to use for any modulation. More details and registration info is at this link: https://dsprelated.com/courses
