Distinguishing Nyquist Criterion and Sampling Theorem

4.8 Distinguishing Nyquist Criterion and Sampling Theorem

Eq. (4.25) is similar to $F_{s} > 2 f_{max}$ , posed by the sampling theorem, especially in case one interprets $f_{max}$ as the bandwidth $BW$ of the signal.

But the reader should distinguish the criterion for zero ISI from the sampling theorem. For example, assume a channel of bandwidth $BW = 200$ Hz is used to transmit a signal with $f_{max} = BW$ and the symbol rate is $R_{sym} = 100$ bauds. Because $R_{sym} ≤ 2 BW$ is obeyed, a zero ISI operation is possible even with the receiver “sampling” at the symbol rate $R_{sym}$ , that is, its ADC using $F_{s}^{Rx} = 100$ Hz. However, this sampling frequency is below the required $F_{s}^{Rx} > 2W = 400$ Hz of the sampling theorem. This is not surprising because in digital communications one is not concerned with reconstructing the waveform unambiguously and can afford aliasing. Recall that the goal is “simply” unambiguous detection.¹¹

Decreasing the sampling frequency $F_{s}^{Rx}$ reduces the computational cost at the receiver. Hence, a lower-bound on $F_{s}^{Rx}$ is of interest and can be obtained as follows: assuming the symbols are independent, at least one signal sample must be used to represent the respective symbol value, such that $F_{s}^{Rx} ≥ R_{sym}$ . In other words, the minimum $F_{s}^{Rx}$ at the receiver corresponds to an oversampling factor of $L = 1$ , i. e., using only one sample to represent a symbol. However, for improved performance, this minimum value is sometimes exceeded in favor of $L > 1$ . Table 4.2 summarizes the discussion.

Table 4.2: Contrasting the Nyquist criterion for zero ISI and the sampling theorem for real-valued signals. It is assumed that the signals are baseband and

f_{max} = BW


Context	Equation	Interpretation

Sampling theorem (Eq. (C.30))	$F_{s} > 2 BW$	At least $F_{s}$ to reconstruct the original signal

Zero-ISI Nyquist criterion	$R_{sym} ≤2 BW$	BW Hz allows a maximum of $2 BW$ bauds

Processing at receiver	$F_{s}^{Rx} ≥ R_{sym}$	At least one sample representing a symbol

Table 4.2 can be expanded when one considers complex-valued signals. The alternative definitions of bandwidth discussed in Section E.7.2.0 are important here, together with Eq. (E.39). The results for both real and complex-valued signals are very similar when one considers that an analytic signal of bandwidth BW has a double-sided bandwidth that is half of the one considered for real-valued signals.

Table 4.3: Contrasting the Nyquist criterion for zero ISI and the sampling theorem for analytic complex-valued signals. The signals are baseband with support from 0 to

f_{max} = BW


Context	Equation	Interpretation

Sampling theorem (Eq. (E.39))	$F_{s} > BW$	At least $F_{s}$ to reconstruct the original signal

Zero-ISI Nyquist criterion	$R_{sym} ≤ BW$	BW Hz allows a maximum of $BW$ bauds

Processing at receiver	$F_{s}^{Rx} ≥ R_{sym}$	At least one sample representing a symbol

For example, assuming a complex envelope with negligible energy outside the frequency range $[0, F_{max}]$ and calling its bandwidth $BW = F_{max}$ , Eq. (4.25) can be written as

R_{sym} ≤ BW .

(4.34)

For the results in Table 4.3, two things to keep in mind are that each sample or symbol corresponds to two real values and BW is half of the double-sided bandwidth.

¹¹ The notation $F_{s}^{Rx}$ emphasizes that the ADC is being used as part of a digital communication system, and not for creating a sampled version that will allow perfect signal reconstruction.