I Introduction
Innovative solutions such as orthogonal frequencydivision multiplexing (OFDM) are needed to quench the exponential growth of the data traffic related to mobile devices [1]. One major drawback of OFDM, however, is its high peaktoaverage power ratio (PAPR) [2]. As OFDM subdivides the frequency bands into several subcarriers, the amplitude of the transmitted signals has a large dynamic range. This results in band distortions when the input exceeds the linear region of the power amplifier. While the banddistortion problem can sometimes be resolved by highly linear power amplifiers, their deployment typically reduces the power efficiency of the hardware and increases costs.
PAPRreduction techniques have been proposed such as clipping [3, 4], phase optimization [5], tone reservation/injection [6], partial transmission sequence [7], and selective mapping [8]. The last two methods reduce the PAPR at the transmitter side while distortions are compensated by transmitting additional bits through a side channel.
An alternative PAPRreduction approach that makes the economy of a side channel is to detect the occurrence of saturated samples and then substitute them with estimates based on prior assumptions. For instance, Marvasti et al. [9, 10, 11] make the assumption that the underlying OFDM signal is bandlimited and then devise an iterative scheme to reconstruct the saturated values^{1}^{1}1The reconstruction of bandlimited signals is a classical problem [12, 13, 14]; it can be traced back to the PapoulisGerchberg algorithm [15, 16]. More recently, a Bayesian reconstruction that takes advantage of the sparse nature of the underlying signals and combines it with compressed sensing was proposed in [17]. Extending a single OFDM reconstruction to the multiuser OFDM reconstructions that take place in modern singleinput multipleoutput systems was further proposed in [18]. However, the aforementioned iterative solutions [9, 17, 18] are not hardwarefriendly and lack in robustness, being highly sensitive to the tuning of their parameters.
In [19]
, the pseudoinversion of a partial discreteFouriertransform matrix constructed with the locations of saturation (time) and socalled “reliable components” (frequency) is used to estimate the values of the true signal. This requires one to assume that the reliable components are known to every underlying OFDM receiver. In practice, however, there is no such information at the receiver and the reliable components have to be guessed.
In this paper, we propose a direct, noniterative algorithm that compensates for the saturation of bandlimited OFDM signals, thus attenuating band distortions and resulting in an improved biterrorratio of the transmitted bit streams. We derive the theory of minimumnorm reconstructions with respect to sampling consistency and reproducing kernel in the space of bandlimited functions. The theory reveals that our approach gives a solution that is unique, freed from the uncertainty that was tied to the reliablecomponent method of [19]
. Our algorithm consists of two steps. The first step (regression) amounts to a matrix inversion and the second step (interpolation) to a simple matrixvector multiplication. The fact that our algorithm is direct makes it very affordable in practice since this leads to a saving of critical resources such as computing cycles. Moreover, it turns out that our algorithm is able to obtain a coefficient estimate from fewer samples than is required by ideal sampling. This property leads to a further reduction of the computational effort in case of long signals. Finally, our algorithm shows stable performance with respect to its tuning parameters in a variety of experimental scenarios.
This paper is organized as follows: In Section II, we provide a fundamental theorem for nonuniform interpolation under the assumption of bandlimitedness. In Section III, we describe our algorithm and its regression and interpolation steps. In Section IV, we demonstrate the capability of our algorithm to recover the saturated samples of bandlimited signals in numerical and realistic OFDM simulations.
Ii Theory
Iia Terminology
We illustrate in Figure 1 the terminology that we shall use throughout this paper.

True signal: analog, unknown signal.

Sample: a (location, value) pair. We assume that all locations are quantized on a regular grid.

True value: value of a sample of the true signal.

Unsaturated sample: sample of the true signal, taken within the linear range of the amplifier. The values of unsaturated samples coincide with true values.

Saturated sample: the (location, undersaturation threshold) or (location, oversaturation threshold) pair that is returned by the hardware whenever the signal exceeds the linear domain of the amplifier. The detection of saturated samples is achieved by monitoring the sample values; no side channel is required.

Estimated value: outcome of our approach.
IiB Assumptions
The strategy for this work is to fit unsaturated samples with a bandlimited curve and then determine (reconstruct) estimated values by resampling the fit at the locations where saturation did occur. When stated in full generality, this is an illposed problem—think of the case when there are but saturated samples. To make the solution tractable, we need additional hypotheses. We consider here the conjunction of three assumptions that are realistic in the OFDM context: (i) the saturated samples are few and their spacing lacks regularity; (ii) the unsaturated samples have low noise; (iii) the true signal is bandlimited.
IiC Development
We are going to show that our reconstruction problem admits a closedform solution that is related to the kernel methods used in machine learning
[20, 21]. The enabling property is that the Shannon space of bandlimited functions can be endowed with the structure of a reproducingkernel Hilbert space [22, 23]. Within this setting, we shall derive an extension of the traditional sampling theory to recover bandpass functions from nonuniform samples. The nominal band of the signal is a (not necessarily connected) set that is symmetric around the origin and whose measure (or bandwidth) is finite. We define the corresponding space of bandpass functions as(1) 
where is the Fourier transform of the continuousdomain signal and is Lebesgue’s space of finiteenergy functions. With this notation, coincides with Shannon’s space of bandlimited functions [24], which is spanned^{2}^{2}2The dot notation “” is a placeholder for mute variables, oftentimes temporal ones. by the orthonormal basis . The main extension here is that the spectral band can be arbitrary and that we do not require the existence of a basis for , but rather a frame [25]. More precisely, letting be an adequate sampling set, this frame is an overcomplete family of functions that span .
IiD Underlying HilbertSpace Structure
We now prove that is a Hilbert space by identifying its reproducing kernel. To that end, we consider the indicator function
(2) 
of the set in the frequencydomain, along with its inverse Fourier transform
(3) 
Both are illustrated in Figure 2.
Given the kernel , we then define the convolution operator as
(4) 
which acts as an ideal bandpass filter. By invoking properties of the ideal filter, we then show idempotence
(5) 
and orthogonality of the error
(6) 
for any , which proves that is the orthogonal projector of onto . Likewise, we readily verify that is a unitary operator when is equipped with the usual inner product
(7) 
This suggests that the space is its own topological dual and that is the corresponding Riesz map whose generalized impulse response is precisely the soughtafter reproducing kernel.
Proposition 1 (Identification of the reproducing kernel).
Let . Then, the bivariate function is such that, for any fixed and ,
(8) 
(9) 
Hence, is the reproducing kernel of the Hilbert space equipped with the inner product.
Proof.
By definition, the reproducing kernel of a Hilbert space is the unique bivariate function such that and such that for all and any fixed [26]. In the present scenario, the kernel is shiftinvariant and the two conditions are established by considering inclusion and reproduction.

Inclusion property: The Fourier transform of is , . This is a finiteenergy function supported in , which proves that .

Reproduction property: The function is symmetric and realvalued, so that the same holds true for its inverse Fourier transform . Consequently, the central part of (9) is equivalent to the convolution product . Its Fourierdomain counterpart is since is the indicator function of and for all . The conclusion ensues.
∎
IiE MinimumNorm Reconstruction
Since the function is an infinitedimensional entity, its reconstruction from a finite number of (possibly nonuniform) samples is illposed in general. We regularize the problem by searching for the minimumnorm solution within the space of bandpass functions. The remarkable outcome of our formulation is that this solution is always well defined, irrespectively of the sampling locations . Furthermore, it can be found efficiently by solving a linear system of equations.
Theorem 1 (Bandpass signal reconstruction).
Let be a list of realvalued samples. The minimumnorm interpolant of in , in the sense that
(10) 
is given by
(11) 
where and , with and . Moreover, is guaranteed to be invertible whenever the sample locations are distinct, which ensures that the solution is unique and welldefined.
Proof.
First, we define the basis fonctions for . They are such that
(12) 
(13) 
as a direct consequence of (9). We then decompose the space as the direct sum of and its orthogonal complement in in , with
(14) 
Consequently, any has the unique decomposition with and . By construction, we also have the Pythagorean relation , due to the equivalence of the norm and the norm. The condition is equivalent to for which, in reason of the minimumnorm requirement, leads to .
Finally, we apply the interpolation constraint to the parametric representation of which, owing to (12) and (13), results in the set of linear equations given by
(15)  
in terms of , with . The important additional ingredient is the positive definiteness of , which follows from the extension of Bochner’s theorem given in Theorem 2. ∎
Definition 1 (Definiteness).
A function is said to be positive semidefinite if
(16) 
for every , , and any positive integer . Likewise, is said to be positive definite if the nonnegative relation in (16) can be replaced by a positive relation for any nonvanishing and distinct .
Theorem 2 (Extended Bochner’s theorem [27]).
A continuous function is positive semidefinite if and only if it is the Fourier transform of a positive Borel measure, so that , with for all . Moreover, it is positive definite if and only if the carrier of is nonzero; in other words, if and only if there is a measurable set such that for all .
In our case, the last condition is met trivially since the carrier of is precisely . Finally, the positive definiteness of implies the invertibility of the symmetric matrix for any given choice of distinct sampling locations .
IiF Particular Cases
For the Nyquist band, when , we have that is the sinus cardinalis for . This yields the solution , which is reminiscent of Shannon’s sampling expansion. The caveat, of course, is that the optimal expansion coefficients are usually distinct from , unless the sampling locations build a regular grid.
In the bandpass scenario, we have that
(17) 
with . This corresponds to the reproducing kernel
(18) 
The relevant functions for this construction are illustrated in Figure 2.
Iii Proposed Method
In this section, we describe the method to obtain in Theorem 1 in the context of few saturated samples coming in an irregular sequence. The overall OFDM system is illustrated in Figure 3. The input and output bit streams are encoded with quadrature amplitude modulation (QAM) and the encoded signal is described by its frequency components. The multicarrier OFDM signal is generated by an inverse discrete Fourier transform (IDFT). The OFDM signal is transmitted in the time domain. The effect of saturation (clipping) is modeled by a sampling operation combined with an analogtodigital converter (ADC). The undersaturation threshold and the oversaturation threshold depend on the specifications of the ADC; once the unknown signal is saturated and sampled, the accessible values are
(19) 
It is now crucial to realize that the theory developed in Section II does not require any specific temporal ordering of the batch of samples we have at our disposal. We take advantage of this opportunity to reindex them and partition them in two groups. The first group is made of unsaturated samples indexed by such that . Meanwhile, the second group contains the remaining saturated samples indexed by such that . By notational convention, we collect the unsaturated OFDM signal values in the vector .
Iiia Regression
(20) 
Applying (15) to the group of all unsaturated samples, we derive that . This regression step is written explicitly in (20). Due to the symmetric structure of assumed in (17), the matrix to invert is symmetric, too, which facilitates computations. Bad conditioning can be further avoided by an parametric regularization [28, 29], as in
(21) 
IiiB Interpolation
(22) 
IiiC Shortcomings
In the presence of regularization (i.e., ), the equality is not necessarily valid anymore if we use (21) to determine in (11). Thus, the combined list of samples is not consistent with our bandpass model. Despite this discrepancy, and for sheer convenience, we shall disregard this state of affairs in the interest of simplicity and reconstruct only the saturated samples, leaving the unsaturated ones pristine.
Likewise, we do not attempt to confirm whether the estimated values do indeed exceed the saturation thresholds. While special configurations of saturated values exist where ringing can mislead our method to return estimated values within the interval
, these configurations never arise in practice. Solving for these rare cases would require linearprogramming methods, which come at an unacceptable costbenefit tradeoff.
IiiD Sliding Window
So far, we have assumed that we have collected a batch of samples before to apply regression and interpolation, and that we estimate values jointly. This implies a long delay if is large, which is highly undesirable in practice, aggravated by the complexity. To reduce delay and complexity, we propose now a samplebysample method in which we deploy the regression and interpolation steps locally over a sliding window centered on the saturated sample , with and small. Then, we build out of its nearest (anterior and posterior) unsaturated samples. While building , no skipping of samples is needed for isolated saturation events, while either a recursive application of the procedure or an augmentedmargin window is needed when their density is higher. The pseudocode that corresponds to the augmentedmargin approach (window length , margin length ) is given in Algorithm 1.
Because we apply the slidingwindow approach once per saturated sample, the average computational effort of its regression step is , where is the average density of saturated samples—the expected value of the saturatedtounsaturated ratio. The cost of the sorting operation implied by Step 14 of Algorithm 1 is , with . The computational effort of the interpolation step is , which is negligible. The cost of finding unsaturated samples is . In the favorable case where all saturation events are isolated, the cost is finally dominated by the peak computational demand .
IiiE Wireline versus Wireless
In wireline applications (top part of Figure 4), the frontend receiver is sometimes called a discrete multitone transceiver. The symbols of the subcarriers, also known as Fourier coefficients, honor a Hermitian symmetry. Consequently, the temporal signal is realvalued and Algorithm 1 can be applied directly.
By contrast, in wireless applications (bottom part of Figure 4), the symbols of the subcarriers do not follow a Hermitian symmetry. Instead, the temporal signal is split in two components transmitted separately trough the inphase channel and the quadrature channel. Together, they represent a complexvalued timedomain signal whose real and imaginary parts are recovered by multiplication with the phaseshifted carrier frequency. But each part retains the property of being bandlimited, which allows our proposed solution to be applied independently to the real and to the imaginary channel.
Iv Results
We now describe our experimental setups and simulation results for five scenarios that each illustrate one aspect of our approach:

Bandpass illustrative case (Section IVA1).

Dependence of the quality of the reconstruction on the number of nearest neighbors (Section IVA2).

Robustness with respect to additive Gaussian noise in wireless applications (Section IVA3).

Dependence of the computational cost on the number of nearest neighbors (Section IVA4).

Recovery of saturated samples in discrete multitone wireline seriallink transceivers. In this simulation, we consider realistic perturbations such as channel distortion, quantization error, and cyclic prefix (Section IVB).
Also note that the first case is bandpass, while all other cases are lowpass, with and , where is the Nyquist rate. The main measure of our quantitative evaluations is the biterrorratio (BER). We also provide the clipping ratio (CR) defined as
(23) 
where is the rootmeansquare norm (i.e., the norm) and is a threshold value.
Iva Numeric Simulations
IvA1 Ideal Case
We assume here that the original signal is noiseless and bandlimited, which we enforce by Fourier synthesis through a tap IDFT. We simulate a wireless (complexvalued) OFDM model characterized by a zeroforced DC component, the use of sixteen subcarriers out of a total of thirtytwo, and symbol encoding by QAM. We show in the top part of Figure 5 the sampled groundtruth data observed in the imaginary (Q channel) of the simulated system. Its saturated counterpart is shown in the middle part of Figure 5. It is apparent from that saturation produces inband and outofband distortions in OFDM tones. Unless cared for, these distortions prevent the receiver from correctly retrieving symbols.
We then deploy our method with , which compensates for saturation by minimizing the inband and outofband distortions. The outcome is illustrated in the bottom part of Figure 5. The PAPR of the original and saturated signal is and , respectively. The BER of the saturated and reconstructed signals is and out of the transmitted bits, respectively.
IvA2 Quality
In this section, we investigate how the BER depends on the number of unsaturated neighbors. We performed random simulations, both in a wireline and a wireless context. Each trial involved an OFDM stream based on a tap IDFT, a bandwidth of taps in the discrete domain. The DC component was maintained to zero. As we want now to focus on the sole impact of , we refrained from including distortions such as insertion loss in the wireline case, additive noise, or multipath fading in the wireless case.
The results for the wireline context are presented in Figure 6. There, the Hermitian symmetry reduces to the number of independent subcarriers. We observe that the proposed scheme is able to robustly estimate up to three saturated sample values (approximately, CR=1.66) with a BER less than while using no more than neighboring unsaturated samples. It is known [30] that our relatively high raw BER could be reduced much further to, say, when combined with a forward errorcorrection code such as ReedSolomon [30]. Together, they satisfy the usual BER requirements of an OFDMbased highspeed pointtopoint wireline serial link. When a more stringent BER target is imposed, it is enough to raise . For instance, would guarantee a robust reconstruction even in the presence of five saturated samples each thirtytwo samples.
The results in the wireless context are presented in Figure 7. We observe again that neighboring unsaturated samples are enough to successfully estimate up to saturations from CR 1.31. The increase in performance of the wireless over the wireline case results from the fact that the complexvalued signal of the former is split into two separate channels, thus decreasing the effective number of saturated samples per channel.
IvA3 Robustness
It is typical for wireless longdistance communications to be corrupted by several sources of noise. We simulate now their impact by the channelwise addition of white Gaussian noise, with a controlled signaltonoise ratio. Like in our previous experiments, a tap signal is used for the forward/backward model of OFDM, the bandwidth is half of the total length of the signal, and QAM is used for symbol encoding; the DC component is zeroed. We report the resulting BER in Figure 8. Once again, we observe that leads to a performance that is very similar to what would be obtained in the absence of saturation and confirms the robustness of our method in the presence of noise.
IvA4 Computational Cost
In Figure 9, we report the computation times observed for  to tap signals and various clipping ratios. We see that they correlate weakly with CR while, in first approximation, they increase linearly with the number of taps. We attribute this dependence to the combined cost of sorting and matrix inversion, given that the number of values to estimate (the number of saturated samples) increases in proportion of the number of taps.
Time savings can be easily obtained if the sample locations sit on a grid. Then, the entries of the matrix to invert in (20) and the matrix to apply in (22) can all be precomputed. When the saturation events are either isolated or in simple configurations, it is convenient to store preinverted matrices in a configurationdependent lookup table. The computational cost is then dominated by the sorting step needed to find the neighbors nearest to each saturation event.
IvB Wireline Communication Links
In electrical wireline transceiver systems, the insertion loss of the transmission line (channel) is the strongest source of noise when the channel is lossy. We illustrate in Figure 10 the actual frequency response of real fourlane inch/inch backplane channels, a longreach lossy FR4 channel, and three different inch twinaxial copper cables.
When the OFDM signal is transmitted through such channels, the worstcase PAPR grows as the signal gets farther away from the emitter. In particular, while the nonzero spectral components are spread evenly over the entire frequency range, the DC component might still be responsible for large amplitudes because it undergoes much less attenuation that the highfrequency components. At the input of the receiver, this results in a wide dynamic range that is difficult to quantize since it is desired to faithfully represent, on one hand, the small signal variations that encode the information carried by highfrequency components and, on the other hand, the large variations related to the lowfrequency components. For a finiterange linear quantizer, the generally accepted tradeoff is to tolerate some saturation in return for a finer representation of the leastsignificant bits. This mechanism explains why the reduction of the PAPR of an OFDM signal is of high interest.
Figure 11 contains a snapshot of the OFDM signal at the receiver frontend, before and after the ADC. Out of , this signal is made of lowfrequency OFDM carriers (including DC); the cyclic prefix length is chosen to be . The constellation is fixed to QAM for all subchannels. The effective number of bits of the ADC is assumed to be , which means that the signaltonoise and distortion ratio is about , for a bandwidth of up to . The ADC sampling rate is for a final data rate. The inch backplane channel exhibits an attenuation of about due to the transmission medium.
We provide in Figure 12 a simulation of the various signals that play a role in our proposed method. Virtual samples of the unspoiled, continuously defined signal (solid line) that enters the ADC are shown with circles; these samples are not accessible in practice. What is accessible is their saturated (i.e., clipped) and quantized version (mixed line and triangles). The ADC hardware also provides a mask signal that encodes the saturation events (top solid line without markers). The saturationcompensated reconstructed signal () we propose is shown with a dashed line and cross markers. It follows closely the input signal.
Constellation diagrams after frequencydomain equalization for channel inversion are shown in Figure 13. It can be observed visually that the method we propose to recover estimated values of the true samples leads to a significant improvement of the SNR of the recovered signal.
V Conclusion
We have proposed a direct, noniterative method to recover value estimates for those samples of OFDM signals that are saturated. The noniterative nature of our algorithm dramatically reduces its implementation cost, while we showed through simulations that our solution performs well and offers a strong mechanism to compensate for information losses related to saturation.
Our main prior assumption is that the original signal is bandlimited. We have developed a theoretical framework that justifies our construction under this specific assumption. This theory extends the scope of our proposed solution beyond the sole application of reconstructing OFDM saturated samples.
Although the most expensive computations of our method are related to a matrixinversion step, we were able to show that the handling of a small matrix is often sufficient in realistic wireline/wireless digitalcommunication systems. As the matrix components do not depend on data, future versions of the the algorithm will also rely on lookup tables so that the matrixinversion step can be bypassed altogether.
The low computing complexity of our proposal shows potential for an efficient hardware implementation, possibly running at hundreds of MHz or even in the GHz domain once implemented on silicon as an applicationspecific integrated circuit (ASIC), aiming at realtime operations.
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