1 Introduction
Analog to Digital Converters (ADCs) are among the most utilized components in digital systems today. The majority of today’s samplers require highly accurate clock support [1]. It is considered a difficult task to design highly precise clock samplers [2]. As a result, asynchronous sampling systems began to gain favor as a suitable choice for successfully replacing synchronous clock samplers [3, 4]. Asynchronous samplers pave the way toward robust and energyefficient analog sampling circuits. These systems are typically based on changes in the input signal rather than sampling at regular intervals [3]. In contrast to synchronous sampling circuits, asynchronous systems can encode time intervals only rather than timeamplitude pairs [5, 6].
Integrateandfire timeencodingmachine (IFTEM) is one of the most prominent timebased asynchronous energyefficient samplers [7, 8, 6, 9, 10]
. The signal is integrated, and when it crosses a given threshold, the time is encoded. IFTEM sampler is known to be lowpower and operates very similarly to neurons in the human body
[7, 8]. In recent years, the IFTEM sampler has been used for sampling and reconstruction of a variety of signal classes, including bandlimited (BL) [8, 11, 6, 12, 13, 14], finiterateofinnovation (FRI) [10, 15, 16], and shiftinvariant signals [17]. Quantification’s effect on the quality of reconstruction utilizing an IFTEM is also addressed [7, 14]. However, prior studies did not leverage the IFTEM’s natural gain as a lowpass filter (LPF). In particular, the IFTEM outputs are stationary, which permits efficient analog compression throughout the sampling period before quantization.In this paper, we present a compressed IFTEM (CIFTEM) sampler, a design methodology for analog compression, in which analog bandlimited (BL) signals are sampled and quantized at the Nyquist rate using the IFTEM ADC. Based on our prior work [14], we analyze BL signals that have finite energy, and their amplitudes that are constrained by the frequency and energy [18]. Our method relies on subdividing the known IFTEM dynamic range into tighter subranges with are then quantized. Leveraging the stationarity of the IFTEM output, the samples would be analogy compressed prior to quantization. Compared to the IFTEM without compression using the same number of samples, the reconstruction MSE is improved while the total number of bits is reduced. To the best of our knowledge, this is the first work to establish analog compression in IFTEM systems.
Our contribution is threefold: first, we provide an analog compression technique based on the stationarity of the samples. By subdividing the known IFTEM dynamic range into fixedsize narrower windows, this method enables efficient analog and digital implementation as well as low bitrate reconstruction. Second, we incorporate an adaptive component in the compression process so that it can better adapt to the sampler system, resulting in compression savings of at least 12 bits compared to sampling and recovering BL signals using IFTEM quantization without an adaptive component. Thirdly, we demonstrate numerically that by utilizing our proposed timebased analog compression technique, the MSE of the reconstructed BL signal is lower than that proposed by [14] while using the same amount of samples and fewer bits.
2 Problem Formulation and Preliminaries
We begin by presenting relevant known results on IFTEM sampling, perfect reconstruction, and quantization, followed by the problem formulation.
2.1 IFTEM for BL signals
An IFTEM sampler is characterized by a bias , scaling , and threshold , as depicted in Fig. 1.
The input is assumed to fulfill the condition . As shown in Fig. 2, before timeencoding the signal , a bias is added. The resultant signal is integrated after being scaled by . Either the time instances or their differences , at which the integral crosses the threshold are recorded, and the integrator is reset to zero.
The following relationship between system parameters:
(1) 
Alternatively, the following amplitude measurements can be computed:
(2) 
Using (2) and [7, 8] it can be shown that is bounded by:
(3) 
The reconstruction of BL signals from IFTEM outputs has been studied for scenarios when the input signal is BL and bounded with finite energy , where is the frequency upper bound [14, 6, 7, 8]. As in [14], we assume BL signals with maximal energy ; in this case, the relation between and the maximal signal’s amplitude is , as shown in [18].
We consider the IFTEM sampling and reconstruction technique as in [8], which showed that BL can be perfectly recovered using an IFTEM with parameters if and [8]
(4) 
The bound in (4) necessitates a bandwidth that is inversely proportional to the time difference between the time instances. Therefore, the BL input signal can be perfectly recovered if the overall firing rate of the IFTEM is greater than the Nyquist rate. In this case, the signal is reconstructed similarly to a BL signal recorded with irregularly spaced amplitude samples (see [8, 6] for information on the IFTEM recovery techniques).
Using an IFTEM as depicted in Fig. 1, the authors in [7] studied the MSE theoretical bounds of the reconstruction of signals sampled using IFTEM and uniform quantizer. The authors in [14] studied the problem of quantizing the time instances differences samples of BL signals. They considered a level uniform design for the quantization levels and demonstrated that the quantization using IFTEM results in better reconstruction for BL signals compared to conventional uniform sampling. They proved that if the bias is chosen such that , where , the stepsize is given by
(5) 
In particular, by experimental study, they showed that the quantizer step size as well as dynamic range of the samples decrease, resulting in a lower MSE compared with the amplitude Nyquist sampling and reconstruction method. In addition, they demonstrated that step size decreases as signals bandwidth increases.
2.2 Problem Statement
We consider the problem of recovering a BL and bounded signal with a fixed maximal energy , from its quantized samples for the IFTEM setup. According to [18], such signals are amplitude bounded and meet the following condition:
(6) 
A generalized CIFTEM sampling with quantization scheme is shown in Figures 3 and 4. The IFTEM sampler with parameters computes the stationary timebased measurements of the signal
, which are then quantized using the compression and estimator blocks, resulting in
, where is the quantizer, and denotes compressed and then quantized, respectively. Due to quantization, it is impossible to fully recover the signal . The authors of [14] demonstrated that the IFTEM sampler can result in a superior MSE of the recovered BL signal from quantized measurements as compared to the standard Nyquist reconstruction.We study the problem of recovering a signal from its quantized samples for both the conventional IFTEM with quantization system proposed by [14] and our proposed CIFTEM method, which employs compression at the quantization phase (see Fig. 3). We would like to establish the advantages of CIFTEM over IFTEM with quantization proposed by [14] in terms of MSE and bits.
3 CIFTEM: compressed IFTEM
In this section, we analyze the problem of recovering a signal from its quantized samples for both the conventional IFTEM with quantization system proposed by [14] and our proposed CIFTEM method, which uses analog compression before the quantization phase. Prior to the quantization operation for the IFTEM, an estimator and compression blocks are added. The advantages of CIFTEM stem from the fact that the IFTEM sampler acts as a LPF and, as such, its output is stationary in time, the measured sample values are close to one another, and analog compression can be efficiently applied. We propose two methods for dividing the dynamic range of quantized samples into time windows. We determine the window number, which may be constant or fluctuate dynamically. We refer to these approaches, as constant CIFTEM (CCIFTEM) and dynamic CIFTEM (DCIFTEM), respectively.
As depicted in Fig. 3, the IFTEM output is the estimator block input. The estimator block calculates , the number of windows which divides the dynamic range of the encoded signal
for higher quantization resolution. The compression block is then divides the dynamic range into windows, and determines to which of the windows the sample belongs. Then bits are assigned to quantize the sample in that specific subdynamic range regime . The output of the compression block , represents the sub dynamic range regime of the window . Finally, the quantizer quantize . By decreasing the quantization step size and employing subdynamic range regimes, the reconstruction is improved in terms of MSE utilizing the CIFTEM compared to [14, 7]. Note that the window number is encoded separately from the samples. Since it is assumed that the encoded samples are stationary in time, the bulk of consecutive samples will fall within the same time windows. Consequently, the value of will fluctuate every few samples. Thus, is only encoded if its value has changed since the last sample. Despite the fact that CIFTEM techniques require more bits for window encoding, as demonstrated numerically in the next section, they can achieve the same MSE as IFTEM with quantization while using less bits overall. An example of a practical implementation for the analog compression scheme in CIFTEM is illustrated in Fig. 4.
3.1 CcifTem
In the CCIFTEM scheme, the number of windows is a positive constant number. The sample’s
statistics are assumed to be known constants. Namely, the sample’s variance
and mean . Given the sample’s variance, the constant number of windows is determined as(7) 
Lemma 1.
Consider a 2 bounded BL signal and an CCIFTEM sampler with parameters , such as , where . Let be the known variance and mean samples statistics. Using a level uniform quantizer, the CCIFTEM quantization step size is
(8) 
Proof.
Using (3), the size of the samples dynamic range is:
(9) 
Since the dynamic range size is divided into uniform subranges and then quantized using level quantizer, the CCIFTEM quantization step size is
(10) 
Using the relation , where results
∎
Note that and are inversely proportional to each other. Using (7) and (8), it follows that decreasing will increase , which will decrease the quantization step size . Next, we demonstrate that when , the quantization step size of the CCIFTEM is smaller than that of the IFTEM, hence potentially decreasing the MSE. This outcome is summarized in the following theorem.
Theorem 1.
Let be the quantization step size of an IFTEM sampler with a level uniform quantizer proposed by [14]. Let be the quantization step size of a CCIFTEM sampler with a level uniform quantizer. Both samplers have the same parameters and the same uniform quantizer with quantization levels. Let the sample’s variance be a constant positive number. In this case, .
Proof.
Note that quantization error can be reduced based on dense quantization, and the CCIFTEM framework results in a lower quantization error than the IFTEM scheme.
Since the CCIFTEM compression utilizes a constant using the signal statistics, we want to be able to estimate the statistics in real time and adjust accordingly, as we suggest in the following subsection.
3.2 DcifTem
In this section, we present the DCIFTEM approach, which permits the window size to alter dynamically during sample acquisition by estimating the signal statistics in realtime. In contrast to the CCIFTEM, where the constant number of windows is determined using previous knowledge about the input signal, we do not have access to this information in the DCIFTEM. The system is initiated with a default value for , and is converging time.
In particular, the estimator block computes for each fixed IFTEM samples based on the variance of the previous samples. The variance is denoted by and is given as:
(12) 
where is the the average of the last samples and is calculated as:
(13) 
In the DCIFTEM scheme, the number of windows is given as
(14) 
Note that increasing decreases , which increases the quantization step size . Similar to the CCIFTEM sampler, we show that when , the quantization step size of the DCIFTEM is smaller than that of the IFTEM, potentially reducing the MSE.
Theorem 2.
Let be the quantization step size of an IFTEM sampler and be the quantization step size of a DCIFTEM sampler. Let also both samplers have the parameters and the same uniform quantizer with quantization levels. Then .
Proof.
Using (3) and Popoviciu inequality in [19], implies that the variance of any set of consecutive samples such that upholds
Since we conclude that . The dynamic range of the IFTEM sampler is defined as and the quantization step size is . DCIFTEM works with sub ranges each of size such that the quantization step size is
Following that we conclude that
Since the condition upholds. ∎
Note that since the samples are stationary, we expect that the size of the window will not vary much between samples. Thus, is computed and updated every samples, where . Thus, the number of windows only changes when the variance, , varies significantly. Signal statistics are computed dynamically in DCIFTEM, whereas they are assumed in CCIFTEM. Knowing rather than estimating the exact sample statistics results in a reduced MSE. Consequently, the CCIFTEM has a lower MSE than the DCIFTEM, which contains less prior information.
Corollary 1.
Based on Lemma 1 and Theorems 1 and 2, in order to reach or , one can use uniform quantizer with quantization levels in the CCIFTEM and DCIFTEM samplers such that .
Since the number of bits utilized to store the signal is directly proportional to , the CIFTEM systems compress signal bits more than IFTEM systems. Note that CIFTEM systems also encode window numbers, which introduces an additional piece of cost. It is assumed that we know the sample’s statistics or can dynamically estimate them. Therefore we expect that consecutive samples will belong to the same time windows, resulting in a low number of window number changes. Our simulations demonstrate that using fewer quantization levels in the quantizers of CIFTEM systems maintains the same MSE as an IFTEM sampler while performing bit compression. In future work, we will drive analytical results for this overhead.
4 Evaluation results
In this section, we exemplify our main result in an experimental study using simulations. First, we demonstrate that the CIFTEM schemes outperform the IFTEM quantization approach proposed in [14] in terms of MSE, by employing the same amount of IFTEM samples and overall bits. Secondly, we show that CCIFTEM produces a lower MSE than DCIFTEM. It is shown, however, that by using the DCIFTEM rather than the CCIFTEM to encode the same amount of samples, less bits may be employed, resulting in the same MSE.
We verify our main result using a BL signal as input. We consider 100 randomly selected bandlimited signals , which is bounded in time, i.e., , for with maximal and varying from Hz. The quantization bit range . We investigate the recovery after quantization for the IFTEM and the CIFTEM methods. The IFTEM parameters are selected as follows; we use fixed values of and . To have a sufficient number of samples needed for recovery, the bias is selected in two ways resulting in a maximal oversampling factor of 3.5: first, a fixed for all signals according to [7, Section 2], and second, choosing with according to [14, Section 3].
In Fig. 5, the suggested CIFTEM sampling frameworks with quantization (CCIFTEM and DCIFTEM) are evaluated in terms of MSE and compared to the IFTEM approach suggested by [14]. The MSE is calculated as:
(15) 
As can be shown, better MSE is reached using less bits in CIFTEM systems compared to the IFTEM systems. The CCIFTEM surpasses the DCIFTEM by employing fewer quantization bits for better reconstruction for the same number of samples. This result may be related to the fact that in the CCIFTEM, the constant number of windows is determined using prior information about the input signal, whereas in the DCIFTEM, this information is unavailable. The DCIFTEM parameters are and . Using the DCIFTEM sampler, the number of windows is calculated as the average number of windows used to encode the signal, , i.e., 23 bits are needed to encode the window.
In Fig. 6 and Table 1, we show the overhead bits which are required for encoding the window numbers in CIFTEM. On average, only 5 additional bits are required to encode window number values. In addition, it can be observed that, on average, DCIFTEM results in a lower overhead. Let be defined as the overhead in the percentage of bits used in the CIFTEM sampler, for a given levels quantizer. Since IFTEM systems require two additional bits to achieve the same MSE as CIFTEM systems, we can compute the compression obtained by using CIFTEM as follows:
Table 1 presents the compression percentage in bits in using the CIFTEM system while preserving the same MSE as in IFTEM, i.e., Table 1 shows the compression in bits [] of encoding the entire signal using CIFTEM systems with levels uniform quantizers compared to encoding the entire signal using IFTEM sampler with level uniform quantizer. As seen in Table 1, for the same MSE in both types of systems, the compression in bits [] between the samplers ranges between 10 and 20. Consequently, the CIFTEM sampler necessitates the use of extra bits to encode the window numbers of the samples, however, it reduces the overall number of bits that encode the signal. Note that, increasing the number of quantization levels , the compression we gain reduces. This happens because as we increase , the MSE decreases and approaches its saturation value.
Quantizer bits  Overall system compression []  

CIFTEM  IFTEM  DCIFTEM  CCIFTEM 
6  8  20.86%  19.84% 
7  9  18.61%  17.37% 
8  10  16.7%  16.02% 
9  11  15.26%  14.11% 
10  12  13.96%  13.05% 
11  13  12.89%  12.25% 
12  14  11.84%  11.17% 
13  15  11.05%  10.75% 
5 Conclusion
Using the IFTEM sampler for BL signals, we introduced methods for time encoding and decoding with analog compression. Prior to the quantizer, analog compression is performed based on the stationarity of the encoded signal, a basic aspect of IFTEM processing. Lowbitrate reconstruction is accomplished by subdividing the known IFTEM dynamic range into tighter windows, which can be of fixed or variable size, and determining which window the sample resides within. We empirically demonstrate that utilizing the same number of samples and up to 7 more bits than the traditional IFTEM improves MSE by 520dB. Using the compressed IFTEM enables the usage of 12 less bits compared to the traditional IFTEM, given the same reconstruction MSE target and amount of samples.
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