I Introduction
The acquisition theory of common analog-to-digital (A/D) converters is based on discrete time and analog amplitude. The classical sampling theorem [1, 2, 3] provides a theoretical foundation by representing a bandlimited signal by its analog amplitude values at a uniform time grid. Alternatively, it is also possible to consider an acquisition procedure where signal representation is based on quantized amplitude and analog time, i.e. amplitude sampling [4]. The latter represents the input signal, for analysis purposes, by the time instants at which a reversible transformation of the input crosses equally-spaced amplitude values.
In practice, however, the sequences generated by an A/D converter implicitly represent signals that lie between the two sampling paradigms described above, i.e. quantization is both present in the amplitude and time domains. This manifests itself in uniform time sampling, which results in discrete-time processing, and amplitude quantization, i.e. digital representation. Note that it is also possible to perform digital signal processing in continuous time when quantization is introduced [5].
In this paper, we view A/D conversion from the perspective of sampling theory. We consider the quantized discrete-time sequences generated by an A/D converter as coming from samples of a function that takes amplitude values on a uniform grid at uniform instants of time. We refer to these signals as lattice functions and its acquisition as lattice sampling. In particular, they can be interpreted as integral-valued entire functions of exponential type.
Based on the analysis of the spectral properties of these functions, we derive a deterministic result about the influence of the quantization error in the spectrum of discrete-time digital sequences. In particular, irrespective of the resolution of the quantizer and the bandwidth of the input signal to an A/D converter, the discrete-time Fourier transform (DTFT) of the digital sequence contains frequency components
rad/s. It is common, for theoretical purposes, to consider the set of bandlimited functions in the analysis of A/D conversion. However, in practice, an A/D converter implicitly considers lattice functions. Thus, we derive results about the cardinality of bandlimited lattice functions and examples of functions within this set in order to provide a deeper understanding of the subset of bandlimited signals that an actual A/D converter is implicitly considering.The approach taken here is in contrast with the traditional analysis of A/D converters based on amplitude quantization as a source of error. The analysis in this context has been mainly addressed from two different perspectives. On the one hand, the probabilistic interpretation considers discrete-time sequences where the quantizer—a nonlinear system—is modeled as a linear system which adds uniformly distributed white noise, in general with a range related to the quantization step
[6][7]. This noise model is a useful analytic tool used in oversampled A/D conversion, and oversampled A/D conversion with noise shaping.The deterministic approach has been considered in several scenarios. In the specific case of limit-cycle oscillations in digital filters, it was used to determine bounds on the limit-cycle amplitude for fixed-point implementations of recursive digital filters [8]. In [6]
, the spectrum of a quantized sinusoid was also studied in a deterministic way where it was derived the well-known result about the output consisting of the odd harmonics of the input frequency. Within the context of digital control systems incorporating quantization, it is derived, assuming a worst-case scenario, an upper bound on the dynamic quantization error
[9, 10]. The bounds derived are, in general, too loose compared with the stochastic approach and experimental data [8].In Section II
, we formally introduce lattice functions as the bandlimited interpolation of discrete-time sequences output by and A/D converter. If both the quantization step and sampling period are integers, these functions are known as integral-valued entire functions. We use the interpretation of bandlimited signals as entire functions of exponential type—which can be informally seen as polynomials of infinite degree. Section
III studies the properties of the set of functions that a common discrete-time signal processing system deals with due to the transformation performed by an A/D converter. In particular, it shows that lattice functions with a given bandwidth form a countable set within the set of signals with the same bandwidth. Additionally, it gives some examples of a family of functions belonging to this set with the same cardinality. Section IV provides results about the connection between integral-valued bandlimited functions and a lower bound on their maximum frequency component. Finally, Section V generalizes the previous results to arbitrary lattice functions and provides a result regarding the Discrete-Time Fourier Transform of quantized sequences with the corresponding connection to the quantization error.Ii Lattice Functions
We represent an A/D converter by the block diagram depicted in Fig. 1. The continuous-to-discrete (C/D) block outputs an analog value every seconds for an input signal and . We consider a uniform quantizer, denoted by for some , with quantization levels . Thus, both systems represented in Fig. 1 are input-output equivalent.
Note that any interpolation —discrete to continous transformation—of the sequence of values generated by the A/D converter that leads to consistent resampling satisfies for . In other words, the A/D converter, assuming consistent resampling, implicitly maps any input signal to a signal that takes amplitude values at multiples of at precisely every sampling instant.
A useful approach to study the class of signals represented by an A/D converter is to first consider a sampling process where quantization is both present in time and amplitude, we refer to it as lattice sampling. This consists of taking samples whenever the source signal crosses exactly the points defined by a two-dimensional grid, i.e. a lattice. In particular, consider the procedure illustrated in Fig. 2 where the input signals are sampled at the points where represent the offsets. Samples are taken only when the input signal passes through one of these points in a lattice. This approach lies between the traditional sampling theorem and amplitude sampling when considered from the point of view of which domains are quantized or assumed to be analog.
Although the sampling points are uniformly separated in time and amplitude, lattice sampling can be nonuniform as Fig. 2 illustrates. It is true, however, that these sampling instants are always related by a multiple of . Our approach throughout this paper is to restrict ourselves to those functions that cross a lattice point at every multiple of , i.e. they take values multiples of at every instant multiple of . We refer to a function satisfying the latter as a lattice function.
Definition 1
A continuous signal is a lattice function if for all where , , and .
Note that any input signal is represented by an A/D converter as a discrete sequence of numbers lying in the lattice described above at every sampling instant. Equivalently, any input signal to an A/D converter is implicitly mapped to a signal that can be sampled at the lattice —hence the name lattice function. Thus, any function generated as the interpolation, assuming consistent resampling, of a sequence produced by an A/D converter is a lattice function. Fig. 3 shows the case when the bandlimited interpolation—represented by the D/C block—is used.
For ease of illustration and without loss of generality, we consider the lattice , i.e. the amplitude levels and the sampling instants correspond, in both cases, to the integers. The generalization to an arbitrary lattice will be carried out in the last section. However, first considering this choice of parameters makes the development and notation less cumbersome. Thus, we are interested in lattice functions that take integer values at the integers, i.e. integral-valued functions.
Definition 2
A function is called integral valued if for all .
The set of integral-valued functions contains lattice functions as a proper subset since the latter are necessarily bandlimited. In order to understand the mapping performed by A/D converters, the question arises as to whether we can characterize bandlimited integral-valued functions—equivalently, bandlimited lattice functions for an appropriate choice of parameters. Due to the connection between polynomials and bandlimited functions, we will first discuss in the next section integral-valued polynomials that will provide insight into the properties of the set of lattice functions.
Iii Integral-Valued Bandlimited Functions in
This section provides an insight into the set of functions an A/D converter implicitly converts to, i.e. the underlying bandlimited signals used in discrete-time LTI processing of continuous-time signals. In fact, we will see, by cardinality arguments, that actually this set of functions is very small compared to the common assumption of considering the whole set of bandlimited functions of a certain bandwidth. The later part of the section illustrates this set of functions by providing an examples of a subset with the same cardinality.
Without loss of generality, we first explore the properties of lattice functions assuming the lattice points have integer coordinates, i.e. integral-valued bandlimited functions. We initially focus on polynomials. This motivation lies in the fact that continuous bandlimited functions bear a resemblance to polynomials in the sense that they admit a factorization based on their roots due to Hadamard’s formula. Roughly speaking, bandlimited functions can be seen as infinite-degree polynomials and may thus be viewed as a limiting case of finite-degree polynomials. Then, we use this set of integral-valued polynomials to prove the countability of the set of lattice functions providing some examples that belong to this set.
Iii-a Integral-Valued Polynomials
Our interest then lies in polynomials that take integer values at the integers [11, Chapter 1]. In particular, we introduce a characterization of these polynomials in terms of the difference operator defined for a function as for . It can be shown that by applying it times, we obtain the -th difference operator
(1) |
If we choose we arrive at
(2) |
which expresses in terms of the values . From (2), we can observe that for if the function is a polynomial of degree . It is also possible to reverse (2) and write in terms of the finite differences at zero in the following way
(3) |
for .
The next result [11, Corollary 1.9.3] provides necessary and sufficient conditions for a polynomial to be integral valued.
Proposition 1
Let be a polynomial of degree . Then, for all if and only if , .
A sufficient condition for a polynomial to take integer values at the integers is considering integer coefficients. However, the backward assumption in Proposition 1 is more general than the latter. In particular, the coefficients are not required to be integers and, in fact, a closer inspection of (3) reveals that they may belong to .
Example 1
Consider the polynomial where and . By Proposition 1, the polynomial is integral valued. Indeed, we can use (3) to arrive at the expression
(4) |
where on its right-hand side, we recognize the identity of the sum of the first positive integers. Since for this particular example the polynomial is an even function, we conclude that for .
Iii-B Integral-Valued Bandlimited Functions in
We use the properties of integral-valued polynomials to prove results about the cardinality of lattice functions. We first show that the entire space of functions in and bandlimited to some cannot be mapped in an unambiguous fashion to a the set of integral-valued polynomials.
Proposition 2
There does not exist a bijection between the set of integral-valued polynomials and the space of functions bandlimited to some .
Proof 1
The space of functions bandlimited to some is a separable Hilbert space, thus it can be identified with . By Cantor’s diagonal argument, the set of all square-summable sequences is uncountable, thus the aforementioned space of bandlimited functions is uncountable.
By Proposition 1 and (3), we see that integral-valued polynomials are a subset of the set of polynomials with rational coefficients, i.e. where for all and . We can identify the latter with the set of finite sequences of the form noting that is a countable set. Then, a subset can be identified with integral-valued polynomials. Consider now the following function
N | (5) | |||||
p_0^a_0⋅…⋅p_n^a_N-1 |
where is the -th prime number. The fundamental theorem of arithmetic—i.e. the unique-prime-factorization theorem—implies that the expression in (5) is a bijection between finite sequences and natural numbers. We can now state that the set is countable and so is since . This implies that the set of integral-valued polynomials in countable. Therefore, there does not exist a bijection between integral-valued polynomials and bandlimited functions to some .
In other words, the previous proposition indicates that the size of square-integrable bandlimited functions is larger than that of integral-valued polynomials. The previous result was, up to some extent, expected. However, we can intuitively expect now that the cardinality of integral-valued bandlimited functions in is the same as integral-valued polynomials. In effect, we show in the next proposition that this intuition is correct by demonstrating that the former set is countable. Let us first denote the set of functions bandlimited to some by where represents the Fourier Transform of the function .
Proposition 3
Consider the set . Then, the set is countable.
Proof 2
Assume , then we have since is square integrable. Moreover, for all , which together with the previous statement implies that there exist and such that for all . Due to the canonical series for functions bandlimited to , we can identify the set with finite sequences of the form where for . As shown in the proof of Proposition 2, this set of finite sequences with integer-valued elements is countable and the conclusion follows.
It is implied in the proof of Proposition 3 that the functions in possess infinitely many zeros at the integers since they decay at infinity and they are forced to take integer values, or equivalently they take nonzero values at the integers only at a finite number of times. Additionally, it is clear that do not form a dense subset within . This would be true for for example.
We can build upon the idea of finitely many zeros to construct an infinite countable subset of that can help us get closer to understanding the set . It is possible to construct a family of these functions without resorting to the canonical series. Consider the sine function that introduces the necessary rate of zeros divided by an appropriate function to force the decay at infinity according to . At the same time, the resulting function should necessarily be an entire function since we want a bandlimited function. We can choose, for example, polynomials and construct a function like where is a finite-degree polynomial. It is necessary then that the zeros of the sine function cancel those of . Therefore, it is sufficient that this polynomial presents simple zeros at the integers.
In the following result, we show how a family of functions of the form are contained in as an infinite countable proper subset. Let us first denote the least common multiple of a finite set of natural numbers by .
Proposition 4
Let be a function of the form
(6) |
where and . Then, . Moreover, is integral valued if and only if
(7) |
for some integer .
Proof 3
The polynomial in the denominator has the form
(8) |
where for and for . It is clear that the function has removable singularities at the zeros of , thus it is an entire function.
Note that there exist a and such that for . Then, we have the following
(9) |
for some . Then, it is immediate to see that .
Let us first denote the order and type of by and respectively. The function is of order one and type , thus, using Euler’s identity we have that and . It is straightforward to see that for every and there exist a large enough such that
(10) |
where . This implies that , and consequently, . By the Paley-Wiener Theorem, the function is bandlimited to .
The function has removable singularities at , thus we can define the values at these points as
(11) |
where we have applied L’Hôpital’s rule. Note that (11) is well defined as for all . In order to see this, consider some and write the derivative of the polynomial as
(12) |
Since the polynomial has simple roots, the first term in (12) will be different from zero for and the second term will vanish for all . In view of the preceding, we can also write
(13) |
Note that the product is an integer for all . In fact, it clear that it is necessary and sufficient that is a multiple of the values in order for to take integral values at the integers.
We have seen that if we properly choose the roots of and its leading coefficient , the function is an integral-valued bandlimited function in , i.e. a square-integrable lattice function for . Denote this family of functions by
(14) |
that, based on the previous result, satisfies . Note again that the generalization to an arbitrary quantizer step and sampling rate can be performed by an appropriate time warping an scaling. The set in effect forms a countable subset of square-integrable lattice functions as the next result shows.
Corollary 1
There exists a bijection between the set and .
Proof 4
Roughly speaking, the size of is the same as , however, lives within as a proper subset, i.e. where the inclusion is strict. In order to see this, let us illustrate it in the following example.
Example 2
Assume we have the functions
(15) |
where and . Note that the function satisfies the conditions imposed by Proposition 4, in fact, for any we have that . We can easily see in Fig. 4 that there does not exist such in such a way that both functions are equal for all . Thus, this counterexample clearly implies that is a proper subset of .
This proposition allows us to extract an identity—proven from a sampling-theoretic point of view—involving functions of the form (6) and finite linear combinations of sinc functions when the coefficients are appropriately chosen.
Corollary 2
Assume where satisfies (7) and . If the values are chosen such that there exists a nonzero integer satisfying for all , then the following identity holds
(16) |
Proof 5
Both sides of expression (16) vanish at . Since is chosen such that (7) is satisfied, the values of the right-hand side of the previous expression at the roots of are precisely for . The latter implies that the expression is valid for the integers. Moreover, since both sides are bandlimited to , by the sampling theorem, they agree for all .
Iv Integral-Valued Entire Functions
The main results regarding the spectrum of integral-valued bandlimited functions are based upon the interpretation of bandlimited signals as entire functions of exponential type. Loosely speaking, bandlimited signals correspond to entire functions—complex differentiable functions at every point in —with an exponential growth on the whole complex plane. In particular, a signal bandlimited to rad/s can be analytically continued to the whole complex plane in such a way that for all [12, Theorem X]. This result also establishes the connection between the bandwidth and the rate of exponential growth. The type can also be defined as
(17) |
where for .
The first results that connect the rate of growth of integral-valued entire functions with their structure can be found in [13, 14, 15]. In particular, it can be shown that if an entire function takes integer values for nonnegative integers and , this function has to be a polynomial [16, Theorem 11]. This result was extended and refined in many instances in the literature [17, 18, 19, 20, 21, 22]. We now state the result in [23] that we will be using later in connection with bandlimited signals.
Theorem 1
Assume is an entire function of exponential type , with for integers . If the type satisfies
(18) |
then is of the form , where and are polynomials. Moreover, if , then
(19) |
Essentially, these functions take the shape of a sum of finitely many terms of the form , where is an algebraic integer (a complex number which is a root of a monic polynomial with integer coefficients). Notice that it is not possible to increase the type up to since is an integral-valued entire function of exponential type .
Theorem 1 relates the type and the structure of an integral-valued entire function. However, we are interested in integral-valued bandlimited signals that are square integrable, thus we can use the previous theorem to derive a result regarding the bandwidth of such signals.
Proposition 5
If is a bandlimited function with for all integers , then the type of satisfies .
Proof 6
Due to the connection between the bandwidth of a square-integrable bandlimited signal and the type of its analytic extension, we can interpret the previous result as providing as a lower bound for the maximum frequency component of integral-valued bandlimited signals. Since these signals satisfy for all , Proposition 5 implies that their maximum frequency component has to be always greater or equal than rad/s. In summary, there are no square-integrable bandlimited signals that take integer values at integer points whose spectrum is confined to the interval rad/s.
In the following, we show how to utilize this result to construct functions with a certain bandwidth by choosing its sample values. In the next section, we will extend this result to lattice functions and will make the connection to the quantization error introduced by A/D converters.
Iv-a Constructing functions with bandwidth between and multiples of .
Proposition 5 establishes a relationship between sample values, in this case integers, and the bandwidth of a signal. We now show how to construct a function whose maximum frequency component lies in an interval based solely on appropriately choosing a subset of its sample values as integers. We make this notion precise in the following result.
Corollary 3
If is a complex square-summable sequence such that for all , then the type of the entire function constructed as
(20) |
satisfies . Moreover, there are finitely many for .
Proof 7
Since the sequence is square-summable, by Parseval’s identity we have that . The latter implies that cannot be a sum with terms of the form . According to Lemma 5, the function is of type . By construction, we also have the upper bound . The square-summability of the sequence implies that as . This implies that there exists an such that for all and the conclusion follows.
The previous result shows a way of constructing a function whose maximum frequency component is between and rad/s. We only have to construct a square-summable sequence with any value for negative integers and finitely many nonzero integer values for setting the remaining ones to zero. Notice that it is not guaranteed that any square-summable sequence in (20) generates a function whose bandwidth is . Indeed, a straightforward counterexample is oversampling. If we take the values of a sinewave at the integers whose frequency is strictly smaller than rad/s, then the sequence together with (20) will result in a function whose bandwidth is strictly smaller than .
Interestingly, we can easily extend the procedure and consider as interpolating functions for some integer . In this case, we can choose for integers . The latter are the sample values that correspond to integer sampling points in the sampling sequence. In this situation, we ensure that the bandwidth of the resulting function is between and . In other words, we decimate by a factor of the sequence for nonnegative integers and restrict those to be integers.
V Lattice Functions: Spectral Properties
We develop in this section the main result concerning the effect of quantization in the spectrum of the sequences generated by an A/D converter. In particular, we frame the result from the perspective of the Discrete-Time Fourier Transform (DTFT) of quantized sequences. We present a lower bound on the maximum frequency component of quantized sequences which represents the effect of the quantization error.
The previous sections focused on integral-valued bandlimited signals to cleanly introduce the results. Now, let us generalize these results to arbitrary bandlimited lattice functions. Assume that the lattice points are for and . The lattice functions in this case take values for some at every instant for . Assume further that these functions are bandlimited with appropriate decay conditions on the real line. It is now possible to relate their properties to those of integral-valued bandlimited functions. In order to do so, consider the lattice function for the lattice described above and construct the function
(21) |
where it follows that is an integral-valued bandlimited function. The Fourier transforms are then related by
(22) |
where is the Dirac delta function. In view of (22), the results in the previous sections can be easily extended to general lattice functions by scaling the bandwidth by a factor of . In particular, the bound in Proposition 5 will be . Similarly, the bounds derived in Corollary 3 will be of the form .
Thus, any bandlimited lattice function on a grid has spectral content at least up to rad/s irrespective of the value of or , i.e. irrespective of the resolution of the quantizer.
V-a DTFT of Quantized Discrete-Time Signals
The previous results state that any bandlimited lattice function for the lattice points has frequency components that extend at least up to rad/s. We can also take the discrete-time perspective and consider the underlying sequences involved. In particular, for the chosen parameters, bandlimited lattice functions can be seen as the bandlimited interpolation of a sequence generated by an A/D converter with appropriate sampling period and quantization step (see Fig. 5). This means that the discrete-time Fourier transform (DTFT) of the maximum frequency component of such sequences lies between and .
Corollary 4
Let be a sequence such that for , , , and all . Then, the Discrete-Time Fourier transform of defined by
(23) |
for all , satisfies for some interval .
It should be emphasized that the previous result is independent of the resolution of the quantizer, and it only depends on the fact that the sequence is quantized. Fig. 5 illustrates how any analog signal—with appropriate decay conditions—passed through an A/D converter produces a quantized sequence whose bandwidth lies in the interval described above. In other words, it is not possible to construct a quantized square-summable sequence with a bandwidth, in the sense of the DTFT, smaller than 0.8 rad/s. This also provides a fundamental limit for the bandwidth introduced by the quantization error. Assume the DTFT of the sequence following the C/D block is only allowed to be nonzero in an interval rad/s where rad/s. After the quantizer, the quantized sequence must have frequency components greater or equal than rad/s. As explained before, the fact that the quantization error includes frequency components above this frequency does not depend on the resolution of the quantizer.
Example 3
For analysis purposes, sampling and quantization can be interchanged without altering the result. Thus, it is well known that a sinusoid with frequency passed through a symmetric quantizer contains components at odd harmonics for . Sampling causes these harmonics to be aliased at for and a sampling frequency . Thus, it is clear that this quantization noise, which here takes the form of aliased harmonic distortion, is in agreement with Corollary 4.
Vi Conclusion
We presented a deterministic theoretical analysis of the signals that common A/D converters output, i.e. digital discrete-time signals. We placed the bandlimited interpolation of these signals—i.e. bandlimited lattice functions—within the framework of integral-valued entire functions to analyze its set and spectral properties. We showed their structure within the space of bandlimited functions and prove a lower bound on their maximum frequency component. This allows to interpret the influence of the quantization error in the spectrum of quantized discrete-time signals. The work shown here suggests that viewing digital discrete-time signals as integral-valued entire functions may provide a theoretical framework where robust deterministic analysis of quantization effects can be performed.
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