A CMOS Tailed Tent Map for the Generation of Uniformly Distributed Chaotic Sequences

Let $I=[a,b]$ be an interval of the real line ${\mathcal{R}}$ and consider the one-dimensional dynamical system

where $M$ is a nonsingular transformation of $I$ into itself. When designing a pseudo random number or a noise generator based on system (1) two properties are particularly important, namely i) chaotic behavior and ii) uniform probability density.

1. i)

   Chaotic behavior is assured when the system (1) has a positive Lyapunov exponent, defined as [5]:

   $$h=\lim_{T\rightarrow\infty}\frac{1}{T}\sum_{i=0}^{T-1}\ln|M^{\prime}(x_{i})|$$

   (2)

   Roughly speaking, the Lyapunov exponent is an index of the sensitivity of the system to initial conditions and therefore of its level of unpredictability. From (2) it immediately follows that systems based on everywhere expanding maps (for which $\mbox{inf}_{x\in I}|M^{\prime}(x)|>1$) are chaotic.

2. ii)

   Consider an interval $S\subseteq I$ and let $\chi_{S}$ be the relative characteristic function. Then, the “average time” spent in $S$ by an orbit originating in $x_{0}$ may be defined as

   $$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=0}^{n-1}\chi_{S}(M^{i}(x_{0})),$$

   (3)

   where $M^{i}$ is the $i$-th iterate of the map. Note that if limit (3) exists, it depends on the choice of the initial condition $x_{0}$. Birkhoff ergodic theorem [6] gives a sufficient condition for the existence of limit (3) “almost” independently from $x_{0}$. In particular, it states that, if $M$ is measure preserving¹¹1$M$ is said to be measure preserving if a measure $\mu$ exists such that $\mu(M^{-1}(A))=\mu(A)$ for any measurable set $A\subseteq I$. and ergodic then there is a unique PDF $\rho$ for $M$ such that

   $$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=0}^{n-1}\chi_{S}(M^{i}(x_{0}))=\int_{S}\rho(\xi)d\xi\ \ \ \mbox{a.e.}$$

   (4)

   for any measurable $S\subseteq I$. Basically, ergodicity provides that any initial condition randomly chosen in $I$ leads to a trajectory characterized by the same statistical properties. Furthermore, ergodicity enables extraction of information about time-based statistics from the a priori knowledge of the PDF. In particular, it can be shown that, if $M$ is ergodic, the PDF $\rho$ is the unique invariant under the Perron-Frobenius operator [6] ${\mathcal{P}}_{M}:L^{1}\rightarrow L^{1}$ defined as

   $$\int_{S}{\mathcal{P}}_{M}[\rho](\xi)d\xi=\int_{M^{-1}(S)}\rho(\xi)\,d\xi\ \ \ \forall S\subseteq I,$$

   (5)

   namely ${\mathcal{P}}_{M}[\rho]=\rho$.

   In the following, we will consider the design of the tailed tent map in order to obtain an ergodic system and we will thus determine its uniform PDF as a fixed point of ${\mathcal{P}}_{M}$. Eventually, we will verify its robustness to breakpoints misplacement both via analytic considerations and by approximating $\rho$ using time series data extracted from simulations.

When the map is implemented using an analog circuit, some misplacement of the breakpoint $B_{P}$ is unavoidable. If the ordinate of the breakpoint exceeds its assigned value due to a slope error, a parasitic stable equilibrium is introduced. In order to see how this happens, it is necessary to consider the circuit static characteristic outside the map domain. Due to the fact that the current mirrors or the current rectifier transistors (Fig. 2) reach their maximum allowable current, the map slope decreases and flattens, as shown in Fig. 3A. As a consequence a parasitic equilibrium point exists (point E in Fig. 3) characterized by $M^{\prime}(x_{E})<1$, i.e. the equilibrium is stable. Although this equilibrium always exists, it ideally lies out of the map invariant set. On the contrary, if the breakpoint ordinate is greater than its assigned value (independently of the magnitude of the misplacement), the map invariant set abruptly expands, including also point $E$. Therefore, after a transient phase, the system trajectory will converge towards it, as schematically represented in Fig. 3B. The abrupt expansion of the invariant set is due to the fact that its closure contains the unstable equilibrium point $F$. As soon as the invariant set grows to include $F$ in its interior it must also contain the whole interval $[x_{E},x_{F}]$.

Note that besides breakpoint misplacement, also noise added to the system state may allow the system trajectory to converge to E.

In order to avoid this problem, several strategies can be used. All of them are based on small modifications of the map²²2 Without loss of generality we will refer our considerations to normalized maps defined in $I=[0,1]$. and are equivalent to designing a system whose breakpoint ordinate is less than 1, to assure an adequate margin. Two typical solutions are shown in Fig. 4.

The margin one must allow mainly depends on three factors:

1. 1.

   the maximum random error one can expect for the circuit static characteristic. This originates mainly from transistors mismatching;

2. 2.

   the error introduced by the two SI sample and hold performing the delay operation. This is mainly due to clock feedthrough and appears as an undesired signal added to the system state;

3. 3.

   the error due to the dynamic response of the current comparator used to implement the breakpoint (Fig. 2). This error manifests itself as a sort of hysteresis and appears when the circuit has to switch immediately over or below threshold.

In particular, if speed is an important requirement, transistors dimensions must be reduced in order to contain parasitic capacitances effects and the first factor will have great importance. On the other hand, if a low-power circuit is desired, one has to accept a worsening of the current comparator dynamic response and a consequent increased relevance of the third factor. Therefore, particularly in fast, low power systems the required margin can be appreciable.

Adopting one of the above mentioned strategies implies sacrificing PDF uniformity. For instance, Fig. 5 shows the consequences on the PDF of modifying the map as shown in Fig. 4A.

The graphs are obtained simulating system (1) for a tent map in which the ordinate of $B_{P}$ has been decreased by 2%, i.e. for a system tolerant to breakpoint misplacements up to 2%. In Fig. 5A it is represented the case in which no misplacement occurs (ideal case), while Fig. 5B shows the worst case in which a 2% misplacement takes the breakpoint even further down.

By similar considerations, it can be seen that systems based on the Bernoulli shift suffer from the same inconveniences mentioned above [4].

A  piecewise linear map $M:I\rightarrow I$ is also a Markov map if it takes partition points into partition points, namely a partition ${\mathcal{W}}=\{(a_{0},a_{1}),\ldots,(a_{N-1},a_{N})\}$, $a_{0}=a$, $a_{N}=b$, exists such that if $Q=\{a_{0},\ldots,a_{N}\}$ then $M(Q)\subseteq Q$. It is easy to show that whenever $x_{b2}$ is a rational number, (7) is a Markov map. Moreover, in this case, $M$ satisfies also the conditions $1-4$ of Definition 3 in [7] which assure the PDF unicity and the ergodicity of the map. Thus, it is easy to prove that

is the invariant PDF. In fact, it can be easily seen that as any $x\in I$ has a finite number of counterimages, then ${\mathcal{P}}_{M}$ can be also written as

Thus, by substituting (8) in (9) one immediately gets

Knowing the invariant PDF, the Lyapunov exponent for the map can be computed as [5]

which leads to $h=x_{b2}\ln{2}$, proving system chaoticity.

First of all, note that one can design the map in such a way that point E does not exist, as shown in Fig. 6A (dot-dashed line). Furthermore, even if E exists, the alteration in the invariant set consequent to breakpoint misplacement is now graceful, i.e. the smaller the misplacement the smaller the invariant set change, as long as this set does not include the unstable equilibrium point $F$. Therefore, up to a certain misplacement the invariant set can grow without including E (Fig. 6B). More precisely, the lower $x_{b2}$, the higher the system tolerance.

As a consequence, a system based on the tailed tent map can be designed without having to compromise on the PDF in principle.

The effect of implementation errors is shown in Fig. 7.

In particular, Fig. 7A is the ideal case (i.e. the map is implemented correctly) and Fig. 7B represents the case of a 2% $B_{P1}$ misplacement. The superiority of the case in Fig. 7A over the one in Fig. 5A is evident and also the case in Fig. 7B is more similar to an ideal uniform PDF than the one in Fig. 5B, since the latter even presents a much larger range of values characterized by null probability density. The performance improvement becomes even more visible when one considers probability distributions rather than densities.

Fig. 8 shows a schematic of the proposed circuit implementation for the tailed tent map. Its left side is the same as in Fig. 2, while it right side allows to obtain the slope variation necessary to carry out the “tail.” Assuming that current mirrors $CM_{p2}$ and $CM_{n3}$ have a $1:1$ ratio, the output current is $I_{out}=I_{ref}-2|I_{in}|+\mbox{rect}(I_{in}-I_{ref2})$, where $\mbox{rect}(x)=1/2(x+|x|)$, namely a tailed tent map apart from the usual change of system reference.

Note that the extra silicon area requested to generate the tailed tent map is only a small part of the area used for the complete circuit. In fact, this includes two sample and hold circuits which use a large proportion of the total area.

Fig. 9 shows the results of device level simulations for the circuit in Fig. 8 designed using a standard $1\mu m$ CMOS technology. To obtain high linearity for the map branches, high swing cascode mirrors have been used while, to improve maximum frequency operation, a careful design of the sample and hold stage has also been considered. Fig. 9A shows a typical chaotic waveform for $I_{out}(k)$. Fig. 9B represents the approximation of the PDF obtained by a post-processing executed on the transient analysis output, for a 750 KHz operating frequency. Deviation from the ideal line is not only due to circuit misbehavior, but also to the finite number of samples used to extract the PDF.

In conclusion, the new proposed solution shows several improvements compared to previously published results, with respect to the achievable PDF [1] and parameter deviation robustness [4].