This chapter goes into the details of some of the circuit blocks used in the silicon cochleas discussed in Chapter 4. It looks at some of the basic circuit structures used in the design of various one-dimensional (1D) and two-dimensional (2D) silicon cochleas. Nearly all silicon cochleas are built around second-order low-pass or band-pass filters. These filters are usually described as second-order sections.
For the 1D cochlea, voltage-domain filters have been most commonly used, largely because of historical reasons – voltage-domain circuits were well understood – and partly because there was no compelling reason for using current-domain circuits. The first 2D cochlea (Watts et al. 1992) indeed was also designed using voltage mode second-order filters, but the use of current mode second-order filters greatly simplifies the implementation of the resistive grid, modeling the fluid in the 2D cochlea (see Chapter 4). Hence, later versions of the 2D cochlea have used current-domain (i.e., log-domain) filters.
In this chapter, we will first look at the voltage-domain second-order filters, followed by the log-domain second-order filter. Both the 1D and the 2D cochleas, whether implemented in the voltage or current domain, need exponentially decreasing currents to bias the filters, which we discuss in a separate section. As discussed in Chapter 4, the inner hair cells (IHC) stimulate the auditory nerve neurons and cause them to fire. At the end of this chapter an implementation of the IHC will be presented.
The second-order filters of the 1D voltage-domain cochlea are built from three transconductance amplifiers. We will first introduce the transconductance amplifier and then discuss the second-order filters. We also give a detailed analysis of the operation of these circuits.
The transconductance amplifier in its most basic version is shown in Figure 9.1a. When biased in weak inversion it has a hyperbolic tangent transfer function given by
with Ibias, V+, and V_ as in Figure 9.1, and n is the weak inversion slope factor which depends on the technology and normally has a value somewhere between 1 and 2. The thermal voltage UT is given by UT = kT/q, with k the Boltzmann constant, T the temperature in Kelvin, and q the charge of an electron. UT is about 25 mV at room temperature.
The transconductance amplifier is biased in weak inversion when the current through the differential pair transistors is much smaller than the specific current of these transistors, that is,
where W is the channel width of the transistor, L the length, μ the mobility of the minority carriers, and Cox the gate oxide capacitance per unit area. The specific current depends on process parameters, the geometry of the transistor, and temperature (Vittoz 1994). For small inputs (|V+ − V_| < 60 mV) we can approximate the amplifier as a linear transconductance:
with the transconductance given by
This last equation shows that the transconductance of the amplifier is proportional to the bias current when the amplifier operates in weak inversion.
As we will see in Section 9.2.4, we will also need a transconductance amplifier with an enlarged linear range as shown in Figure 9.1b. The transfer function of this amplifier is given by
and
The linear range of this amplifier is thus enlarged by a factor n + 1, where n is about 1.5, and the transconductance is reduced by a factor n + 1 with respect to the normal transconductance amplifier with the same bias current. The transfer functions of both amplifiers are shown in Figure 9.2.
A second-order low-pass filter can be made with three transconductance amplifiers (A1, A2, A3 as shown in Figure 9.3) and two capacitors. The transfer function of this filter is given in the Laplace domain by
where s = jω, j2 = −1, and ω is the angular frequency, the time constant τ = C/gm when both A1 and A2 have a conductance gτ and both capacitors have capacitance C. The quality factor Q of the filter can be expressed as
where gQ is the conductance of the amplifier A3. The gain and phase response of the filter described by Eq. (9.7) are shown in Figure 9.4 for two values of Q.
From Eq. (9.8), it is clear that Q becomes infinite when gQ = 2gτ, and from Eq. (9.7) we can then see that when ω = 1/τ, the gain of the filter also becomes infinite and the filter will be unstable. This gives gQ < 2gτ as the small-signal stability limit of the filter. However, the filter of Figure 9.3 also has a large-signal stability limit, which puts a more stringent constraint on the value of Q. In order to obtain the transfer function of Eq. (9.7), we have treated the filter as a linear system. This approximation is only valid for small input signals. It has been shown by Mead (1989) that large transient input signals can create a sustained oscillation in this filter. During most of this oscillation, all three amplifiers are saturated so that their output is either plus or minus their bias current. We can therefore adopt a piece-wise linear approach to analyze this situation in which we treat the amplifiers as current sources. The following analysis is mostly adapted from Mead’s, but is more general since it does not assume that the amplitude of the oscillation is equal to the supply voltage.
When Vin suddenly increases by a large amount, the amplifier A1 will saturate and will charge the capacitor at its output with its maximum output current Iτ. If, at the same time, V1 is larger than Vout, A3 will also charge the capacitor with its maximum output current IQ and we can write for V1:
where Vout ≪ V1 ≪ Vin. V1 will thus rise at its maximum rate. Once V1 catches up with Vin, the output current of A1 changes sign and we write for V1:
where Vout ≪ Vin ≪ V1.
In order to have Q larger than one, IQ has to be larger than Iτ so that in this case V1 will continue to increase with a smaller slope until it reaches the positive power supply Vdd or until Vout catches up with V1. As long as V1 stays larger than Vout, we can write in our piece-wise linear approach for Vout:
where Vout ≪ V1.
Once Vout catches up with V1, the sign of the output of A3 will change, and V1 will start its steep descent, until V1 goes below Vin, when the sign of the output of A1 changes and V1 descends more slowly to the negative power supply Vss, or until Vout catches up again. Figure 9.5 sketches the behavior of the circuit according to the above equations. The thick line shows the evolution of V1 and the thin line shows the same for Vout. Whenever Vout catches up with V1, the change in both voltages will switch direction.
By comparing the voltages at which Vout catches up with V1 at the start and the end of a single rise and fall cycle, we can determine the nature of the oscillation. If ΔV (see Figure 9.5) is positive, the amplitude of the oscillation will decrease during each period and the oscillation will cease after a certain time. The limit of stability is reached when ΔV becomes zero, so that the amplitude of the oscillation stays constant. For the rising part of the oscillation we write:
where VL and VH are as shown in Figure 9.5. Similarly, for the falling part, we write when ΔV equals zero:
Equations (9.12) and (9.13) can only be satisfied when VL = VH. Substituting V for VL and VH in either equation, and dividing by CV/Iτ yields
Rewriting this equation, we obtain the following solution:
This gives the critical value for large-signal stability of the low-pass filter of Figure 9.3. Since the conductance of the amplifiers is directly proportional to the bias currents, this large-signal stability condition also limits gQ/gτ to this value, and thus limits Q to a maximum value of 2.62 (see Eq. 9.8). The large-signal stability limit therefore severely limits the maximum quality factor of the filter when using the basic transconductance amplifier.
The circuit can be improved by using two wide-range transconductance amplifiers to implement A1 and A2 (Figure 9.6) and a basic transconductance amplifier for A3 (Watts et al. 1992) (see Figure 9.1b). In this case we can write for the conductance ratio:
which ensures that gQ/gτ becomes 2, that is, Q becomes infinite, before IQ/Iτ becomes 1.62, since n is larger than 1. Thus the filter is always large-signal stable whenever it is small-signal stable.
With the two wide-range transconductance amplifiers A1 and A2, and the one basic transconductance amplifier A3, we have a second-order low-pass filter for which we can set the cutoff frequency and the quality factor using the bias currents of these amplifiers. By cascading these filters and biasing the amplifiers with exponentially decreasing currents, we can then create a model of the basilar membrane. The limited input linear range of the second-order LPF in Figure 9.6 can be increased by modifying the amplifiers A1 and A2 so that the inputs go to the well terminals instead of the gate terminals of the input differential pair transistors (Sarpeshkar et al. 1997).
The voltage Vout (Figure 9.6) at the output of each second-order stage in the cochlear filter cascade represents the displacement of a small section of the basilar membrane. However, since the stimulation of the inner hair cells in the biological cochlea is proportional to the velocity of the basilar membrane, the output of each second-order stage has to be differentiated. This can be done by creating a copy of the output current Idif of amplifier A2 at every stage as in Watts et al. (1992). Since the voltage on a capacitor is proportional to the integral of the current onto the capacitor, Idif is effectively proportional to the basilar membrane velocity. Yet, with equal displacement amplitudes, velocity will be much larger for high frequencies than for low frequencies, yielding output signals with amplitudes that decrease from the beginning of the cochlea to the end. This can be corrected by normalizing Idif to give equal amplitude at every output. A resistive line controlling the gain of the current mirrors that create the copies of Idif at each stage is used for this purpose by Watts et al. (1992). However, this resistive line introduces an extra source of mismatch in the circuit.
An alternative solution which does not need normalization is to take the difference between Vout and V1 (see Figure 9.7). We can rewrite Eq. (9.3) applied to A2 as
or
This is equivalent to differentiating Vout, with 0 dB gain at the cutoff frequency for all stages. Figure 9.8 shows the gain and phase response of the filter after differentiation. We can see that a single band-pass filter only has a shallow high frequency cutoff slope of 20 dB per decade. In the filter cascade, however, the 40 dB per decade cut-off slopes of the individual low-pass filters will be accumulated (Figure 9.4). This can yield very steep high frequency cut-off slopes, as we will see in the measurements later on in Figure 9.12.
A current output Idif can be taken from the output of the section by adding an additional transconductance amplifier A4 as shown in Figure 9.7, which ensure that:
where gm is the transconductance of this amplifier.
In this section, we introduce the log-domain circuits used in the implementation of the 2D silicon cochlea presented in Hamilton et al. (2008a, 2008c). While these circuits are not specific to the 2D silicon cochlea, they provide a good introduction to log-domain filters and to operating circuits in the current mode; a good counterpoint to the circuits presented in Section 9.2.
Log-domain filters are based on translinear loops. The translinear loop is a fundamental concept in log-domain circuit design (Gilbert 1975). It is described in Gilbert (1990) as follows:
In a closed loop containing an even number of forward-biased junctions, arranged so that there are an equal number of clockwise-facing and counter-clockwise-facing polarities, the product of the current-densities in the clockwise direction is equal to the product of the current densities in the counter-clockwise direction.
This concept is based on the relationship
and, as such, is only valid when current through a device is an exponential function of the voltages at the terminals of the device, as is the case for BJTs and MOSFETs operating in the sub-threshold region. This concept can be best described when looking at the circuit in Figure 9.9.
In Figure 9.9, M1 and M3 comprise the counter-clockwise junctions while M2 and M4 comprise the clockwise junctions. Transistor M5 is necessary to correctly bias the source
voltage of both M2 and M3 and sink the current running through M2 and M3. Hence, given the definition above we can say:
or, defining the current through M2 as the output of the circuit, we can write:
Equation (9.21) demonstrates the ease with which a multiplier circuit can be constructed when operating in the log-domain. Understanding the principle of translinear loops can also be an important tool when analyzing the function of other log-domain circuits.
The tau cell (van Schaik and Jin 2003) is a basic building block representing a class of log-domain filters. A schematic of the tau cell is shown in Figure 9.10. The tau cell is designed for complete programmability via the time constant, τ, and the current feedback, Aiv. It can be used as a building block to create a number of more complex, higher order filters.
The tau cell is based on the principle of translinear loops and its core structure is identical to that of Figure 9.9. In Figure 9.10, the closed loop of gate-source junctions necessary to form a translinear loop is created by transistors M1 to M4 and hence,
The capacitor C introduces dynamics into the translinear loop resulting in a filter. The current through the capacitor is given by
If we assume that the gate capacitance of M3 and M4 is significantly smaller than C, that is, if we assume that the dynamics at the gate of M3 is much faster than at source of M3, we can simplify Eq. (9.24) with IM3 = I0 and using Eq. (9.23) to eliminate IM2:
In addition, since M3 and M4 have the same gate voltage and assuming the voltage at the drain of M4 is at least 100 mV larger than Vref, we can write:
Equation (9.26) implies that
We can use this result to eliminate VC from Eq. (9.25):
In the Laplace domain, the transfer function for a single tau cell can thus be written as
where τ = is the time constant, τi is the time constant of stage i, Ii−1 is the input current, Ii is the output current of stage i and Ii+1 = Ti+1Ii. If there is no next stage then Ti+1 = 0 and Ai = 0 by definition.
A second-order low-pass filter can be realized by connecting two tau cells as illustrated in Figure 9.11. Here we see that the first cell has a feedback gain A1, while the second tau cell has no feedback. The current feedback, A1I0Ii+1/Ii, can be implemented using the multiplier of Figure 9.9. A more optimized method to implement the current feedback is described in Hamilton et al. (2008b). The schematic view of the second-order low-pass filter is given in Figure 9.12. The general equation for the second-order low-pass filter in Figure 9.12 is given by
where τ is the time constant and Q is the quality factor.
In order to construct a band-pass filter similar to that described in Section 9.2.5, the output from the second tau cell in Figure 9.12, Iout2, must be subtracted from the output of the first tau cell in Figure 9.12, Iout1, such that
where Iout is the output from the band-pass filter. Figure 9.13 shows a band-pass filter structure that is described by Eq. (9.31).
Since there is an exponential relationship between position along the basilar membrane and best frequency in the real cochlea, we will need to use filters with exponentially decreasing cut-off frequencies in our model. In all the silicon cochlear models mentioned in Chapter 4, the exponential dependency is obtained using a linear decreasing voltage on the gates of MOS transistors operating in weak-inversion. In weak-inversion, the drain current of a saturated nFET with its source tied to the bulk and its gate voltage referred to the same bulk can be expressed by
with IS as defined in Eq. (9.2) and VT0 the threshold voltage of the transistor. This shows that the drain current depends exponentially on the gate voltage. A spatial voltage distribution which decreases linearly with distance is easily created using a resistive polysilicon line; if there is a voltage difference between the two ends of the line, the voltage on the line will decrease linearly all along its length. It is therefore possible to create a filter cascade with an exponentially decreasing cut-off frequency by biasing the amplifiers of Figure 9.6 using MOS transistors whose gates are connected by equal lengths of the polysilicon line (Lyon and Mead 1988). As we can see in Eq. (9.32), however, the drain current also depends exponentially on the threshold voltage and small variations in VT0 will introduce large variations in the drain current. Because both the cut-off frequency and the quality factor of the filters are proportional to these drain currents, large parameter variations are generated by small VT0 variations. A root mean square (RMS) mismatch of 12% in the drain current of two identical transistors with equal gate and source voltages is not exceptional (Vittoz 1985), even when sufficient precautions are taken. We can circumvent this problem by using CMOS compatible lateral bipolar transistors (CLBTs) as bias transistors. A CLBT is obtained if the drain or source junction of a MOS transistor is forward-biased in order to inject minority carriers into the local substrate. If the gate voltage is negative enough (for an n-channel device), then no current can flow at the surface and the operation is purely bipolar (Arreguit 1989; Vittoz 1983). Figure 9.14 shows the major flows of current carriers in this mode of operation, with the source, drain, and well terminals renamed emitter E, collector C, and base B.
Since there is no p+ buried layer to prevent injection to the substrate, this lateral npn bipolar transistor is combined with a vertical npn. The emitter current IE is thus split into a base current IB, a lateral collector current IC, and a substrate collector current ISub. Therefore, the common-base current gain B = −IC/IE cannot be close to 1. However, due to the very small rate of recombination inside the well and to the high emitter efficiency, the common-emitter current gain E = IC/IB can be large. Maximum values of E and B are obtained in concentric structures using a minimum size emitter surrounded by the collector and a minimum lateral base width. For VCE = VBE − VBC larger than a few hundred millivolts, this transistor is in active mode and the collector current is given, as for a normal bipolar transistor, by
where ISb is the specific current in bipolar mode, proportional to the cross section of the emitter-to-collector flow of carriers. Since IC is independent of the MOS transistor threshold voltage VT0, the main source of mismatch of distributed MOS current sources is suppressed, when CLBTs are used to create the current sources. A disadvantage of the CLBT is its low early voltage, that is, the device has a low output resistance. Therefore, it is preferable to use a cascode circuit as shown in Fig. 9.15a. This yields an output resistance several hundred times larger than that of the single CLBT; whereas the area penalty, in a layout as shown in Figure 9.15b, is acceptable (Arreguit 1989).
Another disadvantage of CLBTs, when biased using a resistive line, is their base current, which introduces an additional voltage drop on the resistive line. However, since the cutoff frequencies in the cochlea are controlled by the output current of the CLBTs and since these cut-off frequencies are relatively small (typically 20 kHz or less), the output current of the CLBTs will be small. If the common-emitter current gain E is much larger than 1, the base current of these CLBTs will be very small compared to the current flowing through the resistive line and the voltage error introduced by the small base currents will be negligible. Furthermore, since the cut-off frequencies of the cochlea will typically span two decades with an exponentially decreasing cut-off frequency from the beginning to the end, only the first few filters will have any noticeable influence on the current drawn from the resistive line.
As shown in Section 9.2.5 and Figure 9.7, both the voltage-domain and the current-domain silicon cochleae typically have a current output signal for each section representing the velocity of the BM vibration. This current then serves as the input to an IHC circuit, modeling the transduction from vibration to a neural signal. There have been a number of IHC circuit models developed since the first, proposed in Lazzaro and Mead (1989). The more complex circuit models, such as the one presented in McEwan and van Schaik (2004), aim to reproduce the IHC behavior in great detail, including all the various time constants. Here we discuss the IHC model proposed in Chan et al. (2006) that models the two main properties of the IHC, namely, half-wave rectification and low-pass filtering. The circuit for this IHC model is shown in Figure 9.16.
In this circuit a current which represents the band-pass output of a cochlea section, Ic, is half-wave rectified by a current mirror. A DC offset can be added by using Vioff to set the current Ioff. This half wave-rectified current is, as a first approximation, given by
The output of the half-wave rectifier is passed through a first-order log-domain low-pass filter that is based on the tau cell of Figure 9.10, except that it uses pFETs rather than nFETS. The transfer function is nonetheless the same and is given in the Laplace domain by
where G is given by
The time constant of the low-pass filter is given τ = , where C is modeled by an MOS capacitor. The cut-off is set around 1 kHz as in the biological IHC, modeling the reduction in phase-locking observed on real auditory nerves at frequencies greater than 1kHz. The two control signals Vref and Vref0 are slightly below Vdd to allow the two pFETs providing 2I0 to operate in saturation. Any voltage difference between Vref and Vref0 will show up as a current gain depending exponentially on this difference as given in Eq. (9.36).
As shown in Chapter 4, the biological IHC exhibits adaptation to an ongoing stimulus. As a result it responds more strongly to the onset rather than the sustained part of a stimulus. This is facilitated by temporarily suppressing its response after the offset of stimulation. This adaptation can be directly modeled in a more complex IHC circuit, as in McEwan and van Schaik (2004), or it can be emulated by using the output of the IHC to stimulate a neuron with an adaptive threshold. In the second case it is actually the neuron that has a stronger response to the onset of a sound, rather than the IHC, but the end result is very similar. Silicon neurons are discussed in great detail in Chapter 7.
In this chapter we have described circuits for the main building blocks of the silicon cochlea models described in Chapter 4. As discussed in Chapter 4, there are still many open issues in analog silicon cochlea design today; for example, whether the operating domain of these circuits should be current or voltage. The circuits in this chapter face the same design challenges usually of analog designs: noise, dynamic range, and mismatch to name but a few. While current-mode circuits are usually more compact than voltage-mode circuits and have a larger dynamic range, they are more susceptible to variations in threshold voltages. To reduce the mismatch in the fabricated devices, transistors have to be sized properly and calibration circuits are needed. These tradeoffs are taken by the designer in the various cochlea designs. While the design of silicon cochleas has come a long way since the first implementation by Lyon and Mead (1988), we still have a long way to go to match the remarkable performance of biological cochleas. Several research groups worldwide are actively researching ways to get closer to this goal.
Arreguit X. 1989. Compatible Lateral Bipolar Transistors in CMOS Technology : Model and Applications. PhD thesis Ecole Polytechnique Fédérale Lausanne, Switzerland.
Chan V, Liu SC, and van Schaik A. 2006. AER EAR: a matched silicon cochlea pair with address event representation interface. IEEE Trans. Circuits Syst. I: Special Issue on Smart Sensors 54(1), 48–59.
Gilbert B. 1975. Translinear circuits: a proposed classification. Electron Lett. 11, 14–16.
Gilbert B. 1990. Current-mode circuits from a translinear viewpoint: a tutorial. In: Analogue IC Design: The Current-Mode Approach (eds Toumazou C, Lidgley FJ, and Haigh DG). Peter Peregrinus Ltd. pp. 11–93.
Hamilton TJ, Jin C, van Schaik A, and Tapson J. 2008a. A 2-D silicon cochlea with an improved automatic quality factor control-loop. Proc. IEEE Int. Symp. Circuits Syst. (ISCAS), pp. 1772–1775.
Hamilton TJ, Jin C, van Schaik A, and Tapson J. 2008b. An active 2-D silicon cochlea. IEEE Trans. Biomed. Circuits Syst. 2(1), 30–43.
Hamilton TJ, Tapson J, Jin C, and van Schaik A. 2008c. Analogue VLSI implementations of two dimensional, nonlinear, active cochlea models. Proc. IEEE Biomed. Circuits Syst. Conf. (BIOCAS), pp. 153–156.
Lazzaro J and Mead C. 1989. Circuit models of sensory transduction in the cochlea. In: Analog VLSI Implementations of Neural Networks (eds Mead C and Ismail M). Kluwer Academic Publishers. pp. 85–101.
Lyon RF and Mead CA. 1988. An analog electronic cochlea. IEEE Trans. Acoust. Speech Signal Process. 36(7), 1119–1134.
McEwan A and van Schaik A. 2004. An alternative analog VLSI implementation of the Meddis inner hair cell model. Proc. IEEE Int. Symp. Circuits Syst. (ISCAS), pp. 928–931.
Mead CA. 1989. Analog VLSI and Neural Systems. Addison-Wesley, Reading, MA.
Sarpeshkar R, Lyon RF, and Mead CA. 1997. A low-power wide-linear-range transconductance amplifier. Analog Integr. Circuits Signal Process. 13, 123–151.
van Schaik A, Fragnière E and Vittoz E. 1996. Improved silicon cochlea using compatible lateral bipolar transistors. In: Advances in Neural Information Processing Systems 11 (NIPS) (eds. Touretzky DS, Mozer MC, and Hasselmo, ME). MIT Press, Cambridge, MA. pp. 671–677.
van Schaik A and Jin C. 2003. The tau-cell: a new method for the implementation of arbitrary differential equations. Proc. IEEE Int. Symp. Circuits Syst. (ISCAS), pp. 569–572.
Vittoz E. 1983. MOS transistors operated in the lateral bipolar mode and their application in CMOS technology. IEEE J. Solid-State Circuits SC-24, 273–279.
Vittoz E. 1985. The design of high-performance analog circuits on digital CMOS chips. IEEE J. Solid-State Circuits SC-20, 657–665.
Vittoz EA. 1994. Analog VLSI signal processing: why, where, and how? J.VLSI Signal Process. Syst. Signal Image Video Technol. 8(1), 27–44.
Watts L, Kerns D, Lyon R, and Mead C. 1992. Improved implementation of the silicon cochlea. IEEE J. Solid-State Circuits 27(5), 692–700.