Analog Computation Circuits

Analog computation circuits perform mathematical operations directly on continuous electrical signals, enabling real-time processing without the latency and complexity of digital conversion. These circuits form the foundation of analog computers, signal processing systems, and measurement instrumentation where continuous-time computation offers advantages in speed, simplicity, or power efficiency over digital approaches.

From basic operations like multiplication and division to complex functions including logarithms, trigonometric calculations, and statistical operations, analog computation circuits exploit the physical relationships between voltage, current, and component behavior to implement mathematical transformations. Understanding these circuits enables engineers to design sophisticated signal processing systems, implement control algorithms in hardware, and create measurement instruments that extract meaningful parameters from complex waveforms.

Fundamentals of Analog Computation

Analog computation represents mathematical variables as continuous physical quantities, typically voltages or currents, and performs operations using the natural behavior of electronic components. Unlike digital computation, which discretizes values into binary representations, analog circuits maintain continuous signal fidelity throughout the computation process.

Representation and Scaling

In analog computation systems, voltages typically represent variables with a defined scale factor. For example, a system might represent physical quantities using a scale of 1 volt per unit, where a 3.7-volt signal represents a value of 3.7. The choice of scale factor involves trade-offs:

Dynamic range: The ratio between the largest and smallest signals that can be accurately represented, limited by power supply rails and noise floor
Signal-to-noise ratio: Larger signals provide better noise immunity but risk saturation
Linearity: Most components exhibit better linearity over limited voltage ranges
Power consumption: Higher voltage swings generally require more power

Time Representation

Analog computers can operate in real-time, where one second of computation time corresponds to one second of problem time, or in scaled time, where computations run faster or slower than real-time. Time scaling proves valuable for studying slow phenomena quickly or analyzing fast transients in slow motion for detailed observation.

Accuracy Considerations

Analog computation accuracy depends on component tolerances, thermal drift, noise, and nonlinearities. While digital systems can achieve arbitrary precision through word length selection, analog accuracy typically ranges from 0.01% to 1% depending on circuit complexity and component quality. For many applications, this accuracy proves sufficient while offering advantages in speed and simplicity.

Analog Multipliers and Dividers

Multiplication and division of analog signals enable a wide range of computations including power measurement, modulation, automatic gain control, and implementation of mathematical functions. Several circuit techniques achieve these nonlinear operations with varying performance characteristics.

Multiplier Fundamentals

An ideal analog multiplier produces an output proportional to the product of two input signals: Vout = K * Vx * Vy, where K is the scale factor (typically 0.1 V^-1 for 10-volt systems). Key performance parameters include:

Bandwidth: The frequency range over which multiplication accuracy is maintained
Total error: Deviation from ideal multiplication, including offset, linearity, and feedthrough errors
Quadrant operation: Whether the multiplier handles one, two, or four polarities of input combinations
Feedthrough: Output signals appearing when one input is zero

Transconductance Multipliers

Transconductance multipliers exploit the linear relationship between collector current and transconductance in bipolar transistors. The Gilbert cell, invented by Barrie Gilbert in 1968, forms the basis of most integrated analog multipliers. This topology uses a cross-coupled differential pair whose tail current is controlled by the second input signal:

Four-quadrant operation: Handles all combinations of positive and negative inputs
Wide bandwidth: Limited primarily by transistor fT, enabling operation to hundreds of megahertz
Good linearity: Achieves total errors below 1% with proper design
Temperature stability: Differential operation provides first-order temperature cancellation

Log-Antilog Multipliers

Log-antilog multipliers use the logarithmic relationship between transistor base-emitter voltage and collector current. The multiplication process follows three steps:

Logarithmic conversion: Both inputs pass through log amplifiers, producing voltages proportional to the logarithm of each input
Addition: The log outputs are summed, equivalent to multiplying the original signals (log A + log B = log AB)
Antilog conversion: An exponential amplifier converts the sum back to a linear voltage representing the product

Log-antilog multipliers achieve excellent accuracy but are limited to single-quadrant operation (positive inputs only) and lower bandwidths compared to transconductance designs.

Variable-Transconductance Multipliers

Operational transconductance amplifiers (OTAs) provide another multiplication approach. The OTA output current is the product of the input voltage and a bias-controlled transconductance: Iout = gm * Vin, where gm is proportional to the bias current. Using one signal to control the bias current and the other as the input voltage creates a multiplier:

Simple implementation: Requires only an OTA and current-to-voltage converter
Two-quadrant operation: Vin can be bipolar, but bias current must remain positive
Electronic tunability: Well-suited for voltage-controlled applications

Analog Dividers

Division circuits produce an output proportional to the ratio of two inputs: Vout = Vx / Vy * Vref. Division is inherently more difficult than multiplication because the output must increase without bound as the denominator approaches zero. Practical implementations include:

Feedback multiplier: Places a multiplier in the feedback path of an operational amplifier, forcing the output to equal Vx / Vy
Log-antilog divider: Subtracts logarithms rather than adding them (log A - log B = log A/B)
Variable gain amplifier: Uses the divisor to control amplifier gain with the dividend as input

All division techniques must limit the denominator to prevent overflow when Vy approaches zero. Most practical dividers specify a minimum denominator value below which accuracy degrades or the output saturates.

Multiplier Applications

Analog multipliers find extensive use in:

Power measurement: Multiplying voltage and current to compute instantaneous power
Modulation and mixing: AM modulation, frequency mixing, and phase detection
Automatic gain control: Adjusting signal amplitude based on feedback
Trigonometric functions: Computing squares, square roots, and ratios
Control systems: Implementing nonlinear control laws and compensation

Square Root and RMS Circuits

Square root extraction and root-mean-square (RMS) computation represent essential operations in measurement systems, enabling accurate characterization of AC signals and power calculations regardless of waveform shape.

Square Root Extraction

Square root circuits produce an output whose square equals the input: Vout^2 = Vin * Vref. Implementation approaches include:

Feedback multiplier: Connecting a multiplier to square its output and comparing to the input forces the output to equal the square root
Log-antilog: Taking the logarithm, dividing by two, and exponentiating: antilog(log(Vin)/2) = sqrt(Vin)
Piecewise linear approximation: Using breakpoint circuits to approximate the square root function

Square root circuits find applications in vector magnitude computation (computing sqrt(X^2 + Y^2)), linearizing sensors with square-law characteristics, and demodulating frequency-modulated signals.

RMS-to-DC Conversion

The RMS (root-mean-square) value of a signal represents its equivalent DC heating value and provides a meaningful measure of AC signal amplitude regardless of waveform. The RMS value is defined as:

Vrms = sqrt((1/T) * integral(V(t)^2 dt))

True RMS converters implement this definition directly through three operations:

Squaring: An analog multiplier computes V^2(t)
Averaging: A low-pass filter computes the mean of the squared signal
Square root: A square root circuit extracts the final RMS value

Implicit RMS Computation

An elegant alternative topology uses implicit computation through feedback. A squarer/divider computes Vin^2/Vout, and this result passes through an averaging filter whose output becomes the circuit output. At equilibrium, the average of Vin^2/Vout equals Vout, implying Vout = sqrt(average(Vin^2)) = Vrms. This approach offers advantages:

Wide dynamic range: Division by the output provides automatic ranging
Simplified square root: No explicit square root circuit required
Good crest factor handling: Maintains accuracy with peaked waveforms

Thermal RMS Converters

For the highest accuracy, thermal RMS converters use matched heating elements and temperature sensors. The input signal heats one element while a DC feedback signal heats a matched element. A servo loop adjusts the DC signal until both temperatures match, at which point the DC value equals the true RMS of the input. Thermal converters achieve accuracy better than 0.01% and respond correctly to signals with arbitrarily high crest factors, making them the reference standard for AC voltage measurement.

RMS Converter Specifications

Key parameters for RMS converter selection include:

Bandwidth: The frequency range over which RMS accuracy is maintained
Crest factor: The ratio of peak to RMS value that the converter handles accurately
Averaging time constant: Determines settling time and low-frequency response
DC accuracy: Performance with pure DC input signals

Integrators and Differentiators

Integration and differentiation, the fundamental operations of calculus, find extensive application in analog computation for solving differential equations, implementing control algorithms, and processing signals in ways that emphasize particular signal characteristics.

Operational Amplifier Integrators

The Miller integrator uses a capacitor as the feedback element in an inverting amplifier configuration. The output voltage represents the time integral of the input:

Vout = -(1/RC) * integral(Vin dt) + Vinitial

where RC is the integration time constant. Key design considerations include:

Time constant selection: RC determines the integration rate; larger values produce slower integration
Initial conditions: A reset switch across the capacitor establishes starting voltage
DC stability: Op-amp offset and bias currents cause output drift; requires periodic resetting or DC stabilization techniques
Capacitor selection: Low-leakage capacitors (polypropylene, polystyrene) minimize droop

Practical Integrator Enhancements

Several techniques improve integrator performance in practical applications:

Parallel reset resistor: A large resistor in parallel with the capacitor limits DC gain and prevents saturation from offset currents, converting the ideal integrator to a low-pass filter at DC
Chopper stabilization: Modulating the signal before integration and demodulating afterward cancels DC errors
Switched reset: Periodically shorting the capacitor clears accumulated errors, useful in repetitive measurement applications
Servo feedback: A slow feedback loop corrects for drift while allowing fast integration

Differentiators

A differentiator produces an output proportional to the rate of change of the input:

Vout = -RC * d(Vin)/dt

The basic differentiator places a capacitor in the input path of an inverting amplifier. However, ideal differentiation presents practical challenges:

Noise amplification: High-frequency noise is amplified proportionally to frequency
Stability: The increasing gain with frequency can cause oscillation
Bandwidth limitation: Practical differentiators must limit high-frequency gain

Practical Differentiator Design

Real differentiators include high-frequency rolloff to ensure stability and limit noise:

Series input resistor: Adding a resistor in series with the input capacitor creates a high-frequency breakpoint
Feedback capacitor: A small capacitor in parallel with the feedback resistor provides additional rolloff
Design rule: Keep the differentiator bandwidth at least a decade below the op-amp's unity-gain frequency

The resulting circuit differentiates accurately at low frequencies while rolling off gain at high frequencies, providing a practical compromise between mathematical accuracy and circuit stability.

Applications of Integrators and Differentiators

These circuits enable numerous applications:

Analog computation: Solving ordinary differential equations in real-time
PID control: Integral and derivative terms in control loops
Waveform generation: Triangle waves from square waves (integration) or pulse generation from triangles (differentiation)
Phase shifting: Integrators and differentiators shift phase by 90 degrees
Velocity and acceleration: Differentiating position gives velocity; integrating acceleration gives velocity

Analog-to-Logarithmic Converters

Logarithmic amplifiers produce an output proportional to the logarithm of the input, compressing wide dynamic ranges into manageable voltage spans. This nonlinear operation proves valuable in measurement systems, automatic gain control, and as a building block for multiplication and other mathematical functions.

Logarithmic Conversion Principle

The base-emitter voltage of a bipolar transistor varies logarithmically with collector current over many decades:

Vbe = (kT/q) * ln(Ic/Is)

where k is Boltzmann's constant, T is absolute temperature, q is electron charge, and Is is the saturation current. At room temperature, kT/q equals approximately 26 mV, producing about 60 mV per decade of current change.

Basic Log Amplifier

A basic logarithmic amplifier places a transistor in the feedback path of an operational amplifier. The input voltage creates a current through the input resistor that becomes the collector current of the feedback transistor. The output voltage equals:

Vout = -(kT/q) * ln(Vin / (Rin * Is))

This configuration accepts only positive input voltages (or currents) and produces a negative logarithmic output.

Temperature Compensation

The kT/q factor causes the logarithmic slope to vary with temperature, changing by approximately 0.3% per degree Celsius. Additionally, the saturation current Is doubles roughly every 10 degrees, shifting the intercept. Compensation techniques include:

Matched transistor pairs: Using two transistors with matched Is values in a ratiometric configuration cancels Is variation
Temperature-proportional gain: A gain stage with positive temperature coefficient compensates for the kT/q variation
Reference current: Comparing the input to a temperature-stable reference current through matched devices provides inherent compensation

Antilog (Exponential) Amplifiers

The inverse operation, antilog or exponential amplification, produces an output that is the exponential of the input:

Vout = Vref * exp(Vin / Vt)

where Vt = kT/q. Interchanging the positions of the resistor and transistor in the log amplifier topology creates an antilog amplifier. The same temperature compensation techniques apply but in reverse roles.

Multi-Decade Log Amplifiers

For applications requiring accurate logarithmic conversion over many decades (such as optical power measurement or spectrum analyzer displays), specialized multi-stage log amplifiers extend the useful range:

Successive detection: Multiple limiting amplifier stages, each detecting when its output limits, combine to create a piecewise-logarithmic approximation
True log stages: Cascaded transistor-based stages with appropriate scaling cover different amplitude ranges
Dynamic range: Modern log amplifiers achieve 80 dB or more of accurate logarithmic response

Applications

Logarithmic amplifiers serve diverse applications:

Decibel scaling: Converting linear amplitude to logarithmic dB representation
Dynamic range compression: Reducing the amplitude range of signals for display or transmission
Analog computation: Multiplication and division through log addition and subtraction
Optical power measurement: Fiber optic receivers and laser monitoring
AGC systems: Deriving control signals proportional to signal level in dB

Trigonometric Function Generators

Generating sine, cosine, tangent, and inverse trigonometric functions enables coordinate transformations, vector computations, phase control, and the solution of geometric problems in analog hardware.

Sine and Cosine Generation

Several techniques produce trigonometric functions from DC or low-frequency input signals representing angles:

Piecewise linear approximation: Breakpoint circuits create straight-line segments that approximate the sine curve, achieving accuracies of 1-2% with sufficient segments
Polynomial approximation: Implementing Taylor series terms (x - x^3/6 + x^5/120 - ...) using multipliers and summers
Triangle-to-sine shaping: Nonlinear networks that convert a triangle wave to an approximate sine wave
Resolver-based: Using the mechanical angle of a resolver as input and its electrical outputs as sine/cosine

Function Fitting Networks

For generating arbitrary single-valued functions, diode function fitting networks use multiple diodes with different bias points to create piecewise-linear approximations. Each diode conducts when the input exceeds its threshold, adding a slope change to the transfer characteristic. By selecting appropriate thresholds and resistors, designers can approximate sine, cosine, logarithmic, or custom transfer functions.

Coordinate Transformations

Rectangular-to-polar and polar-to-rectangular conversions require trigonometric computations:

Polar to rectangular: X = R * cos(theta), Y = R * sin(theta), implemented using analog multipliers with sine/cosine generators
Rectangular to polar: R = sqrt(X^2 + Y^2), theta = arctan(Y/X), requiring vector magnitude and inverse tangent circuits

Specialized integrated circuits implement these conversions for applications in radar display systems, motor control, and navigation equipment.

Vector Magnitude Circuits

Computing the magnitude of a two-dimensional vector requires evaluating sqrt(X^2 + Y^2). Direct implementation uses two squarers, a summer, and a square root circuit. Alternative approaches include:

Successive approximation: Iterative algorithms that converge on the magnitude
CORDIC-inspired: Rotation-based algorithms adapted to analog hardware
Approximation formulas: Simpler expressions like max(|X|,|Y|) + 0.4 * min(|X|,|Y|) that approximate magnitude with modest error

Inverse Trigonometric Functions

Arctangent circuits are particularly important for angle computation in Cartesian-to-polar conversion. Implementation approaches include:

Ratio-to-angle converters: Specialized circuits that accept Y/X and produce the angle
Feedback sine/cosine: A servo loop adjusts an angle voltage until cos(theta) = X/R and sin(theta) = Y/R
Piecewise approximation: Breakpoint networks approximating the arctangent curve

Correlation and Convolution Circuits

Correlation measures the similarity between two signals as a function of time offset, while convolution computes the overlap integral fundamental to linear system analysis. Both operations have analog implementations suited for real-time signal processing applications.

Correlation Fundamentals

The cross-correlation of signals x(t) and y(t) is defined as:

Rxy(tau) = integral(x(t) * y(t + tau) dt)

This function measures how much y resembles a time-shifted version of x. For two identical signals, the autocorrelation Rxx(0) equals the signal power, and the correlation function peaks when tau equals zero.

Analog Correlator Architecture

A basic analog correlator consists of:

Multiplier: Computes the instantaneous product x(t) * y(t + tau)
Integrator or averaging filter: Accumulates the product over the measurement interval
Delay element: For swept-tau correlation, provides variable delay to one signal

Fixed-tau correlators are simpler, requiring only multiplication and averaging at a single delay value. Swept-tau correlators scan through a range of delays to map out the complete correlation function.

Applications of Correlation

Correlation serves numerous signal processing functions:

Time delay measurement: The correlation peak location indicates the time delay between two signals
Signal detection: Cross-correlating a received signal with a known pattern detects the pattern in noise
Velocity measurement: In ultrasonic or radar systems, correlation determines target velocity from Doppler shift
Synchronization: Lock-in amplifiers use correlation with a reference to extract signals from noise

Convolution Circuits

Convolution of h(t) and x(t) is mathematically related to correlation:

y(t) = integral(h(tau) * x(t - tau) dtau)

Convolution represents the output of a linear system with impulse response h(t) when excited by input x(t). Analog convolution implementations include:

Tapped delay lines: Multiple delay taps weighted by h(n) coefficients and summed
CCD or BBD analog delay: Charge-coupled or bucket-brigade devices providing multiple taps
Surface acoustic wave (SAW) devices: Integrated structures with built-in weighting for matched filtering

Lock-In Amplification

Lock-in amplifiers represent a specialized application of correlation, extracting small signals buried in noise by correlating with a known reference frequency. The technique achieves extraordinary sensitivity:

Principle: Multiplying the signal by a reference and low-pass filtering yields a DC output proportional to the signal component at the reference frequency
Phase sensitivity: Both in-phase and quadrature outputs reveal signal amplitude and phase
Noise rejection: The narrow effective bandwidth (set by the output filter) rejects broadband noise
Dynamic reserve: Ability to extract small signals in the presence of much larger interfering signals

Analog Delay Lines

Delay elements store analog signals and reproduce them after a specified time interval, enabling temporal manipulation essential for correlation, echo generation, comb filtering, and analog signal processing algorithms requiring access to past signal values.

LC Delay Lines

Distributed LC structures approximate transmission lines that delay signals by their propagation time. Lumped-element LC delay lines cascade multiple LC sections:

Delay: Td = N * sqrt(LC) for N sections with inductance L and capacitance C each
Bandwidth: Limited to fc = 1 / (pi * sqrt(LC)), above which signals are severely attenuated
Characteristics: Passive, linear, with characteristic impedance Z0 = sqrt(L/C)

LC delay lines provide continuous delay with no sampling artifacts but become bulky and expensive for long delays.

Charge-Coupled Devices (CCDs)

CCD delay lines store analog samples as charge packets that shift through a chain of capacitors under clock control:

Operation: Each clock cycle transfers charge from one stage to the next
Delay: Td = N / fclk, where N is the number of stages and fclk is the clock frequency
Variable delay: Changing the clock frequency adjusts the delay
Multiple taps: Access to intermediate stages enables transversal filter implementation

CCDs provide long delays in compact form but introduce sampling effects and require anti-aliasing filtering.

Bucket-Brigade Devices (BBDs)

Similar to CCDs, bucket-brigade devices transfer charge through a chain of capacitors using a different clocking scheme. BBDs were historically popular in audio applications like chorus effects and echo units:

Audio applications: Guitar effects, reverb, and delay units
Limitations: Higher noise and distortion compared to CCDs
Simplicity: Easier to manufacture than CCDs

Surface Acoustic Wave (SAW) Devices

SAW delay lines convert electrical signals to acoustic waves that propagate along a piezoelectric substrate before reconversion to electrical signals:

Delay mechanism: Acoustic velocity (roughly 3000 m/s) is much slower than electromagnetic propagation
Compact long delays: Microsecond delays achieved in small packages
High frequency: Operate from MHz to GHz frequencies
Fixed characteristics: Delay determined by physical dimensions during manufacturing

SAW devices find application in radar, communications, and specialized signal processing where fixed delays at high frequencies are needed.

All-Pass Filter Delays

Cascaded all-pass filter sections provide group delay with constant amplitude response. Unlike true delay lines that delay all frequencies equally, all-pass delays introduce frequency-dependent group delay but can approximate constant delay over limited bandwidth:

Continuous-time: No sampling required
Tunable: Component adjustment changes delay
Limited delay-bandwidth product: Long delays require many stages

Sample Correlation Techniques

Sample correlation extends correlation concepts to sampled signals, implementing discrete-time versions of continuous correlation operations. These techniques bridge analog and digital signal processing, often using analog circuits to process sampled data.

Sampled Analog Correlation

Sampled correlators multiply corresponding samples of two signals and accumulate the products. Using sample-and-hold circuits and analog multipliers, these systems compute:

R[k] = sum(x[n] * y[n + k])

where x[n] and y[n] are sampled versions of the input signals. The correlation lag k determines which samples are paired for multiplication.

Sample-and-Hold in Correlation

Sample-and-hold circuits capture signal values at discrete instants for correlation processing:

Acquisition: The capacitor charges to the input voltage during the sample phase
Hold: The switch opens, and the capacitor maintains the sampled voltage
Accuracy factors: Aperture jitter, droop rate, and feedthrough affect correlation accuracy

Boxcar Integrators

Boxcar integrators (gated integrators) accumulate signal values only during defined time windows, implementing sampled integration useful for correlation with pulsed signals:

Operation: An electronic switch gates the signal to an integrator during the measurement window
Noise reduction: Averaging many gated windows reduces random noise
Synchronous detection: When triggered synchronously with a repetitive signal, boxcar integration extracts the signal from noise

Photon Correlation

In optical applications, photon correlation spectroscopy measures correlations in light intensity fluctuations to determine particle sizes and dynamics. The hardware correlator:

Counts photons: Detects individual photon arrivals and generates digital pulses
Computes autocorrelation: Determines the probability of detecting a photon at time t + tau given detection at time t
Multiple tau architecture: Uses a combination of linear and logarithmically-spaced delay channels to cover wide time ranges efficiently

Cross-Correlation Timing

Precision timing measurements use sample correlation between received signals and reference waveforms. Applications include:

Time-of-flight measurement: Determining distance from propagation delay
Synchronization: Aligning local timing to received signals
Multipath resolution: Separating direct and reflected signal paths

Sub-sample timing resolution is achieved by interpolating between correlation samples to find the precise peak location.

Practical Design Considerations

Implementing analog computation circuits requires attention to numerous practical factors that affect accuracy, stability, and reliability.

Dynamic Range and Scaling

Proper signal scaling ensures signals remain within the linear operating range of all circuit stages:

Maximum signals: Must not exceed supply voltages or amplifier output swing limits
Minimum signals: Must exceed noise floor and offset voltages for acceptable signal-to-noise ratio
Intermediate scaling: Multiplier outputs scale as Vx * Vy / Vref; proper reference selection prevents overflow
Integration limits: Integrator outputs must be bounded or reset to prevent saturation

Temperature Stability

Analog computation accuracy often depends strongly on temperature:

Transistor parameters: Base-emitter voltage and current gain vary with temperature
Component drift: Resistors and capacitors change value with temperature
Offset drift: Operational amplifier offsets typically increase with temperature
Compensation techniques: Matched components, temperature-compensating networks, and temperature-controlled enclosures

Noise Considerations

Noise limits the accuracy of analog computation, especially at small signal levels:

Thermal noise: Generated by all resistive elements
Shot noise: Arises from current flow through semiconductor junctions
Flicker (1/f) noise: Dominates at low frequencies, particularly in MOSFETs and carbon resistors
Noise accumulation: Each computation stage adds noise to the signal

Bandwidth and Speed

Analog computation speed depends on the bandwidth of constituent circuits:

Amplifier bandwidth: Gain-bandwidth product limits high-frequency accuracy
Multiplier bandwidth: Decreases with increasing signal amplitude in some topologies
Settling time: Integrators and filters require time to settle after transients
Phase errors: Frequency-dependent phase shifts affect computation accuracy

Calibration and Adjustment

Practical analog computation systems often include adjustment provisions:

Offset nulling: Trimming amplifier and multiplier offsets to zero
Gain calibration: Setting precise scale factors
Linearity adjustment: Optimizing nonlinear circuit accuracy
Periodic recalibration: Compensating for long-term drift

Modern Applications and Integration

While digital computation dominates many signal processing applications, analog computation continues to serve important roles where its inherent characteristics provide advantages.

Hybrid Analog-Digital Systems

Modern systems often combine analog and digital computation:

Analog front ends: Signal conditioning and preprocessing before A/D conversion
Digital control of analog: Microprocessors adjusting analog circuit parameters
Real-time analog, precise digital: Fast analog approximations refined by digital post-processing
Power optimization: Analog preprocessing reduces digital processing load

Neuromorphic Computing

Analog computation principles inspire neuromorphic processors that emulate neural networks:

Synaptic weights: Analog multipliers or variable resistors represent connection strengths
Summation: Current summing implements neuron input aggregation
Activation functions: Nonlinear circuits implement sigmoid or ReLU responses
Energy efficiency: Analog approaches offer potential power advantages for inference tasks

Sensor Signal Processing

Analog computation remains prevalent in sensor interfaces:

RMS measurement: True-RMS conversion for AC signals
Power computation: Instantaneous V*I products for energy metering
Vector operations: Magnitude and angle from multi-axis sensors
Ratio computation: Ratiometric measurements for improved accuracy

Integrated Analog Computing

Application-specific integrated circuits (ASICs) and programmable analog devices enable complex analog computation:

Multiplier/divider ICs: Dedicated analog multiplier and divider integrated circuits
RMS converter ICs: Complete true-RMS conversion in single packages
Analog signal processors: Configurable analog computation blocks
System-on-chip integration: Analog computation combined with digital processing

Conclusion

Analog computation circuits provide powerful tools for performing mathematical operations directly on continuous signals. From fundamental operations like multiplication and integration to sophisticated functions including RMS conversion, logarithmic transformation, and correlation, these circuits enable real-time signal processing with inherent speed advantages and often simpler implementations than equivalent digital approaches.

Understanding analog computation requires familiarity with both the mathematical operations being implemented and the electronic circuit techniques that realize them. The designer must balance accuracy requirements against practical constraints of noise, temperature stability, component tolerances, and bandwidth limitations. Careful attention to scaling, calibration, and circuit topology selection ensures that analog computation circuits meet their intended performance specifications.

While digital signal processing has assumed many roles previously filled by analog computation, the analog approach continues to thrive in applications demanding real-time response, ultra-low power consumption, or natural integration with analog sensor and actuator systems. As electronic systems become more sophisticated, the combination of analog computation for appropriate tasks with digital processing for others often provides the optimal solution, leveraging the strengths of both approaches.