Control and Communication Systemsfor BME

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This section covers the modelling of physical and physiological systems by differential equations, their conversion into transfer functions, block diagram representation, and state-space models in matrix notation.

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Systems and Control Terminology • A system is a combination of components acting together to perform an objective; a model is a mathematical description of its behaviour, used for analysis, design, prediction and simulation before anything is built. • Open-loop control has no feedback: the output has no effect on the control action, so the system is simple, cheap and stable, but cannot correct for disturbances or for changes in the plant — a domestic toaster, or an infusion pump set to a fixed rate.

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Closed-loop (feedback) control measures the output, compares it with the desired value to form an error, and acts to reduce that error: it gives accuracy, rejection of disturbances, reduced sensitivity to parameter variation and controllable transient response, at the cost of complexity, expense and the possibility of instability.

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Physiology is the great exemplar of closed-loop control, as 1.1 set out — thermoregulation, blood pressure and glucose control are all negative feedback loops, and the same structure recurs in the closed-loop infusion systems of 6.5. • Classification: linear or non-linear; time-invariant or time-varying; continuous or discrete; deterministic or stochastic;

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SISO or MIMO; lumped or distributed.

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Analysis assumes linear time-invariant (LTI) behaviour, which permits superposition — the response to a sum of inputs is the sum of the responses. • Standard test inputs: the impulse (δ), whose response is the impulse response h(t) and completely characterises an LTI system; the step, from which rise time, delay time, peak time, percentage overshoot, settling time and steady-state error are read; the ramp; and the sinusoid, which gives the frequency response of 9.3.

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Differential Equations and the Transfer Function • Physical systems are modelled by writing the governing physical laws — Kirchhoff's laws for circuits, Newton's laws for mechanics, conservation of mass and energy for thermal and fluid systems — which yields an ordinary differential equation with constant coefficients relating output to input. • The Laplace transform converts that differential equation into an algebraic one:

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Lf(t) = F(s) = ∫₀∞ f(t)e−st dt.

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Its value is that differentiation becomes multiplication by s and integration division by s (with zero initial conditions), so calculus becomes algebra.

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Useful pairs: δ(t) → 1, u(t) → 1/s, t → 1/s², e−at → 1/(s + a), sin ωt → ω/(s² + ω²), and the final value theorem, limt→∞ f(t) = lims→0 sF(s), which gives the steady-state value without inverting the transform. • The transfer function G(s) is defined as the ratio of the Laplace transform of the output to that of the input, with all initial conditions zero.

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It is a property of the system alone, independent of the input, and applies only to linear time-invariant systems. • Written as G(s) = N(s)/D(s), the roots of the numerator are the zeros and the roots of the denominator the poles;

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D(s) = 0 is the characteristic equation, and the order of the system is the highest power of s in the denominator.

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The poles determine the nature of the transient response and the stability: a system is stable if and only if all its poles have negative real parts, that is lie in the left half of the s-plane — a pole in the right half gives an exponentially growing response, and poles on the imaginary axis give sustained oscillation. • First-order system:

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G(s) = K/(τs + 1), with time constant τ; its step response rises exponentially, reaching 63.2 % in one time constant and about 98 % in four.

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A thermometer, an RC filter and a single-compartment drug model all follow this form. • Second-order system:

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G(s) = Kωn²/(s² + 2ζωns + ωn²), with natural frequency ωn and damping ratio ζ — the form that governs the pressure transducers of 5.2 and the mechanical systems of 9.2.

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Block Diagrams and Signal Flow • A block diagram represents the system graphically, with blocks (transfer functions), arrows (signals), summing junctions and take-off points. • Reduction rules: blocks in cascade (series) multiply, G₁G₂; blocks in parallel add, G₁ + G₂; and the closed-loop transfer function of a feedback system is T(s) = G(s)/[1 ± G(s)H(s)], with the plus sign for negative feedback and the minus sign for positive feedback, where G(s)H(s) is the open-loop (loop) transfer function.

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For unity feedback, H = 1 and T = G/(1 + G). • Mason's gain formula gives the same result directly from a signal flow graph, and is used where the diagram is too tangled to reduce by hand.

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State-Space and Matrix Notation • The transfer function approach is limited to linear, time-invariant, single-input single-output systems with zero initial conditions.

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The state-space approach removes all of these restrictions and is what computer packages use. • Definitions: the state is the smallest set of variables that, with the input from time t₀ onwards, completely determines the future behaviour; the state variables are those quantities, usually chosen as the energy-storage variables — capacitor voltages, inductor currents, positions and velocities, or compartment concentrations; and the order of the system equals the number of state variables. • The standard form is a pair of matrix equations: the state equation ẋ = Ax + Bu and the output equation y = Cx + Du, where x is the n × 1 state vector, u the input vector and y the output vector, and A is the n × n system (state) matrix, B the input matrix, C the output matrix and D the feedthrough matrix.

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The great advantage is that a system of any order, with any number of inputs and outputs, is described by the same two compact equations. • Connections: the transfer function is recovered as G(s) = C(sI − A)−1B + D; the poles of the system are the eigenvalues of A, that is the roots of |sI − A| = 0 — so stability requires all eigenvalues of A to have negative real parts.

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State-space also supports the concepts of controllability and observability, and is the basis of modern control, optimal control and the Kalman filter. • Physiological and biomedical modelling: the same framework describes compartmental pharmacokinetic models (one-, two- and three-compartment drug distribution — the basis of the target-controlled infusion of 6.5), the Windkessel model of the arterial system, respiratory mechanics as a resistance-compliance system, the Hodgkin-Huxley model of the action potential, glucose-insulin dynamics for the artificial pancreas, and models of thermoregulation and of the cardiovascular baroreflex.

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The universal caution is that physiological systems are non-linear, time-varying and highly variable between individuals, so every model is a linearised approximation valid over a limited operating range — which is exactly the subject of 9.3.

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This section covers the three basic mechanical elements — mass, spring and damper — the modelling of translational and rotational mechanical systems, the electrical analogies, and the resulting second-order response.

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The Three Translational Elements Element Relation Role and energy Mass (m, kg) f = m·a = m·d²x/dt² = m·dv/dt Inertia — resists change of velocity; stores kinetic energy ½mv²; force is proportional to acceleration Spring (k, N/m) f = k·x (Hooke's law) Elasticity/stiffness — resists deformation; stores potential energy ½kx²; force is proportional to displacement.

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Compliance is 1/k Damper / dashpot (b or c, N·s/m) f = b·v = b·dx/dt Viscous friction — resists velocity; dissipates energy as heat; force is proportional to velocity • Rotational equivalents: moment of inertia J (T = J·d²θ/dt²), torsional spring K (T = Kθ) and rotational damper B (T = B·dθ/dt), with torque replacing force and angular displacement replacing linear displacement.

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A gear train or lever transforms these quantities much as a transformer does electrically. • Combinations: springs in parallel add their stiffnesses (k = k₁ + k₂), while springs in series add their compliances (1/k = 1/k₁ + 1/k₂) — the reverse of the rule for resistors, and a favourite examination point.

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Dampers follow the same pattern as springs.

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Modelling a Mechanical System • The procedure is: draw a free-body diagram for each mass → apply Newton's second law, ΣF = m·a, taking care with the sign of each element's force → obtain one differential equation per mass → take Laplace transforms with zero initial conditions → solve for the transfer function. • For the classic mass-spring-damper system with an applied force f(t) and displacement x(t): • m·d²x/dt² + b·dx/dt + k·x = f(t), which transforms to (ms² + bs + k)X(s) = F(s), giving the transfer function X(s)/F(s) = 1/(ms² + bs + k). • Writing this in standard form gives natural frequency ωn = √(k/m) and damping ratio ζ = b/(2√(km)), with the damped natural frequency ωd = ωn√(1 − ζ²).

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These three relations are worth memorising, because every second-order instrument in the paper — the fluid-filled pressure line of 5.2, the galvanometer, the ultrasound transducer, the limb segment in gait analysis — reduces to them.

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Second-Order Response Damping ratio Poles Response to a step ζ = 0 (undamped) Purely imaginary Sustained oscillation at ωn — never reaches equilibrium 0 < ζ < 1 (underdamped) Complex conjugate pair Overshoots and oscillates with decaying amplitude; fast rise time; percentage overshoot depends only on ζ ζ = 1 (critically damped) Two equal real poles Reaches the final value in the shortest possible time without any overshoot ζ > 1 (overdamped) Two distinct real poles Slow, sluggish approach with no overshoot • The instrument designer's compromise: a critically damped system never overshoots but is slower to respond than one that is slightly underdamped, so measuring instruments are usually designed for ζ ≈ 0.6-0.7, which gives the fastest response consistent with a small acceptable overshoot (about 5-10 %) and the flattest frequency response.

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This is exactly the value quoted for the arterial pressure monitoring system in 5.2, where under-damping causes resonant overshoot and an artificially high systolic reading, and over-damping blunts the waveform and underestimates it. • Step response specifications read from the curve: delay time (to 50 %), rise time (10-90 % or 0-100 %), peak time, percentage overshoot, settling time (to within 2 % or 5 %) and steady-state error.

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Electrical Analogies • Because the mathematics is identical, mechanical systems can be analysed as equivalent circuits, and this is the reason a single second-order theory serves the whole of instrumentation. • Force-voltage (direct) analogy: force ↔ voltage, velocity ↔ current, mass ↔ inductance, damper ↔ resistance, spring compliance (1/k) ↔ capacitance, displacement ↔ charge.

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The mechanical equation m·ẍ + b·ẋ + kx = f then corresponds exactly to the series RLC equation L·q̈ + R·q̇ + q/C = v. • Force-current (mobility) analogy: force ↔ current, velocity ↔ voltage, mass ↔ capacitance, damper ↔ conductance, spring ↔ 1/inductance, corresponding to the parallel RLC circuit and preserving the topology of the original diagram. • Analogous quantities across domains extend further: fluid systems (pressure ↔ voltage, flow ↔ current, fluid resistance, inertance and compliance) — which is how the Windkessel model represents the arterial tree as an RC circuit, and how respiratory mechanics is modelled as airway resistance in series with lung compliance; and thermal systems (temperature ↔ voltage, heat flow ↔ current, thermal resistance and capacitance), used to model heating of an implant or of tissue under diathermy. • Biomechanical application: the viscoelastic models of 1.6 are built from exactly these elements — the Maxwell model (spring and damper in series, showing stress relaxation), the Kelvin-Voigt model (spring and damper in parallel, showing creep) and the standard linear solid (three elements, showing both) — which is why tendon, ligament and cartilage behaviour is described in the same language as a control system.

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This section covers linearization, the frequency-domain characterization of systems, Bode magnitude and phase plots, the effects of gain and time constants on those plots, and the proportional, integral and derivative controller.

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Linearization • Almost every real system — and every physiological one — is non-linear, showing saturation, dead zone, backlash, hysteresis, friction and square-law or exponential relations.

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But the whole apparatus of transfer functions, Bode plots and classical control assumes linearity. • Linearization resolves this by expanding the non-linear relation as a Taylor series about a chosen operating (equilibrium) point and retaining only the first-order term, so that the model describes small deviations about that point: Δy ≈ (dy/dx)|x₀·Δx.

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The resulting linear model is accurate only for small excursions around the operating point, and a new linearization is needed if the operating point moves — which is precisely why physiological controllers must be adaptive. • Biomedical examples: the oxyhaemoglobin dissociation curve is markedly non-linear but nearly linear over a limited range; respiratory compliance is linear only in the mid-range of lung volume; drug dose-response curves are sigmoid; and the thermistor of 5.1 is deliberately linearized by a parallel resistor over its working range.

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Frequency Response and Bode Plots • The frequency response is the steady-state response to a sinusoidal input, obtained by substituting s = jω into the transfer function.

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The result G(jω) has a magnitude |G(jω)|, the amplitude ratio, and a phase angle ∠G(jω) — so the output of a linear system to a sine wave is a sine wave of the same frequency, altered only in amplitude and phase. • A Bode plot presents this as two graphs against log frequency: magnitude in decibels (20 log₁₀|G|) and phase in degrees.

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The logarithmic scales are what make the method powerful: multiplication of transfer functions becomes addition of their plots, so the response of a complicated system is built up by adding the contributions of its simple factors, and straight-line asymptotes suffice for design. • The standard factors and their asymptotes: a constant gain K gives a horizontal line at 20 log K dB and 0° phase; a pole at the origin (1/s, an integrator) gives −20 dB/decade through all frequencies and a constant −90°; a zero at the origin (s, a differentiator) gives +20 dB/decade and +90°; a simple pole 1/(1 + jωτ) gives 0 dB below the corner frequency ω = 1/τ and −20 dB/decade above it, with the phase falling from 0° through −45° at the corner to −90°; a simple zero gives the mirror image, +20 dB/decade and +45° at the corner rising to +90°; and a quadratic (second-order) pair gives −40 dB/decade and −180°, with a resonant peak whose height depends on the damping ratio — pronounced for ζ below about 0.5 and absent above 0.707.

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At the corner frequency itself the true magnitude of a simple pole is −3 dB below the asymptote. • Effect of changing the gain, a favourite examination point: increasing K shifts the entire magnitude plot upward by 20 log K decibels without changing its shape, and leaves the phase plot completely unaltered.

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The consequence is that raising the gain moves the gain crossover frequency to a higher frequency, which usually reduces the phase margin and therefore the stability, while improving speed of response and reducing steady-state error — the fundamental trade-off of feedback control. • Effect of the time constant: increasing τ moves the corner frequency ω = 1/τ to a lower frequency, so the system's bandwidth falls and its response becomes slower and more sluggish, with the phase lag beginning earlier.

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A small time constant means a fast system with a wide bandwidth.

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Bandwidth and speed of response are directly related — a wider bandwidth means a faster rise time. • Stability from the Bode plot: the gain margin is the amount by which the gain can be increased before instability, measured in dB at the frequency where the phase is −180°, and the phase margin is the additional phase lag that would cause instability, measured at the frequency where the gain is 0 dB.

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Positive margins indicate stability; typical design targets are a gain margin above 6 dB and a phase margin of 30-60°.

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The related Nyquist criterion examines the polar plot's encirclement of the −1 point, and the Routh-Hurwitz criterion determines stability algebraically from the characteristic equation.

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PID Controllers • The PID controller is by far the commonest control algorithm in industry and in medical equipment.

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It acts on the error e(t) = setpoint − measured output and produces u(t) = Kpe(t) + Ki∫e dt + Kd·de/dt, or in transfer function form Gc(s) = Kp + Ki/s + Kds = Kp(1 + 1/(Tis) + Tds).

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Term What it does Benefits Penalties Proportional (P) Output proportional to the present error Reduces rise time and steady-state error; increases speed Cannot eliminate steady-state error (an offset always remains); excessive gain causes overshoot and instability Integral (I) Output proportional to the accumulated past error Eliminates steady-state error completely Slows the response, increases overshoot and settling time, and reduces stability; subject to integral wind-up when the actuator saturates Derivative (D) Output proportional to the rate of change — an anticipatory or predictive action Improves damping and transient response, reduces overshoot and settling time, increases stability Greatly amplifies high-frequency noise; has no effect on steady-state error; useless on a noisy or step-like signal unless filtered • Common variants:

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P alone where an offset is tolerable;

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PI — the commonest industrial combination, giving zero steady-state error without the noise problems of derivative action;

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PD where speed and damping matter but offset does not; and full PID. • Tuning: classically by the Ziegler-Nichols methods — either the open-loop step-response (reaction curve) method or the closed-loop ultimate-gain method, in which the proportional gain is raised until sustained oscillation begins, and Ku and the oscillation period Pu are used to set the three terms — or by Cohen-Coon, software auto-tuning, or manual trial and error.

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Practical implementations add anti-wind-up limiting, derivative filtering, derivative-on-measurement (to avoid a derivative kick when the setpoint changes), bumpless transfer and output limits. • Biomedical applications: the closed-loop infusion systems of 6.5 — insulin delivery from continuous glucose measurement, vasopressor infusion from blood pressure, anaesthetic infusion from a depth-of-anaesthesia index; ventilator pressure and flow control; temperature control in incubators, blood warmers, autoclaves and hypothermia systems; servo control in imaging gantries, patient tables and surgical robots; and prosthetic and orthotic limb control.

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In every case the practical difficulties are the same: sensor noise and artefact, long and variable time delays between action and effect, large inter-patient variability, and the absolute requirement for safety limits and clinician override — which is why fully closed-loop drug delivery has been adopted only slowly.

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This section covers the elements of a communication system, analogue and digital communication, transmitters, transmission channels and their impairments, and receivers.

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Elements of a Communication System • Every communication system, from a telemetry transmitter to a hospital network, has the same structure: information source → input transducer → transmitter → channel (with noise) → receiver → output transducer → destination. • The transmitter prepares the signal for the channel, performing amplification, filtering, encoding, modulation and power amplification, and feeding an antenna or line driver.

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The channel is the physical medium.

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The receiver reverses the process — selecting the wanted signal, amplifying it, demodulating it and recovering the original information — while rejecting noise and interference. • Analogue versus digital communication: an analogue system transmits a continuously varying waveform, and noise accumulates irreversibly at every stage, because there is no way to distinguish signal from noise.

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A digital system transmits symbols from a finite set, so at each repeater the signal can be regenerated exactly, and noise does not accumulate; it also permits error detection and correction, encryption, compression, time-division multiplexing and direct integration with computers.

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The price is greater bandwidth for a given signal and the need for conversion at both ends.

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Digital techniques have consequently displaced analogue in almost all biomedical telemetry (5.3). • Digital communication chain: sampling (9.6) → quantisation → source encoding (compression) → channel encoding (adding redundancy for error control) → modulation (ASK, FSK, PSK, QAM) → channel → demodulation → decoding → reconstruction.

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Transmitters • Block diagram of an AM transmitter: microphone or transducer → audio amplifier → modulating amplifier; and, in parallel, crystal oscillator → buffer → frequency multiplier → RF power amplifier, where modulation is applied → antenna.

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An FM transmitter replaces the modulating amplifier with a reactance modulator or varactor acting on the oscillator, or uses a phase-locked loop. • Key elements: the oscillator, which must be highly stable and is therefore crystal controlled (7.2); buffer amplifiers to prevent the load from pulling the oscillator frequency; frequency multipliers; the modulator; the power amplifier, usually class C with a tuned load for efficiency (7.2); filters to suppress harmonics and spurious emissions; and the antenna and its matching network. • Antenna considerations: an efficient antenna must be an appreciable fraction of a wavelength — typically λ/4 or λ/2 — which is the fundamental reason low-frequency signals must be translated to a high carrier frequency before radiation (9.5).

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Transmission Channels Channel Characteristics Twisted pair Cheap and simple; limited bandwidth and distance; twisting reduces crosstalk and common-mode pick-up — the basis of the shielded twisted lead used for ECG cables Coaxial cable Wide bandwidth, good shielding against interference; used for RF and for the connections of imaging and ultrasound equipment Optical fibre Enormous bandwidth, very low loss, complete electrical isolation and total immunity to electromagnetic interference — which makes it ideal for carrying signals across a patient isolation barrier (6.6) and through the MRI room Free space (radio) No physical connection, permitting mobility and telemetry (5.3); subject to path loss, multipath fading, shadowing and shared-spectrum interference, and regulated as to frequency and power Waveguide and microwave link High frequencies over line-of-sight paths Satellite Very wide coverage with a long propagation delay; used for telemedicine to remote regions • Channel impairments: attenuation (loss with distance, requiring amplifiers or repeaters); distortion, which may be amplitude (different frequencies attenuated unequally), phase/delay (different frequencies delayed unequally, so waveform shape is lost) or non-linear (harmonics and intermodulation products generated); noise; interference from other transmitters; fading and multipath; dispersion in fibre; and limited bandwidth. • Channel capacity is set by the Shannon-Hartley theorem, C = B·log₂(1 + S/N) bits per second — so capacity rises linearly with bandwidth but only logarithmically with signal-to-noise ratio, which is why increasing transmitter power gives diminishing returns and why bandwidth is the more valuable resource.

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Receivers • A receiver must provide sensitivity (the ability to detect weak signals, limited by its own noise), selectivity (the ability to separate the wanted signal from adjacent ones), fidelity and stability. • The tuned radio frequency (TRF) receiver — several tuned RF stages followed by a detector — is simple but suffers from poor and variable selectivity across the band, difficulty in ganging the tuned circuits, and instability at high gain. • The superheterodyne receiver is the universal solution and the standard examination topic.

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Its principle is to convert every incoming signal, whatever its frequency, to a single fixed intermediate frequency (IF), at which all the gain and selectivity are provided.

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The chain is: antenna → RF amplifier (with preselector) → mixer, into which a local oscillator signal is injected → IF amplifier at the fixed intermediate frequency → detector/demodulator → audio or data amplifier → output transducer, with automatic gain control fed back to the RF and IF stages. • The mixer produces the sum and difference frequencies, and the IF filter selects the difference: fIF = fLO − fRF, the local oscillator being tuned in step (ganged) with the RF stage so that the difference stays constant — for broadcast AM the standard IF is 455 kHz and for FM 10.7 MHz. • Advantages: uniform selectivity and gain across the whole tuning range, since the IF filter is fixed and can be made as sharp as required; high stable gain at a lower frequency; and simpler alignment.

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The characteristic disadvantage is the image frequency: a signal at fRF + 2fIF also produces the correct difference frequency in the mixer and is received as interference.

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It is suppressed by the RF preselector before the mixer and by choosing a sufficiently high IF; the double-conversion receiver uses a high first IF for image rejection and a low second IF for selectivity. • Demodulation: the envelope (diode) detector for AM; the slope detector, Foster-Seeley discriminator, ratio detector, quadrature detector or phase-locked loop for FM, preceded by a limiter that removes amplitude variations and is the basis of FM's noise immunity; and coherent (synchronous) detection where a phase reference is available.

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Software-defined radio now performs most of these functions digitally after an early ADC.

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This section covers the reasons for modulation, amplitude, frequency and phase modulation with their bandwidths and power relations, pulse and digital modulation, and distortion, noise and interference.

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Why Modulate? • Modulation is the process of varying some characteristic of a high-frequency carrier — its amplitude, frequency or phase — in accordance with the instantaneous value of the low-frequency information (baseband) signal.

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The reasons, which are examined as a list: • (1) Reduction of antenna height — the dominant reason.

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An efficient antenna must be about λ/4 long; a 1 kHz audio signal has a wavelength of 300 km, requiring an antenna 75 km high, whereas at a 1 MHz carrier the antenna is 75 m.

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Translating the signal to a high frequency makes radiation physically possible. • (2) Frequency multiplexing — many signals can share one medium by being allocated different carrier frequencies, so that transmissions do not interfere; without modulation every source would occupy the same band. • (3) Improved noise performance — angle modulation in particular trades bandwidth for signal-to-noise ratio. • (4) Practicality of equipment — smaller components and more efficient amplification at radio frequency. • (5) Range and penetration — different carrier frequencies propagate differently, allowing the designer to choose ground wave, sky wave or line-of-sight propagation. • In biomedical telemetry (5.3) all of these apply, and the same reasoning underlies the choice of carrier frequency for implantable devices, where tissue attenuation rises steeply with frequency.

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Amplitude Modulation • In AM the amplitude of the carrier is varied in proportion to the modulating signal while its frequency and phase remain constant.

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For a sinusoidal modulating signal, v(t) = Ac(1 + m·cos ωmt)·cos ωct. • Modulation index m = Am/Ac, also computable from the envelope as m = (Vmax − Vmin)/(Vmax + Vmin).

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It must satisfy m ≤ 1; if m > 1 the carrier is over-modulated, the envelope is clipped and severe distortion with spurious sidebands results. • Spectrum and bandwidth: the modulated wave contains the carrier plus an upper and a lower sideband at fc ± fm, so the bandwidth is twice the highest modulating frequency, BW = 2fm. • Power:

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Ptotal = Pc(1 + m²/2), so at m = 1 the total power is 1.5 times the carrier power and each sideband carries only one-sixth of the total — meaning that at best only 33 % of the transmitted power carries information, and at 100 % modulation two-thirds of the power is in the carrier, which conveys none.

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This inefficiency is AM's fundamental weakness and the reason for DSB-SC (suppressed carrier), SSB (single sideband — half the bandwidth and far better power efficiency, used in point-to-point communication) and VSB (vestigial sideband, used for television video). • Summary of AM: simple and cheap to generate and demodulate (an envelope detector suffices), and bandwidth-efficient, but power-inefficient and highly susceptible to noise, because noise is itself an amplitude variation and cannot be separated from the signal.

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FM and PM • Frequency modulation varies the instantaneous frequency of the carrier in proportion to the amplitude of the modulating signal, the amplitude of the carrier remaining constant.

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Phase modulation varies the phase in proportion to the modulating signal amplitude.

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The two are closely related:

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PM is equivalent to FM of the differentiated modulating signal, and FM to PM of the integrated signal, and both are grouped as angle modulation.

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In FM the frequency deviation Δf is proportional to the amplitude of the modulating signal and independent of its frequency, whereas in PM the deviation depends on both. • Modulation index: for FM, β = Δf/fm, which varies inversely with the modulating frequency; for PM the index is the peak phase deviation and is independent of fm. • Bandwidth: unlike AM, angle modulation generates an infinite number of sidebands, whose amplitudes are given by Bessel functions.

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In practice the bandwidth is estimated by Carson's rule:

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BW ≈ 2(Δf + fm) = 2fm(1 + β).

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Narrowband FM (β < 1) occupies about the same bandwidth as AM, while wideband FM occupies much more — commercial FM broadcasting allows a deviation of 75 kHz and a channel of 200 kHz, against 10 kHz for an AM channel. • Power: because the amplitude is constant, the total transmitted power in FM is constant and independent of the modulation index — modulation merely redistributes power between the carrier and the sidebands.

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This also allows the use of efficient non-linear class C amplifiers. • Noise performance — the decisive advantage: since the information is carried by frequency and not amplitude, a limiter in the receiver can remove amplitude fluctuations, and with them most of the noise.

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FM therefore gives a much better output signal-to-noise ratio than AM for the same transmitted power, provided the input is above the threshold; below a certain input level the advantage collapses abruptly — the capture and threshold effect, in which a receiver locks to the stronger of two signals and suppresses the weaker.

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Pre-emphasis at the transmitter and de-emphasis at the receiver further improve the high-frequency noise performance. • Summary:

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FM and PM give far better noise immunity, constant transmitted power and efficient amplification, at the cost of much greater bandwidth and more complex circuitry — which is exactly why analogue biotelemetry uses FM (5.3) and why FM/FM subcarrier systems became the standard. • Pulse and digital modulation:

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PAM, PWM/PDM and PPM vary the amplitude, width or position of a pulse;

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PCM converts the signal into a binary code (9.6); and digital carrier modulation uses ASK, FSK, PSK, QPSK and QAM, with FSK and PSK preferred for their noise immunity and QAM for spectral efficiency.

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Distortion, Noise and Interference • These three degradations are distinct and are frequently confused in examinations. • Distortion is a deterministic alteration of the signal by the system itself, and is in principle correctable by equalisation.

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Amplitude (frequency) distortion — unequal gain at different frequencies; phase (delay) distortion — unequal delay, which alters waveform shape even when the amplitude spectrum is preserved, and is critical for pulse-like biosignals; non-linear (harmonic and intermodulation) distortion — the generation of new frequencies not present in the input, caused by overdriving an amplifier or by a non-linear device; and cross-over, clipping and slew-rate limiting in amplifiers. • Noise is unwanted random energy, and is not correctable, only minimisable.

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Internal noise: thermal (Johnson-Nyquist) noise, Vn = √(4kTBR), present in every resistance and proportional to temperature and bandwidth; shot noise from the discrete nature of charge carriers; flicker (1/f) noise, dominant at low frequencies and therefore a serious problem in DC-coupled biopotential amplifiers; partition and burst noise.

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External noise: atmospheric, cosmic and man-made.

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The figures of merit are the signal-to-noise ratio and the noise figure of an amplifier; noise is reduced by restricting the bandwidth to that of the signal, cooling, using low-noise devices in the first stage (which dominates the overall noise figure), and signal averaging — which improves the SNR by √n for n averaged sweeps, the principle behind evoked potential recording. • Interference is unwanted energy from an identifiable source — and, unlike noise, it can in principle be eliminated at source.

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In the clinical environment the sources are familiar:

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50 Hz mains and its harmonics, surgical diathermy (6.2), fluorescent lighting and dimmers, motors and lifts, mobile phones and radio transmitters, switching power supplies (7.4), and other medical devices.

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The remedies are shielding (a Faraday cage or screened cable), earthing and equipotential bonding, twisted pairs, differential recording with high CMRR, filtering, physical separation, and regulatory compliance with electromagnetic compatibility (EMC) standards, which require every medical device both to limit its own emissions and to tolerate a defined level of external fields (immunity).

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This section covers the sampling of analogue signals, the spectrum of a sampled signal, the sampling theorem for band-limited signals, aliasing and its prevention, and quantisation.

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Sampling of Analogue Signals • Sampling converts a continuous-time signal into a discrete-time sequence by measuring its amplitude at regular intervals, the sampling interval Ts and the sampling frequency fs = 1/Ts.

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It is the first step of every analogue-to-digital conversion and therefore of every digital medical instrument. • The complete conversion chain: anti-aliasing low-pass filter → sample-and-hold → quantiser → encoder, giving a binary word; the reverse chain is decoder → hold → reconstruction (smoothing) filter. • The sample-and-hold circuit — a switch, a hold capacitor and a buffer amplifier — is necessary because the ADC needs a constant input during its conversion time; without it the signal would change while being converted, producing an aperture error. • Types of sampling: ideal (impulse) sampling, which is the mathematical model; natural sampling, where the pulse top follows the signal; and flat-top sampling, which is what a practical sample-and-hold produces and which introduces a mild high-frequency roll-off known as the aperture effect, correctable by an equalising filter.

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The Spectrum of a Sampled Signal • Sampling in the time domain is equivalent to multiplication by a train of impulses, and multiplication in the time domain corresponds to convolution in the frequency domain.

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The consequence is the central result of the subject: the spectrum of a sampled signal consists of the original baseband spectrum repeated (replicated) at every integer multiple of the sampling frequency — centred at 0, ±fs, ±2fs and so on, each replica being an exact copy scaled by 1/Ts. • It follows immediately that the original signal can be recovered by a low-pass filter provided the replicas do not overlap.

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The replicas are separated by fs and each extends fmax on either side of its centre, so the condition for no overlap is fs − fmax > fmax, that is fs > 2fmax — which is the sampling theorem derived directly from the spectrum.

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The Sampling Theorem • The Nyquist-Shannon sampling theorem: a band-limited continuous signal containing no frequency component higher than fmax can be completely and exactly reconstructed from its samples, provided it is sampled at a rate greater than 2fmax. • Terminology:

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2fmax is the Nyquist rate — the minimum acceptable sampling frequency; fs/2 is the Nyquist frequency (or folding frequency) — the highest signal frequency that a given sampling rate can represent.

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These two are frequently confused and the distinction is worth fixing. • Reconstruction is performed in theory by an ideal low-pass filter, which is equivalent to sinc interpolation; in practice a hold circuit followed by a realisable reconstruction filter is used.

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Because no practical filter is ideal, real systems sample well above the theoretical minimum — typically 2.5 to 10 times fmax, which also relaxes the anti-aliasing filter requirement.

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Aliasing • Aliasing (folding) is what happens when fs ≤ 2fmax: the spectral replicas overlap, and a frequency component above the Nyquist frequency is indistinguishable after sampling from a lower frequency — it 'folds back' into the baseband and masquerades as a legitimate signal. • The apparent (alias) frequency of a component at f, sampled at fs, is falias = |f − k·fs| for the integer k giving the smallest result.

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Thus a 60 Hz component sampled at 100 Hz appears as 40 Hz, and a 1,100 Hz component sampled at 1,000 Hz appears as 100 Hz.

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The everyday illustration is the wagon-wheel effect in film, where a rotating wheel appears to turn slowly backwards. • The critical point, and the reason this is examined so persistently: aliasing is irreversible.

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Once the samples have been taken, the aliased component is arithmetically identical to a genuine signal at that frequency and can never be separated or removed by any amount of subsequent filtering or processing. • Prevention: an anti-aliasing low-pass filter must be placed before the sampler, with a cut-off below fs/2 and sufficient attenuation in the stopband; and the sampling rate must be chosen above twice the highest frequency present — which is not the same as the highest frequency of interest, since out-of-band noise and interference alias just as readily as signal.

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Oversampling followed by digital filtering and decimation relaxes the analogue filter requirement and is what sigma-delta converters exploit.

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Quantisation and Practical Rates • Quantisation assigns each sample to one of a finite number of levels.

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An n-bit converter provides 2ⁿ levels, and for a full-scale range VFS the step size (resolution) q = VFS/2ⁿ.

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The unavoidable rounding error, the quantisation noise, has an RMS value of q/√12, from which follows the standard result SNR ≈ 6.02n + 1.76 dB — so each additional bit improves the signal-to-noise ratio by about 6 dB.

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Typical converters in medical instruments are 12 to 24 bits. • Choosing n and fs in practice: the bit depth is set by the required dynamic range (for ECG, enough to resolve microvolt detail while accommodating a large electrode offset), and the rate by the signal bandwidth of 5.1.

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Standard sampling rates:

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ECG 250-500 Hz for monitoring and up to 1 kHz for diagnostic and pacemaker-spike detection;

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EEG 250-500 Hz (higher for high-frequency oscillations);

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EMG 1-4 kHz; phonocardiogram and speech 8-44.1 kHz; ultrasound RF data tens of megahertz. • Consequences of getting it wrong in a clinical instrument: too low a rate aliases muscle artefact and mains interference into the ECG band, where they cannot be removed and may mimic pathology; too high a rate wastes storage, bandwidth and power, which matters in ambulatory and implantable devices.

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Undersampling (bandpass sampling) is a deliberate exception in which a band-limited signal centred on a high carrier is sampled below its highest frequency but above twice its bandwidth, using aliasing constructively to translate it to baseband — the technique used in software-defined radio and in some ultrasound and MRI receivers.