Sources and References

These notes synthesize material from standard references on linear systems, signals, and biomedical signal processing. Primary sources include Oppenheim and Willsky’s Signals and Systems, Lathi’s Linear Systems and Signals, Proakis and Manolakis’s Digital Signal Processing: Principles, Algorithms, and Applications, open courseware from MIT 6.003 and Stanford EE102, Rangayyan’s Biomedical Signal Analysis, and Semmlow’s Biosignal and Medical Image Processing. No instructor-specific materials or institutional learning platforms are referenced. This course is cross-listed at the University of Waterloo as BME 252, MTE 252, and SYDE 252; content is identical across the three listings.

Chapter 1: Signals, Systems, and the Biomedical Context

A signal is any quantity that varies with an independent variable, most often time. In biomedical engineering the signals of interest arise from the physiology of living systems: the electrocardiogram (ECG) records the electrical depolarization and repolarization of cardiac muscle, the electroencephalogram (EEG) records the aggregate post-synaptic potentials of cortical neurons, and the electromyogram (EMG) records the electrical activity of motor units in skeletal muscle. Each is a voltage measured between electrodes, but each carries information about fundamentally different physical processes, at different time scales and different amplitude ranges. The ECG sits in a band from roughly 0.05 Hz (baseline wander from respiration) to 150 Hz (sharp QRS edges), with amplitudes of about one millivolt at the body surface. The EEG is smaller, typically tens of microvolts, and its clinically relevant rhythms occupy the range from roughly 0.5 Hz (delta) to about 40 Hz (gamma). Surface EMG extends higher, commonly 20 Hz to 500 Hz, with amplitudes up to a few millivolts during strong contractions. One reason to study linear systems and signals is to have a principled way to extract, interpret, and transform these signals despite the noise and artifacts that contaminate them.

A system is any operator that transforms one signal into another. A differential amplifier connected to chest electrodes is a system whose input is the two raw electrode voltages and whose output is the amplified ECG. A digital bandpass filter is a system whose input is the sampled ECG and whose output is a cleaner version. The human body, considered as a volume conductor between a cardiac dipole and a surface electrode, is itself a system. Much of what we do in this course is characterize systems well enough that we can predict their output for any admissible input, and design systems that deliver a desired response.

1.1 Continuous-time and discrete-time signals

A continuous-time signal \( x(t) \) is defined for every real value of the independent variable \( t \). The voltage at an electrode before sampling is continuous in time. A discrete-time signal \( x[n] \) is defined only at integer values of the index \( n \); these signals arise naturally from sampling, and they are the data objects that digital filters, classifiers, and estimators actually manipulate. In these notes we consistently use round parentheses for continuous-time signals and square brackets for discrete-time sequences.

Amplitude can also be continuous or discrete. The output of a 16-bit analog-to-digital converter is discrete in time and quantized in amplitude. In this course we almost always treat amplitude as continuous, setting quantization aside as a secondary (though real) source of error.

Strict periodicity almost never occurs in biology. A resting ECG is approximately periodic beat to beat, but beat-to-beat interval variability (heart rate variability) is itself a signal of diagnostic interest. We nonetheless use the periodic idealization to build intuition, and we extend the theory through Fourier methods to aperiodic signals.

1.2 Signal energy and power

For a continuous-time signal the energy over an interval \( [t_1, t_2] \) is

\[ E = \int_{t_1}^{t_2} \left| x(t) \right|^2 \, dt, \]

and the total energy is the limit as the interval spans the real line. If the total energy is finite, \( x \) is called an energy signal. If the energy diverges but the time-averaged power

\[ P = \lim_{T\to\infty} \frac{1}{2T} \int_{-T}^{T} \left| x(t) \right|^2 \, dt \]

is finite and nonzero, \( x \) is a power signal. Periodic signals are power signals; finite-duration transients are energy signals. A single QRS complex in isolation is an energy signal; the sustained rhythm over minutes is, for modeling purposes, a power signal.

1.3 Elementary signals

Several signals recur throughout the subject.

The unit step \( u(t) \) equals zero for \( t < 0 \) and one for \( t \geq 0 \). The unit impulse (Dirac delta) \( \delta(t) \) is not a function in the classical sense but a distribution, defined by its sifting property: for any continuous test function \( \phi \),

\[ \int_{-\infty}^{\infty} \phi(t) \, \delta(t - t_0) \, dt = \phi(t_0). \]

Formally, \( \delta(t) = \tfrac{d}{dt} u(t) \). In discrete time the Kronecker delta \( \delta[n] \) equals one when \( n = 0 \) and zero otherwise, and is a genuine function.

Complex exponentials \( x(t) = A \, e^{s t} \) with \( s = \sigma + j\omega \) unify sinusoids and exponentials. When \( \sigma = 0 \) we recover pure sinusoids \( e^{j\omega t} = \cos(\omega t) + j \sin(\omega t) \). The centrality of complex exponentials in linear systems theory arises because they are the eigenfunctions of LTI systems, a fact we will prove below.

1.4 System properties

Before studying any particular system we classify its properties, because these properties alone imply much of its behaviour.

A system is linear if it satisfies both additivity and homogeneity; equivalently, it obeys superposition: the response to \( a\,x_1(t) + b\,x_2(t) \) is \( a\,y_1(t) + b\,y_2(t) \), where \( y_i \) is the response to \( x_i \). A system is time-invariant if a shift in the input produces the same shift in the output: if \( y(t) \) is the response to \( x(t) \), then \( y(t-t_0) \) is the response to \( x(t-t_0) \). A system with both properties is a linear time-invariant (LTI) system, and almost the whole machinery of this course applies to LTI systems.

A system is causal if the output at any time depends only on the present and past inputs. All physically realizable real-time systems are causal. A system is memoryless if the output at time \( t \) depends only on the input at the same time \( t \); a multiplier is memoryless, but a filter is not. A system is stable in the bounded-input, bounded-output (BIBO) sense if every bounded input produces a bounded output. A patient monitor must of course be BIBO stable; otherwise a brief artifact could drive the display off scale forever.

Chapter 2: Continuous-Time LTI Systems and Convolution

The single most important result in linear systems theory is that the output of an LTI system is fully determined by its response to an impulse. Once we have that impulse response, we know everything.

2.1 The impulse response

Let \( h(t) \) denote the response of a continuous-time LTI system to the input \( \delta(t) \). By time invariance, its response to \( \delta(t - \tau) \) is \( h(t - \tau) \). By linearity, its response to a superposition of shifted impulses is the corresponding superposition of shifted impulse responses.

Now any reasonable input can be written as such a superposition via the sifting property:

\[ x(t) = \int_{-\infty}^{\infty} x(\tau) \, \delta(t - \tau) \, d\tau. \]

Viewing the right side as a continuous sum of scaled, shifted impulses, the output must be the same continuous sum of scaled, shifted impulse responses:

\[ y(t) = \int_{-\infty}^{\infty} x(\tau) \, h(t - \tau) \, d\tau. \]

This operation is called the convolution of \( x \) and \( h \), written \( y = x * h \).

Convolution has a geometric interpretation that is worth internalizing. To compute \( y(t_0) \), reflect \( h(\tau) \) about the vertical axis to form \( h(-\tau) \), shift the reflected copy by \( t_0 \) to form \( h(t_0 - \tau) \), multiply pointwise by \( x(\tau) \), and integrate over \( \tau \). The integrand is a product of the input, weighted by a mirrored and shifted impulse response. When \( h \) is short compared to features in \( x \), convolution approximates identity; when \( h \) is long, it smooths.

2.2 Properties of LTI systems from the impulse response

Several system properties translate directly into conditions on \( h \).

• Causality: the system is causal if and only if \( h(t) = 0 \) for \( t < 0 \). A future input cannot influence the present output.
• BIBO stability: the system is BIBO stable if and only if \( h \) is absolutely integrable, \[\int_{-\infty}^{\infty} \left| h(t) \right| \, dt < \infty.\]
• Memorylessness: the system is memoryless if and only if \( h(t) = K \, \delta(t) \) for some constant \( K \).

2.3 Step response and convolution with common signals

The step response \( s(t) \) of an LTI system is its response to \( u(t) \). Because \( u(t) = \int_{-\infty}^{t} \delta(\tau) \, d\tau \), we have

\[ s(t) = \int_{-\infty}^{t} h(\tau) \, d\tau, \]

so \( h = ds/dt \). This is practically useful: it is often easier to observe the response to a step (turning on a current, clamping a voltage) than to a true impulse, and we can differentiate numerically to recover \( h \).

Chapter 3: Differential Equations and System Response

Many continuous-time systems are described by linear constant-coefficient differential equations (LCCDEs):

\[ \sum_{k=0}^{N} a_k \, \frac{d^k y(t)}{dt^k} = \sum_{k=0}^{M} b_k \, \frac{d^k x(t)}{dt^k}. \]

The left side governs the natural dynamics of the system; the right side describes how the input drives it. A single RC filter is first order (\( N = 1 \)); a biomechanical second-order system such as a muscle tendon with inertia, damping, and stiffness has \( N = 2 \).

3.1 Zero-input and zero-state response

The solution decomposes linearly into a zero-input response (the solution with \( x = 0 \) but nonzero initial conditions) and a zero-state response (the solution with zero initial conditions and the given input). The zero-input response is a linear combination of the characteristic modes \( e^{\lambda_k t} \), where \( \lambda_k \) are roots of the characteristic polynomial \( \sum_k a_k \lambda^k = 0 \). The zero-state response is the convolution \( x * h \).

The total response can also be split into a natural response (transient, decaying for stable systems) and a forced response (of the same form as the driving input for standard inputs). For a stable LTI system driven by a sinusoid, the natural response dies out and the forced response is a sinusoid at the same frequency — the basis for frequency-response analysis.

3.2 Impulse response from an LCCDE

For an LCCDE with \( M \leq N \) and simple characteristic roots, the impulse response has the form

\[ h(t) = \left( \sum_{k=1}^{N} c_k \, e^{\lambda_k t} \right) u(t), \]

with coefficients \( c_k \) determined by matching the initial conditions implied by the differential equation at \( t = 0^+ \). When roots are repeated, polynomial-in-\( t \) factors appear. When roots are complex conjugate pairs \( \sigma \pm j\omega \), the modes combine into damped sinusoids \( e^{\sigma t} \cos(\omega t + \phi) \).

Chapter 4: The Laplace Transform and Transfer Functions

Solving LCCDEs with convolutions and characteristic roots is manageable for second-order systems and quickly becomes tedious for anything larger. The Laplace transform converts differentiation into multiplication by the complex variable \( s \), turning differential equations into algebraic equations.

4.1 Definition and convergence

Because most physical systems are causal and most signals in this course are defined for \( t \geq 0 \), we typically use the unilateral Laplace transform with lower limit zero. The unilateral form handles initial conditions cleanly and is the right tool for solving causal LCCDEs.

Convergence depends on \( \mathrm{Re}(s) = \sigma \). For \( x(t) = e^{at} u(t) \), \( X(s) = 1/(s - a) \) with ROC \( \sigma > a \). For \( x(t) = -e^{at} u(-t) \), \( X(s) = 1/(s - a) \) with ROC \( \sigma < a \). The algebraic expression is the same; only the ROC distinguishes the two signals. This is why specifying the ROC matters.

4.2 Key properties

Linearity is immediate. Time shifting: \( x(t - t_0) \leftrightarrow e^{-s t_0} X(s) \). Differentiation (unilateral, \( x \) causal with suitable regularity): \( dx/dt \leftrightarrow s X(s) - x(0^-) \), and higher derivatives peel off further initial conditions. Integration: \( \int_0^t x(\tau) d\tau \leftrightarrow X(s)/s \). Convolution in time becomes multiplication in \( s \): \( x * h \leftrightarrow X(s) \, H(s) \). The final-value theorem, when applicable, gives \( \lim_{t\to\infty} x(t) = \lim_{s\to 0} s X(s) \); the initial-value theorem gives \( x(0^+) = \lim_{s\to\infty} s X(s) \).

4.3 Transfer functions, poles, and zeros

For an LTI system described by an LCCDE with zero initial conditions, Laplace-transforming both sides gives \( A(s) Y(s) = B(s) X(s) \), so

\[ H(s) = \frac{Y(s)}{X(s)} = \frac{B(s)}{A(s)} = \frac{\sum_{k=0}^{M} b_k s^k}{\sum_{k=0}^{N} a_k s^k}. \]

\( H(s) \) is the transfer function. The roots of \( B(s) \) are the zeros of the system; the roots of \( A(s) \) are the poles. Poles are the same as the characteristic roots \( \lambda_k \) that govern the natural response.

The pole-zero plot in the complex \( s \)-plane summarizes the system’s dynamics. Poles close to the imaginary axis produce lightly damped oscillations; poles far into the left half-plane produce fast-decaying transients; poles in the right half-plane produce unbounded growth.

4.4 Solving LCCDEs with initial conditions

The unilateral Laplace transform handles initial conditions directly. Consider a first-order model of an RC amplifier’s low-frequency coupling:

\[ \tau \, \frac{dy}{dt} + y(t) = x(t), \qquad y(0^-) = y_0. \]

Transforming gives \( \tau \left[ s Y(s) - y_0 \right] + Y(s) = X(s) \), so

\[ Y(s) = \frac{X(s)}{\tau s + 1} + \frac{\tau y_0}{\tau s + 1}. \]

The first term is the zero-state response, the second the zero-input response. Inverse-transforming (with \( X(s) = 1/s \) for a unit step input) and decomposing into partial fractions returns the familiar \( y(t) = 1 - e^{-t/\tau} + y_0 e^{-t/\tau} \).

Chapter 5: Frequency Response and Bode Plots

Setting \( s = j\omega \) in a stable system’s transfer function evaluates it on the imaginary axis and yields the frequency response \( H(j\omega) \). This object is what engineers usually mean by a system’s “filter shape.”

5.1 Eigenfunction property and frequency response

Apply the input \( x(t) = e^{j\omega t} \) to an LTI system with impulse response \( h(t) \). Then

\[ y(t) = \int_{-\infty}^{\infty} h(\tau) \, e^{j\omega (t-\tau)} \, d\tau = e^{j\omega t} \int_{-\infty}^{\infty} h(\tau) \, e^{-j\omega \tau} \, d\tau = e^{j\omega t} H(j\omega). \]

The output is the same complex exponential, scaled by the complex number \( H(j\omega) \). That is what it means to call complex exponentials the eigenfunctions of LTI systems, and \( H(j\omega) \) the corresponding eigenvalues. For a real-valued sinusoidal input \( A\cos(\omega t + \phi) \), the steady-state response is \( A \left| H(j\omega) \right| \cos\!\left(\omega t + \phi + \angle H(j\omega)\right) \). Amplitude is scaled by \( \left| H(j\omega) \right| \); phase is shifted by \( \angle H(j\omega) \).

5.2 Bode plots

The Bode plot displays magnitude in decibels \( 20 \log_{10} \left| H(j\omega) \right| \) and phase in degrees, both against logarithmic \( \omega \). Its great virtue is that factored transfer functions turn into sums. For \( H(s) = K \prod_i (s - z_i) / \prod_j (s - p_j) \), magnitude in dB is a sum of contributions from each real pole, real zero, and complex-conjugate pair. Each first-order factor contributes a piecewise-linear asymptote with a corner at its pole or zero frequency, sloping at \( \pm 20 \) dB/decade past the corner. Complex pairs contribute \( \pm 40 \) dB/decade with a resonance peak near the natural frequency whose height depends on damping.

Biomedical instrumentation specifications are routinely given in Bode terms. An ECG amplifier is expected to have roughly flat gain from about 0.05 Hz to 150 Hz, with specified roll-off beyond. An EEG recorder is configured with a high-pass corner around 0.5 Hz to reject slow drifts and a low-pass corner near 70 Hz for standard clinical recordings. These are direct statements about \( H(j\omega) \).

Chapter 6: Fourier Series and the Fourier Transform

The Laplace transform is the right tool for dynamics and stability. For understanding the frequency content of a signal itself — the spectrum — the Fourier transform is central.

6.1 Fourier series for periodic signals

Any reasonable periodic continuous-time signal of fundamental period \( T_0 \) and fundamental frequency \( \omega_0 = 2\pi / T_0 \) can be written as

\[ x(t) = \sum_{k=-\infty}^{\infty} c_k \, e^{j k \omega_0 t}, \qquad c_k = \frac{1}{T_0} \int_{T_0} x(t) \, e^{-j k \omega_0 t} \, dt. \]

The \( c_k \) are complex Fourier series coefficients. Parseval’s identity gives the average power as \( \sum_k \left| c_k \right|^2 \), so the \( \left| c_k \right|^2 \) tell us how power is distributed among harmonics.

A periodic square pulse train (period \( T_0 \), pulse width \( T_p \)) has coefficients proportional to \( \mathrm{sinc}(k T_p / T_0) \), a discrete sampling of the sinc envelope. A narrow pulse produces a broad, slowly-decaying spectrum; a wide pulse produces a narrow spectrum concentrated near DC. Any time we sharpen a waveform, we broaden its spectrum.

6.2 The Fourier transform for aperiodic signals

Letting \( T_0 \to \infty \) turns the Fourier series into the Fourier transform:

\[ X(j\omega) = \int_{-\infty}^{\infty} x(t) \, e^{-j\omega t} \, dt, \qquad x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(j\omega) \, e^{j\omega t} \, d\omega. \]

For a real-valued signal, \( X(-j\omega) = \overline{X(j\omega)} \), so \( \left| X(j\omega) \right| \) is even and \( \angle X(j\omega) \) is odd. The energy spectral density is \( \left| X(j\omega) \right|^2 \); Parseval again gives total energy as \( \frac{1}{2\pi} \int \left| X(j\omega) \right|^2 d\omega \).

Important properties: linearity; time shift \( x(t - t_0) \leftrightarrow e^{-j\omega t_0} X(j\omega) \); frequency shift (modulation) \( e^{j\omega_0 t} x(t) \leftrightarrow X(j(\omega - \omega_0)) \); time scaling \( x(at) \leftrightarrow \frac{1}{|a|} X(j\omega/a) \); duality; differentiation \( dx/dt \leftrightarrow j\omega X(j\omega) \); and the convolution property \( x * h \leftrightarrow X(j\omega) \, H(j\omega) \). The last is the frequency-domain statement of the fact that LTI systems work by shaping the spectrum of the input.

6.3 Connection to the Laplace transform

The Fourier transform is the Laplace transform evaluated on the imaginary axis, provided the axis lies in the Laplace ROC. For BIBO-stable causal systems this is always true, and \( H(j\omega) = H(s) \big|_{s = j\omega} \). For unstable systems the Fourier transform may not exist, even though the Laplace transform does; this is yet another sense in which the Laplace transform is the more general tool and the Fourier transform a restriction that exposes steady-state behavior.

Chapter 7: Discrete-Time Signals and Systems

Modern biomedical instruments sample signals and process them digitally. The theory we have built up for continuous time has a parallel in discrete time, structurally identical but with its own subtleties.

7.1 Elementary discrete-time signals and classifications

The discrete-time unit impulse \( \delta[n] \) is one at \( n = 0 \) and zero elsewhere. The unit step \( u[n] \) is zero for \( n < 0 \) and one for \( n \geq 0 \); note \( \delta[n] = u[n] - u[n-1] \). The complex exponential \( x[n] = A \, z^n \) for \( z \in \mathbb{C} \) plays the role of \( e^{st} \). A key distinction from continuous time: the signal \( e^{j\omega n} \) is periodic in \( \omega \) with period \( 2\pi \), because \( e^{j(\omega + 2\pi) n} = e^{j\omega n} \). In discrete time, frequency lives on a circle, not a line.

The classifications of linearity, time invariance, causality, memorylessness, and BIBO stability carry over verbatim to discrete-time systems, with differential operators replaced by shifts.

7.2 Convolution sum

For a discrete-time LTI system with impulse response \( h[n] \) and input \( x[n] \), the output is the convolution sum

\[ y[n] = \sum_{k=-\infty}^{\infty} x[k] \, h[n - k]. \]

The system is causal if and only if \( h[n] = 0 \) for \( n < 0 \) and BIBO stable if and only if \( \sum_n \left| h[n] \right| < \infty \).

7.3 Linear constant-coefficient difference equations

The discrete-time analogue of an LCCDE is the linear constant-coefficient difference equation (LCCDE, same acronym by convention):

\[ \sum_{k=0}^{N} a_k \, y[n-k] = \sum_{k=0}^{M} b_k \, x[n-k]. \]

When \( N = 0 \), \( y[n] \) depends only on the current and past \( M \) samples of \( x \); the impulse response has finite support \( M+1 \), and the system is a finite impulse response (FIR) filter. When \( N > 0 \), the recursion makes the impulse response (generically) infinite, and the system is an infinite impulse response (IIR) filter.

Chapter 8: The Z-Transform

The Z-transform is to discrete-time systems what the Laplace transform is to continuous-time systems.

8.1 Definition and region of convergence

For a causal signal the ROC is the exterior of a disk centered at the origin. The unit circle \( \left| z \right| = 1 \) plays the role of the imaginary axis: the discrete-time Fourier transform (DTFT) \( X(e^{j\Omega}) \) is \( X(z) \) evaluated on the unit circle, provided the unit circle lies in the ROC.

8.2 Properties

Linearity; time shift \( x[n - n_0] \leftrightarrow z^{-n_0} X(z) \); convolution \( x * h \leftrightarrow X(z) H(z) \); scaling in z-domain \( \alpha^n x[n] \leftrightarrow X(z/\alpha) \); differentiation in z \( n \, x[n] \leftrightarrow -z \, dX/dz \). The unilateral Z-transform handles initial conditions analogously to the unilateral Laplace transform.

8.3 Transfer functions, poles, zeros, stability

Transforming the LCCDE \( \sum a_k y[n-k] = \sum b_k x[n-k] \) with zero initial conditions gives

\[ H(z) = \frac{\sum_{k=0}^{M} b_k \, z^{-k}}{\sum_{k=0}^{N} a_k \, z^{-k}}. \]

Multiplying numerator and denominator by \( z^{N} \) (assuming \( M \leq N \)) yields a rational function in \( z \) whose poles and zeros are the roots of the denominator and numerator.

This is the discrete analogue of the left-half-plane condition. The left half-plane maps to the interior of the unit circle under \( z = e^{sT_s} \), where \( T_s \) is the sampling period — a relationship that formalizes the connection between continuous and discrete systems, and underlies several discretization methods such as the bilinear transform.

8.4 Inverse Z-transforms

For rational \( X(z) \) the standard technique is partial-fraction expansion followed by inverse-transforming each term via a table. A common pitfall: the same algebraic \( X(z) \) corresponds to different time-domain signals depending on the ROC, so always track the ROC when inverting.

Chapter 9: Sampling, Reconstruction, and Aliasing

Whenever we represent a continuous-time biosignal inside a computer, we sample it. Sampling is what connects the two parallel worlds we have built.

9.1 Ideal sampling

Define the impulse train \( s(t) = \sum_{n=-\infty}^{\infty} \delta(t - n T_s) \). The ideal sampled signal is \( x_s(t) = x(t) s(t) = \sum_n x(n T_s) \delta(t - n T_s) \). In the frequency domain, multiplication by \( s(t) \) corresponds to convolution with the transform of \( s \), which is itself an impulse train spaced by \( \omega_s = 2\pi / T_s \). Therefore

\[ X_s(j\omega) = \frac{1}{T_s} \sum_{k=-\infty}^{\infty} X\!\left(j(\omega - k \omega_s)\right). \]

Sampling creates infinitely many shifted copies of the original spectrum, each scaled by \( 1/T_s \) and centered at integer multiples of the sampling frequency.

9.2 The sampling theorem

The threshold \( \omega_s = 2 \omega_M \) is the Nyquist rate. Violating it produces aliasing: shifted copies of \( X(j\omega) \) overlap, and high-frequency content is misrepresented as low-frequency content in the sampled sequence.

9.3 Anti-alias filtering and biomedical practice

Because real biosignals are never strictly bandlimited, practical digitization places an anti-alias filter — an analog lowpass filter with cutoff below the Nyquist frequency — before the ADC. For example, to sample an ECG at 500 Hz, one low-passes the analog signal at about 150 Hz so that content above 250 Hz has been attenuated below the quantization floor before it can fold in. Chosen cutoffs are a negotiation between preserving signal fidelity (including sharp QRS edges) and guaranteeing alias-free sampling.

9.4 Reconstruction

Ideal reconstruction passes \( x_s(t) \) through a lowpass filter with gain \( T_s \) and cutoff \( \omega_s/2 \), yielding the sinc interpolation

\[ x(t) = \sum_{n=-\infty}^{\infty} x(n T_s) \, \mathrm{sinc}\!\left( \frac{t - n T_s}{T_s} \right). \]

Practical DACs use zero-order hold followed by smoothing, not an ideal sinc, because an ideal sinc interpolator is non-causal and infinitely long.

Chapter 10: Digital Filters — FIR and IIR

With the Z-transform and sampling theorem in hand we can design digital filters that replace, or augment, analog front-ends.

10.1 FIR filters

An FIR filter of length \( L \) has the form

\[ y[n] = \sum_{k=0}^{L-1} b_k \, x[n - k], \qquad H(z) = \sum_{k=0}^{L-1} b_k \, z^{-k}. \]

FIR filters are always BIBO stable (their impulse response is compactly supported), and they can be designed to have exactly linear phase, which means a constant group delay and no phase distortion across the passband. Linear phase matters in ECG processing because distorting the phase of the QRS complex relative to the T-wave can invalidate clinical features such as ST-segment deviation. The simplest linear-phase design is the moving average, \( b_k = 1/L \) for \( k = 0, \ldots, L-1 \); more sophisticated designs use windowed sinc functions or the Parks–McClellan algorithm to meet arbitrary specifications.

10.2 IIR filters

IIR filters exploit feedback. A first-order IIR lowpass \( y[n] = (1 - \alpha) x[n] + \alpha y[n-1] \) has transfer function \( H(z) = (1 - \alpha)/(1 - \alpha z^{-1}) \), pole at \( z = \alpha \), and is stable for \( \left| \alpha \right| < 1 \). IIR filters typically meet a given magnitude specification with far fewer coefficients than FIR filters, making them attractive for embedded biosignal processors with tight compute budgets, but they generally do not have linear phase.

A common design path for IIR filters is to start from an analog prototype — a Butterworth, Chebyshev, or elliptic lowpass — and map it to discrete time using the bilinear transform, \( s = \frac{2}{T_s} \frac{1 - z^{-1}}{1 + z^{-1}} \), which maps the left half \( s \)-plane bijectively to the interior of the unit circle and so preserves stability.

10.3 Filter specifications and trade-offs

Design begins with specifications: passband edge, stopband edge, allowed passband ripple, required stopband attenuation, and, if relevant, phase linearity. FIR linear-phase filters tend to have higher order; IIR filters are more compact but require attention to phase. When post-processing is acceptable (offline ECG analysis, sleep-stage scoring), noncausal zero-phase filtering — running an IIR filter forward, then backward — gives the compactness of IIR with zero net phase distortion.

Chapter 11: Stability, Feedback, and Control Considerations

Feedback is the recurring structural idea that lets us build stable, high-performance systems out of imperfect components. In biomedical engineering it appears whenever a controller regulates a physiological variable — closed-loop insulin delivery, ventilator pressure control, adaptive stimulation in deep-brain-stimulation devices.

11.1 Feedback configuration

Consider a loop with forward path \( G(s) \) and feedback path \( H(s) \). The closed-loop transfer function from reference \( r \) to output \( y \) is

\[ T(s) = \frac{G(s)}{1 + G(s) H(s)}. \]

The characteristic equation of the loop is \( 1 + G(s) H(s) = 0 \); its roots are the closed-loop poles. Feedback can stabilize a system whose open-loop \( G(s) \) is unstable or marginally stable, and it can desensitize the output to variations in \( G \). It can also destabilize a system that was open-loop stable, if loop gain and phase conspire to push closed-loop poles across the stability boundary.

11.2 Stability criteria

Several tools analyze stability without solving for poles explicitly. The Routh–Hurwitz criterion tests a continuous-time characteristic polynomial for right-half-plane roots from its coefficients alone. The Nyquist criterion relates the number of right-half-plane closed-loop poles to encirclements of the \( -1 \) point by the locus \( G(j\omega)H(j\omega) \). Bode gain and phase margins give distances from instability: gain margin is how much further the loop gain could increase before \( \left| G H \right| \) passes through 1 with phase \( -180^\circ \), and phase margin is how much phase lag could be added at the unity-gain crossover before the same happens. In discrete time the analogous criterion is the Jury test, evaluating whether all roots of the characteristic polynomial lie inside the unit circle.

11.3 Practical implications

In physiological control the plant \( G(s) \) often contains substantial delay (drug transport time, sensor latency). Delay contributes pure phase lag without changing magnitude, and so chews through phase margin. A sensor delay of half a second can turn a nominally stable loop into one that oscillates. Good designs limit loop bandwidth well below the inverse delay, or use predictive schemes to compensate for known transport lags.

Chapter 12: Biosignal Applications and Case Studies

We close by walking through biomedical signal processing scenarios that bring the framework of the course together.

12.1 The ECG processing chain

The raw ECG from chest leads is acquired with an instrumentation amplifier, shaped by an analog front-end (roughly 0.05 Hz high-pass, 150 Hz low-pass, 60 Hz notch), sampled at 500–1000 Hz, then digitally filtered and analyzed. Baseline wander from respiration is removed with a high-pass around 0.5 Hz (or a median-filter baseline estimator). QRS detection classically uses the Pan–Tompkins algorithm: a bandpass filter (approximately 5–15 Hz) emphasizes QRS energy, a derivative enhances the steep slopes, squaring makes the signal positive and further emphasizes peaks, a moving-window integration smooths to a single peak per beat, and thresholding with adaptive logic extracts R-peaks. Every step is an LTI filter (with the exception of squaring and thresholding); the entire pipeline is a composition of operations we have analyzed in this course.

From R-peak times follows the RR interval series, a discrete-time signal at non-uniform times that, after resampling, can be Fourier-analyzed to study heart rate variability. Power in the low-frequency band (0.04–0.15 Hz) and the high-frequency band (0.15–0.4 Hz) reflects autonomic nervous system activity. The whole practice rests on the interpretation of power spectra.

12.2 EEG rhythms and spectral analysis

EEG is conventionally analyzed in frequency bands: delta (0.5–4 Hz), theta (4–8 Hz), alpha (8–13 Hz), beta (13–30 Hz), gamma (30+ Hz). A resting EEG with eyes closed shows prominent alpha over occipital electrodes; the alpha rhythm attenuates when the eyes open. Computing short-time Fourier transforms on windowed segments produces a spectrogram, in which rhythms appear as horizontal bands and events as localized splotches. The choice of window length is a direct instance of the time–frequency trade-off: short windows resolve events in time but smear them in frequency, long windows do the opposite.

12.3 Surface EMG and envelope detection

Surface EMG is a stochastic, approximately zero-mean signal whose amplitude grows with contraction. The standard envelope estimation chain bandpasses the EMG (roughly 20–500 Hz) to remove motion artifact and anti-aliasing residue, rectifies it (full-wave), then lowpass filters with a cutoff of a few hertz to produce a slowly-varying envelope. Each step is interpretable in the framework of this course: the bandpass shapes the spectrum, rectification is a nonlinearity that moves power to baseband, and the lowpass extracts that baseband.

12.4 Evoked potentials and synchronous averaging

Evoked potentials — the brain’s response to a controlled stimulus — are tiny (microvolts) compared to the ongoing EEG (tens of microvolts). One extracts them by synchronous averaging: record many stimulus-locked epochs and average. If the signal is deterministic and the background is zero-mean noise, averaging \( N \) epochs improves the signal-to-noise ratio by \( \sqrt{N} \). This is a linear operation, and the averaging is itself an LTI filter from the perspective of the stimulus-locked ensemble.

12.5 Putting it together

A recurring theme across these applications is that the same few ideas — LTI modeling, impulse responses, transfer functions, frequency responses, sampling, and stability — suffice to understand and design the vast majority of clinical biosignal processing. A working biomedical engineer does not memorize the Pan–Tompkins algorithm; they design or adapt it from first principles by reasoning about the spectra of ECG features and interference, selecting filters accordingly, and verifying stability and phase behavior.

The tools in this course generalize far beyond biopotentials. Imaging systems (MRI, ultrasound, optical coherence tomography) are described by point-spread functions — two-dimensional impulse responses — and analyzed through two-dimensional Fourier transforms. Pharmacokinetic compartment models are LCCDEs whose transfer functions have meaningful pharmacological interpretations. Neural spike trains, once converted to point processes and then to rate functions, are analyzed with the same spectral machinery as continuous biosignals. In every case the underlying theory is the one developed here: linear, time-invariant, characterized by an impulse response, best understood in the frequency domain, and disciplined by the sampling theorem once it enters a digital system.