Filter design proceeds from a transfer function $H(s)$, its pole-zero map, and its Bode response. A biomedical recording, however, is judged on waveform morphology, on spectral content, and on its clinical interpretation. Each section below isolates one mechanism by which $H(s)$ alters the recorded signal, relates it to the governing equation, and states the standard remedy.
Waveform fidelity requires that every spectral component be delayed by the same amount. The relevant quantity is the group delay, $$\tau_g(\omega) = -\frac{d\phi(\omega)}{d\omega},$$ where $\phi(\omega) = \angle H(j\omega)$. Constant $\tau_g$ corresponds to linear phase and produces a pure time shift; non-constant $\tau_g$ shifts components by different amounts and reshapes the waveform.
A fourth-order Butterworth low-pass filter is maximally flat in magnitude, yet $\tau_g(\omega)$ rises near cutoff. Faster ECG features (the QRS complex) are delayed differently from slower ones (the T wave), so inter-beat intervals and ST timing are corrupted. The Bessel response is designed to maximise the flatness of $\tau_g(\omega)$ at the expense of a softer roll-off.
For a rational transfer function $H(s) = N(s)/D(s)$, the impulse response is a sum of modes set by the roots $p_i$ of $D(s)$. A complex pair $p = \sigma \pm j\omega_d$ contributes a term of the form $$h(t) = A\, e^{\sigma t} \cos(\omega_d t + \phi).$$ Bounded-input bounded-output stability therefore requires $\mathrm{Re}(p_i) < 0$ for every pole.
If any pole crosses the imaginary axis, $\sigma \ge 0$, the mode no longer decays and the output either oscillates indefinitely or diverges. In analogue instrumentation this typically arises from excessive loop gain, insufficient phase margin, or parasitic feedback around an op-amp stage. The Routh-Hurwitz criterion gives an algebraic test on the coefficients of $D(s)$ without solving for the roots explicitly.
A second-order low-pass section has the canonical form $$H(s) = \frac{\omega_n^2}{s^2 + \dfrac{\omega_n}{Q}\, s + \omega_n^2},$$ with damping ratio $\zeta = 1/(2Q)$. The poles lie at $p = -\zeta\omega_n \pm j\omega_n\sqrt{1-\zeta^2}$. As $Q$ increases, the poles approach the imaginary axis and the response becomes underdamped.
The fractional overshoot of the step response is $$M_p = \exp\!\left(-\frac{\pi\zeta}{\sqrt{1-\zeta^2}}\right),$$ so a sharper transition is paid for in time-domain ringing. For reference, $Q = 0.707$ (Butterworth) gives a maximally flat magnitude with $\approx 4\%$ overshoot, and $Q \approx 0.58$ (Bessel) gives no overshoot. Above $Q \approx 1$, oscillatory tails appear on QRS edges that are not present in the source.
An ideal low-pass with cutoff $f_c$ has frequency response $H(f) = \mathrm{rect}(f/2f_c)$, and its inverse Fourier transform is $$h(t) = 2 f_c\, \mathrm{sinc}(2 f_c t),\qquad \mathrm{sinc}(x) = \frac{\sin \pi x}{\pi x}.$$ Because $h(t)$ extends over all time and decays only as $1/t$, truncation to a finite FIR length introduces the Gibbs phenomenon: oscillations near the discontinuity whose peak amplitude does not vanish as the length grows.
For a non-causal symmetric truncation, the ringing appears on both sides of a sharp event, including a pre-ringing lobe that precedes the QRS spike in the output. The artifact is purely a property of $h(t)$, not of the underlying physiology. Multiplying $h(t)$ by a smooth window (Hamming, Blackman, Kaiser) trades transition sharpness for reduced sidelobe energy.
A second-order notch centred at $\omega_0$ has the transfer function $$H(s) = \frac{s^2 + \omega_0^2}{s^2 + \dfrac{\omega_0}{Q}\, s + \omega_0^2},$$ with $-3\,\mathrm{dB}$ rejection bandwidth $\mathrm{BW} = f_0/Q$. The notch quality factor $Q$ therefore trades selectivity for residual interference.
For a 50 Hz mains notch on an EMG recording, a low $Q$ widens $\mathrm{BW}$ and removes physiological EMG content adjacent to 50 Hz. A high $Q$ leaves the EMG spectrum intact but is sensitive to drift in the mains frequency: when the interference moves outside the narrow notch, residual leakage remains. The choice is governed by how stationary the interference is.
A Chebyshev type I low-pass of order $N$ has squared magnitude $$|H(j\omega)|^2 = \frac{1}{1 + \varepsilon^2\, T_N^2(\omega/\omega_c)},$$ where $T_N$ is the Chebyshev polynomial and the passband ripple in decibels is $A_p = 10 \log_{10}(1 + \varepsilon^2)$. The polynomial $T_N(x)$ oscillates between $\pm 1$ on $|x| \le 1$, so the magnitude oscillates between $1$ and $1/\sqrt{1+\varepsilon^2}$ across the passband.
The reward is a steeper transition, given by the equiripple property; the cost is that any in-band Fourier component is multiplied by a position-dependent gain. A multi-component biomedical waveform therefore changes shape even when every component lies below the nominal cutoff.
The Nyquist-Shannon theorem requires that a band-limited signal of highest frequency $f_{\max}$ be sampled at $f_s \ge 2 f_{\max}$. The Nyquist frequency $f_N = f_s/2$ is the largest frequency that can be uniquely represented at a sampling rate $f_s$.
A sinusoid at frequency $f$ exceeding $f_N$ is indistinguishable from its alias $$f_{\mathrm{alias}} = \bigl|\, f - k f_s \,\bigr|,\qquad k = \mathrm{round}(f/f_s),$$ so it appears in the sampled record as a low-frequency artifact within $[0, f_N]$. A 40 Hz component superimposed on ECG, sampled at $f_s = 65\,\mathrm{Hz}$, aliases to $25\,\mathrm{Hz}$ and contaminates the diagnostic band. The artifact cannot be removed after sampling; it must be prevented before it.
A first-order high-pass has the transfer function $$H(s) = \frac{s}{s + \omega_c},\qquad \omega_c = 2\pi f_c,$$ with magnitude $|H(j\omega)| = \omega/\sqrt{\omega^2 + \omega_c^2}$. Frequencies satisfying $\omega \ll \omega_c$ are attenuated and phase-shifted; the response is not a simple subtraction of a DC level.
The ST segment of the ECG is a quasi-DC feature whose spectral energy lies below $1\,\mathrm{Hz}$. Raising $f_c$ to suppress baseline drift therefore attenuates the very feature used to detect myocardial ischemia, and the differentiator-like phase response of $H(s)$ tilts the segment, producing apparent elevation or depression that is an artifact of the filter. AHA guidance specifies $f_c \le 0.05\,\mathrm{Hz}$ for diagnostic ECG and tolerates $f_c \le 0.5\,\mathrm{Hz}$ only in monitoring contexts.
An $N$-bit ADC partitions a full-scale range $V_{\mathrm{FS}}$ into $2^N$ codes of width $$q = \frac{V_{\mathrm{FS}}}{2^N}.$$ Modelling the quantization error as uniform on $[-q/2, +q/2]$ gives mean-square error $\sigma_q^2 = q^2/12$. For a full-scale sinusoid the resulting signal-to-noise ratio is $$\mathrm{SNR} = 6.02 N + 1.76\ \mathrm{dB},$$ so each additional bit improves SNR by approximately $6\,\mathrm{dB}$.
At low bit depth the ECG shows a visible staircase, and the spectrum gains a broadband pedestal at $\sigma_q^2 / f_N$ per hertz. Dithering decorrelates the quantization error from the signal at the cost of a small noise increase; oversampling by a factor $M$ followed by decimation reduces in-band noise by $10 \log_{10} M\ \mathrm{dB}$.
A first-order high-pass coupling stage with resistance $R$ and capacitance $C$ has time constant $\tau = RC$. Its response to an initial offset $V_0$ relaxing toward steady state $V_\infty$ is $$v_o(t) = V_\infty + (V_0 - V_\infty)\, e^{-t/\tau}.$$ The transient is set by the circuit, not by the physiology, and it superimposes on every measurement made before settling is complete.
The offset $V_0$ is largest immediately after lead connection, after electrode motion, after recovery from amplifier saturation, or at power-up. The exponential reaches $99.3\%$ of its final value in $5\tau$, so this interval is conventionally treated as the settling time. Acquisition during this window records $v_o(t)$, not the patient signal.
A Sallen-Key second-order section has natural frequency $$f_c = \frac{1}{2\pi\sqrt{R_1 R_2 C_1 C_2}}.$$ Logarithmic differentiation gives the first-order sensitivity $$\frac{\delta f_c}{f_c} \approx -\tfrac{1}{2}\!\left(\frac{\delta R_1}{R_1} + \frac{\delta R_2}{R_2} + \frac{\delta C_1}{C_1} + \frac{\delta C_2}{C_2}\right),$$ so independent ±1% resistors and ±5% capacitors give $\sigma_{f_c}/f_c \approx \tfrac{1}{2}\sqrt{2(0.01)^2 + 2(0.05)^2} \approx 3.6\%$. Temperature coefficients add a slow drift on the same parameters.
The pole quality factor $Q$ depends on the same components, and high-$Q$ sections amplify the spread. The plot shows a Monte Carlo ensemble of magnitude responses for the chosen tolerance and nominal $Q$.
A real op-amp has open-loop gain $A(s) = A_0/(1 + s/\omega_p)$ with gain-bandwidth product $\mathrm{GBW} = A_0 \omega_p / 2\pi$. For a closed-loop stage of noise gain $G$, the loop bandwidth is $$f_{cl} \approx \frac{\mathrm{GBW}}{G},$$ and the realised response of an active filter is the product of the intentional response and the op-amp roll-off at $f_{cl}$.
For a biopotential front end with $G = 100$ driving an intentional low-pass at $f_c = 100\,\mathrm{Hz}$, a $1\,\mathrm{MHz}$ op-amp gives $f_{cl} = 10\,\mathrm{kHz}$, and the realised passband is flat. A $0.1\,\mathrm{MHz}$ op-amp gives $f_{cl} = 1\,\mathrm{kHz}$ and noticeably attenuates the upper passband.
An op-amp has an input offset voltage $V_{os}$ and input bias currents $I_{B+}, I_{B-}$. With a source resistance $R_s$ at the non-inverting terminal, the referred-to-input error is $$V_{e} = V_{os} + I_{B+} R_s,$$ and the stage of gain $G$ delivers an output offset $V_{o,\text{off}} = G\, V_e$. When $|V_{o,\text{off}}|$ exceeds the supply rails the amplifier saturates and the recorded waveform is clipped.
Bioelectrode source impedance is large and time-varying: gel electrodes sit at $10\text{–}100\,\mathrm{k}\Omega$, dry electrodes can reach $1\text{–}10\,\mathrm{M}\Omega$. A $100\,\mathrm{nA}$ bias current and a $1\,\mathrm{M}\Omega$ source produce $100\,\mathrm{mV}$ of input offset before any signal amplification.
The output noise of an active filter is set by three uncorrelated sources referred to the input: the op-amp voltage noise density $e_n$, the current noise $i_n$ flowing in the source resistance $R$, and the Johnson noise of $R$, $$e_R = \sqrt{4 k_B T R}.$$ The total input-referred noise density is $$e_{ni} = \sqrt{e_n^2 + (i_n R)^2 + 4 k_B T R}\ \ [\mathrm{V}/\!\sqrt{\mathrm{Hz}}],$$ and the integrated RMS output noise over the filter bandwidth $\Delta f$ is $V_{n,\text{rms}} = G\sqrt{\int_0^{\infty} |H(j2\pi f)|^2 e_{ni}^2\, df}$.
Choice of $R$ is a trade. Small $R$ minimises $e_R$ and $i_n R$ but loads the previous stage and the op-amp output. Large $R$ relaxes loading but raises both Johnson and current-noise contributions. The optimum depends on $e_n / i_n$ for the chosen op-amp.
An AC-coupled input has a coupling capacitor $C_c$ followed by a bias resistor $R_{in}$ to reference. The intended high-pass corner is $f_c = 1/(2\pi R_{in} C_c)$. Adding a non-zero electrode impedance $Z_e$ in series shifts the realised corner to $$f_c' = \frac{1}{2\pi (R_{in} + Z_e)\, C_c},$$ and reduces the in-band gain by the divider ratio $R_{in}/(R_{in} + Z_e)$.
Skin-electrode impedance is large and time-varying: gel electrodes settle at $10\text{–}100\,\mathrm{k}\Omega$, dry contacts at $1\text{–}10\,\mathrm{M}\Omega$, and motion modulates $Z_e(t)$ on a beat-by-beat scale. The realised filter therefore drifts during the recording, even when the schematic is fixed.
A differential biopotential channel relies on the front end to reject mains and electrosurgical interference that appears equally on both inputs. If a per-channel low-pass with time constants $\tau_1, \tau_2$ is placed before the difference stage, its mismatch $\delta\tau = \tau_1 - \tau_2$ converts a common-mode input $V_{cm}$ into a differential signal of magnitude $$V_{d,\text{cm}}(\omega) = V_{cm}\!\left| \frac{1}{1 + j\omega \tau_1} - \frac{1}{1 + j\omega \tau_2} \right|.$$ The filter's contribution to the system common-mode rejection ratio is therefore $\mathrm{CMRR}_F(\omega) = |H_{\text{diff}}|/|H_{cm \to d}|$, which falls with frequency as the mismatch term grows.
A 1% RC mismatch limits $\mathrm{CMRR}_F$ to roughly $40\,\mathrm{dB}$ at the corner, which is well below the $80\text{–}120\,\mathrm{dB}$ specified for an instrumentation amplifier alone.