Cheat sheet

Every key formula, in canonical order, at a glance. Each formula links to the lesson that develops it.

2 — The audiogram

Pure-tone average

\text{PTA} = \tfrac{1}{3}(\text{HL}_{500} + \text{HL}_{1000} + \text{HL}_{2000})

Decibel (power-like)

L = 10\,\log_{10}(Q/Q_0)

Decibel (amplitude-like)

L = 20\,\log_{10}(A/A_0)

Sound pressure level (dB SPL)

L_p = 20\,\log_{10}(p_\text{rms}/20\,\mu\text{Pa})

Hearing level (dB HL)

L_\text{HL}(f) = L_p - \text{RETSPL}(f)

Sensation level (dB SL)

L_\text{SL}(f) = L_\text{HL}(f) - \text{threshold}(f)

A-weighted level (dB A)

L_A(f) = L_p(f) + W_A(f)

Air-bone gap

\text{ABG}(f) = \text{AC}(f) - \text{BC}(f)

Cross-hearing level at non-test ear

L_\text{NTE} = L_\text{test} - \text{IA}

AC masking needed when

L_\text{AC,test} \ge L_\text{BC,NTE} + \text{IA}

3 — Speech audiometry

Rollover ratio (flags retrocochlear if $> 0.4$ )

\text{peak WRS} - \text{WRS}_\text{max level}

Dynamic range

\text{UCL} - \text{SRT}

Speech intelligibility index

\text{SII} = \sum_i I_i \, A_i

4 — Tympanometry and the middle ear

Specific acoustic impedance

z = p / u

Characteristic impedance of air

z = \rho_0 c \approx 415\ \text{Pa·s/m}

Acoustic impedance (volume velocity)

Z = p / U,\ \ U = u A

Acoustic admittance

Y = 1/Z

Parallel admittance at probe

Y_\text{total} = Y_\text{canal} + Y_\text{eardrum}

Trapped-air admittance (ECV)

Y_\text{air} = j\omega V/(\rho_0 c^2),\ \ V = A L

Tympanogram canal correction

Y_\text{tymp}(p) = Y_\text{ear-canal} + Y_\text{eardrum}(p)

Reflex sensation level (Metz)

\text{SL}_\text{reflex} = \text{ART} - \text{threshold}

5 — Otoacoustic emissions

TEOAE traveling-wave group delay

\tau(f) \approx N / f^{0.5}

Cochlear nonlinearity

y = \alpha x + \beta x^2 + \gamma x^3 + \ldots

Cubic difference tone frequency

2 f_1 - f_2

DP amplitude (pure cubic)

y_{2\omega_1-\omega_2} = \tfrac{3}{4}\,\gamma A_1^2 A_2

DP amplitude scaling

\propto A_1^2 A_2

Scissors protocol

f_2/f_1 = 1.22,\ L_1 = 65,\ L_2 = 55\ \text{dB SPL}

6 — Evoked potentials

Single-trial model

V_k(t) = s(t) + n_k(t)

Averaged response over $N$ trials

\bar V_N(t) = s(t) + \tfrac{1}{N}\sum_{k} n_k(t)

Noise reduction by averaging

\sigma \to \sigma/\sqrt{N}

Post-average SNR ( $+3$ dB per doubling)

\text{SNR}_N = \sqrt{N}\,\lvert s\rvert/\sigma

ABR wave-V latency–intensity slope

d\tau_V/dL \approx 0.04\ \text{ms/dB}

7 — Hearing aids

Compression ratio

\text{CR} = \Delta L_\text{in}/\Delta L_\text{out}

WDRC I/O, linear region

L_\text{out} = L_\text{in} + G_0\ \ (L_\text{in} < \text{CT})

WDRC I/O, compression region

L_\text{out} = \text{CT} + G_0 + (L_\text{in} - \text{CT})/\text{CR}

WDRC I/O, limiting region

L_\text{out} = \text{MPO}

First-order directional pattern

p(\theta) = A + B\cos\theta

Two-mic beamformer output

y(t) = m_1(t) - m_2(t-\tau)

Null-steering delay

\tau = d/c

Front-arrival response

y \approx 2\tau\,\dot m_1(t)

Closed-loop feedback transfer function

y(f)/x(f) = G(f)/\bigl(1 - G(f)H(f)\bigr)

Nyquist stability criterion

\lvert G(f)H(f)\rvert < 1

AFC / LMS filter update

\hat H(t+1) = \hat H(t) + \mu\,\tilde m(t)\,y(t-\tau)

8 — Real-ear measurement and verification

Unaided insertion loss

\text{IL}(f) = \text{REUR}(f) - \text{REOR}(f)

Real-ear-to-coupler difference

\text{RECD}(f) = \text{SPL}_\text{real-ear}(f) - \text{SPL}_\text{coupler}(f)

Real-ear insertion gain

\text{REIG}(f) = \text{REAR}(f) - \text{REUR}(f)

Fitting error (verified within $\pm 5$ dB)

\text{REAR}(f) - \text{target}(f)

9 — Cochlear implants

Point-source current density

J(r) \sim I/(4\pi r^2)

Logarithmic compression mapping

\text{stim}(t) = \alpha\,\log(1 + \beta\,\text{env}(t))

Total stimulation rate

\text{rate}\times N