Swept Along: Measuring Otoacoustic Emissions Using Continuously Varying Stimuli

At the same 2004 Midwinter Meeting of the ARO during which swept-tone DPOAEs made their debut [5], and presented almost side-by-side, just two posters down the aisle, Stephen Neely and his colleagues described a method for continuously varying the primary levels [6‐8].

To appreciate the depth of the times involved, note that the scale along the abscissa in Fig. 4 indicates how twice the expected SNR for a synchronous averaging task varies with the measurement time. Thus, relative to the SNR achieved during the first year, measuring a sinusoid embedded in Gaussian noise for the full 65 million years since the K-T extinction lowers the noise floor by some \(\frac{1}{2}20\log _{10}(65) = 78\,\) dB.

Although echolocation evolved multiple times on Earth—not only in bats but also in toothed whales (e.g., dolphins), as well as in shrews and tenrecs [18], oilbirds and swiftlets [19], and some blind humans [20]—only laryngeal bats employ frequency-modulated, chirp-like signals rather than acoustic clicks.

To estimate the group velocity of the traveling wave, we approximate both the BM frequency response (\(\textrm{TF}\)) and the traveling-wave envelope (\(\textrm{TW}\)) as Gaussians of widths \(\sigma _f\) and \(\sigma _x\), respectively. Thus, \(\textrm{TF}= \exp [-\frac{1}{2}((f-\textrm{CF})/\sigma _f)^2]\) and \(\textrm{TW}= \exp [-\frac{1}{2}((x-\hat{x})/\sigma _x)^2]\), where \(\textrm{CF}\) is the characteristic frequency and \(\hat{x}\) is the best place (peak of the wave). When the cochlear tonotopic map is exponential, the widths \(\sigma _f\) and \(\sigma _x\) are related by \(\sigma _x=(\ell /\textrm{CF})\sigma _f\), where \(\ell\) is the exponential space constant of the map. The equivalent rectangular bandwidth (\(\textrm{ERB}\)) of the Gaussian filter is given by \(\textrm{ERB}=\sqrt{\pi }\sigma _f\). We approximate the group velocity at the peak of the traveling wave by \(v_\textrm{g}\approx \Delta x/\tau\), where \(\Delta x\) is the effective width of the peak region and \(\tau\) is the wave travel time across this region. Since OAE latency is dominated by round-trip delays within the peak region, we approximate \(\tau\) as \(\tau \approx \frac{1}{2}\tau _\textrm{SFE}\), where \(\tau _\textrm{SFE}\) is SFOAE group delay. Approximating the width of the scattering region as \(\Delta x\approx 4\sigma _x\), we have \(\Delta x\approx 4\ell \sigma _f/\textrm{CF}= 4\ell /(\sqrt{\pi }Q_{\textrm{ERB}})\), where \(Q_{\textrm{ERB}}=\textrm{CF}/\textrm{ERB}\). Putting it all together yields \(v_\textrm{g}=8\ell \textrm{CF}/(\sqrt{\pi }N_\textrm{SFE}Q_{\textrm{ERB}})\), where \(N_\textrm{SFE}=\tau _\textrm{SFE}\textrm{CF}\) is SFOAE delay in stimulus periods. Relating \(N_\textrm{SFE}\) and \(Q_{\textrm{ERB}}\) using the tuning ratio (\(Q_{\textrm{ERB}}=rN_\textrm{SFE}\)) yields \(v_\textrm{g}=8\ell \textrm{CF}/(\sqrt{\pi }rN_\textrm{SFE}^2)\). To relate this velocity to sweep rates it is convenient to express \(v_\textrm{g}\) in octaves/s (rather than, say mm/s). For this, we use the relation \(\Delta x_\textrm{oct}=\Delta x/(\ell \ln 2)\). Hence, \(v_\textrm{g}=8\textrm{CF}/(\sqrt{\pi }\ln 2\,rN_\textrm{SFE}^2)\) (oct/s). Figure 5 is computed using published human SFOAE delays [29] and averaged tuning ratios for cat, guinea pig, and chinchilla [30, 31].

The figure shows human DPOAE data. In common laboratory animals, correlation bandwidths are typically a factor of 2 or 3 larger. This is true not only because animal DPOAE fine-structure patterns vary more slowly with frequency (since their OAE delays are shorter than those of humans [54]), but also because the peak-to-peak amplitude of the fine-structure oscillations is generally also reduced.

For a fixed SNR, the potential reduction in measurement time depends on the desired frequency resolution of the measurement. If this resolution is \(\Delta f\) and the chosen analysis bandwidth is \(\textrm{BW}\) (with \(\textrm{BW}>\Delta f\)), then the use of swept tones can speed up the total measurement by roughly a factor of \(\frac{1}{2}(\textrm{BW}/\Delta f)\). If \(\Delta f>\frac{1}{2}\textrm{BW}\) then one may be better off using discrete tones.