Chapter 3:
Frequency selectivity, masking and the critical band
  1. Introduction.
    1. Frequency selectivity refers to the ability of the auditory system to resolve sinusoidal components in a complex sound and plays a role in many aspects of auditory perception.
    2. Frequency selectivity is often demonstrated and measured by studying masking.
      1. Masking is the process by which the absolute threshold for one sound (the signal) is raised by the presence of another sound (the masker).
      2. The usual measure of masking is the dB increase in threshold produced by the masker (dB of masking = 10 log10(IM / I), where IM is the threshold for the sound in the presence of the masker, and I is the threshold in its absence).
    3. It has been known for years that a signal is most easily masked by a sound having frequency components close to those of the signal.
      1. Led to the idea that our ability to separate the components of a complex sound depends on the frequency-resolving power of the basilar membrane.
      2. Also led to the idea that masking reflects the limits of frequency selectivity and provides a way to quantify it.
    4. Time is also important to masking.
      1. We shall initially consider simultaneous masking in which the signal is presented at the same time as the masker.
      2. Later we shall consider forward and backward masking in which the signal is masked by a preceding or a following sound, respectively.
  2. The critical band concept.
    1. Fletcher�s band-widening experiment, the power spectrum model and critical ratios.
      1. Fletcher measured the threshold of a sinusoidal signal as a function of the bandwidth of a noise masker.
        1. The noise was always centered at the signal frequency and the noise power density was held constant.
        2. Thus, the total noise power increased with bandwidth.
      2. Results of a similar study are shown in Fig. 3.1.
        1. The threshold of the signal at first increases as bandwidth (and total noise power) increases, but then flattens off.
        2. That is, addition of more noise at a greater distance from the signal frequency produces no additional masking.
      3. To account for these results, Fletcher suggested that the basilar membrane behaves as if it contains a bank of bandpass filters with overlapping passbands. I.e., each point on the basilar membrane corresponds to a filter with a different center frequency.
        1. When trying to detect a signal in noise, the listener is assumed to make use of a filter with a center frequency close to that of the signal.
        2. This filter passes the signal, but removes much of the noise.
        3. Only the components of the noise passing through the filter have any effect in masking the signal.
        4. Threshold is assumed to correspond to a certain signal-to-noise ratio at the output of the auditory filter.
        5. This set of assumptions has come to be known as the power spectrum model of masking.
          1. Stimuli are represented by their long-term power spectra.
          2. Relative phases of the components and short-term fluctuations in the masker are ignored.
      4. Fletcher called the bandwidth at which the signal threshold ceased to increase the critical bandwidth (CBW). To estimate it, Fletcher made two simplifying assumptions.
        1. The shape of the auditory filter is rectangular. He knew this was not true, but it has little effect on the calculations.
        2. When the noise just masks the tone, the power of the tone, P, divided by the power of the noise inside the critical band is a constant, K, ( P/(W x N0)=K, where W is the width of the critical band and N0 is the noise power density).
        3. Since P and N0 can be measured, if we had an estimate of K, we could compute the CBW as W=P/(K x N0).
        4. Fletcher estimated K to be 1.0 on the basis of the threshold for the tone in a very narrow band of noise.
          1. Using this estimate for K, W=P/N0.
          2. More recent work has shown K to vary somewhat with frequency, but to have a value of about 0.4.
          3. P/N0 is now called the critical ratio and is about 0.4 times the width of the critical band.
      5. Since Fletcher described the critical band concept, many different experiments have shown that listeners� responses to complex sounds differ according to whether the stimuli are wider or narrower than the critical band.
        1. Each of these provides an independent way to estimate the critical bandwidth and how it varies with center frequency.
        2. The fact that these various experiments provide reasonably similar estimates adds considerable strength to the critical band concept.
        3. We now shall examine some of these other kinds of experiments.
    2. The loudness of complex sounds.
      1. For a complex sound of fixed intensity and bandwidth W, if W is less than CBW, then the loudness of the sound is independent of W and is about as loud as a pure tone of equal intensity at the center frequency of the band.
      2. If W is increased beyond the CBW, the loudness of the complex begins to increase.
      3. Typical results are shown in Fig. 3.2.
      4. I do not think the loudness models appealed to by the author of your text will help you understand this phenomenon very much.
      5. The key to understanding why this phenomenon occurs, however, is fairly simple.
        1. The neural activity, and hence the loudness, within a frequency channel (critical band) grows nonlinearly with increases in intensity. E.g., a halving of intensity produces considerably less than a halving of loudness.
        2. The total loudness of a sound results from summing neural activity across frequency channels.
        3. When the energy in the stimulus is spread over more than one channel, the loss in loudness in the center channel is thus less than the gain in loudness in less central channels, so the overall loudness increases.
        4. This does not happen at low overall intensities, because at low levels, loudness changes very rapidly with intensity and the loss in loudness from the central channel is as great as the gain in loudness in less central channels.
    3. The threshold of complex sounds.
      1. Early studies showed that the threshold (in terms of total energy) of multi-tone complexes consisting of evenly spaced sinusoids remains relatively constant as the number of sinusoids is increased, until the overall spacing exceeds the CBW. Beyond this point the threshold increases.
      2. This was interpreted as indicating that only the information from a single critical band is used to detect the complex.
      3. More recent studies have found no clear break at the critical bandwidth and it has been shown that the ear is capable of integrating information over more than one CBW.
    4. Two-tone masking.
      1. When subjects are asked to detect a narrow band of noise in the presence of two tones with frequencies on either side of its center frequency, threshold remains constant as the separation between the tones is increased up to the CBW.
      2. As spacing increases beyond the CBW, threshold for the noise falls sharply.
      3. Interpretation of this result is complicated by the fact that the lower tone may interact with the noise band to produce combination products which may be detected even if the signal is inaudible.
      4. When measures are taken to mask out such combination products, the threshold for the noise increases smoothly with increasing frequency separation between the tones.
      5. This result, like more recent results for the threshold of complex sounds, emphasize that the auditory filter is not rectangular, but has a rounded top and sloping sides.
    5. Sensitivity to phase.
      1. The phenomenon described in this section involves complex acoustics that go beyond the scope of this course.
      2. The results turn out not to be applicable to estimating the CBW.
      3. Therefore, we shall not consider this section.
    6. The discrimination of partials in complex tones.
      1. Subjects� abilities to hear pitches corresponding to the individual sinusoidal components in a complex harmonic sound can be measured.
      2. One way to do this is to present two comparison tones, one corresponding to a partial in the complex and the other one half way between that partial and the one above or below it. The subject is then asked which comparison was part of the complex.
      3. Subjects can do this with 75% accuracy when a partial is separated from its neighboring partials by about 1.25 times the equivalent rectangular bandwidth of the auditory filter.
      4. There may be more involved here than peripheral filtering because musicians do better at this task than nonmusicians, even though musicians do not show narrower CBWs in masking experiments.
      5. The musicians appear to have learned to make more efficient use of the peripheral filtering mechanism.
    7. Interim summary.
      1. A variety of different experiments give consistent estimates of the CBW.
      2. Most do not show a distinct breakpoint corresponding to the CBW because the auditory filter is not rectangular.
      3. We now turn to attempts to measure the shape of the auditory filter.
  3. Estimating the shape of the auditory filter.
    1. Psychophysical tuning curves (PTCs).
      1. Analogous in many ways to neural tuning curves.
        1. The signal (usually a sinusoid) is fixed at a low level to confine the activity it produces to a single auditory filter.
        2. For each of several frequencies of the masker (usually a narrowband noise), the level needed to just mask the signal is determined.
        3. Typical results are shown in Fig. 3.7.
      2. Under the assumptions of the power spectrum model of masking, the PTC shows the level of the masker required to produce a constant output from the auditory filter.
      3. Assuming the auditory filter to be linear, its shape is given by simply inverting the PTC.
      4. One problem with this way of determining the shape of the auditory filter is that subjects improve their performance by using a filter whose center is offset from the frequency of the signal, in a direction away from the masker (off-frequency listening).
      5. This produces a sharper tip on the PTC than if only a single filter were used.
    2. The notched-noise method.
      1. Patterson devised a method of determining auditory filter shape that prevents off frequency listening.
        1. The signal is fixed in frequency and the masker is a noise with a notch centered at the signal frequency.
        2. The width of the notch is varied and the threshold of the signal is measured as a function of notch width, as shown in Fig. 3.8.
      2. Because the notch is symmetric about the signal, the method is not capable of showing asymmetries in filter shape, but PTCs indicate that at least the top part of the filters are reasonably symmetrical.
      3. Off frequency listening is prevented because optimal signal-to-noise ratio is achieved with a filter centered at the signal frequency.
      4. The amount of noise passing through the filter is proportional to the area under the filter in the frequency ranges covered by the noise.
        1. Assuming that threshold corresponds to a constant signal-to-masker, the change in threshold with notch width shows how the area under the filter varies with D f.
        2. The first derivative of the function relating threshold to D f gives the relative height of the filter at each value of D f.
        3. In other word, the relative response of the filter for a given deviation, D f, from the center frequency is equal to the slope of the function relating signal threshold to notch width at that value of D f.
      5. A typical auditory filter derived using this method is shown in Fig. 3.9.
        1. The filter has a round top and steep skirts.
        2. A single number like CBW can not completely describe such a shape, but summary measures such as the 3-dB bandwidth (typically 10-15% of center frequency) or equivalent rectangular bandwidth (typically 11-17% of center frequency) are used.
        3. Fig. 3.10 shows equivalent rectangular bandwidths of auditory filters, as a function of frequency, from a number of experiments using the notched noise method.
      6. Patterson�s method has been extended to conditions in which the notch in the noise is placed asymmetrically about the signal to allow measurement of asymmetry in the auditory filter.
        1. Off frequency listening then has to be taken into account.
        2. The mathematics for analyzing this case are beyond the scope of this course.
        3. Results show that the auditory filter is reasonably symmetrical at moderate sound levels, but becomes increasingly asymmetric at high levels, the low-frequency side becoming shallower than the high-frequency side, as shown in Fig. 3.11.
    3. Some general observations on auditory filters.
      1. Although people can attend to more than one filter at a time, one can predict whether a complex sound (e.g., an alarm) will be detected by computing the signal-to-noise ratio for an auditory filter at its most prominent frequency component. If that ratio is no less than �4 dB, the signal will be detected.
      2. It is important to remember that auditory filters are continuous, not placed end to end.
  4. Masking patterns and excitation patterns.
    1. Many early studies of masking held the frequency of the masker constant and varied the frequency of the signal.
      1. This is not a good way to estimate the shape of the auditory filter because, under the assumptions of the power spectrum model of masking, subjects would use a different auditory filter for each different signal frequency.
      2. The resulting graph plotting masked threshold as a function of the frequency of the signal is called a masking pattern or a masked audiogram and an example is shown in Fig. 3.12.
        1. Such masked audiograms show steep slopes on the low-frequency side.
        2. The high-frequency side is less steep and becomes shallower with increasing level of the masker.
      3. Since the auditory filter is shifted as the signal frequency is altered, these masked audiograms can be used to obtain a crude estimate of the excitation pattern produced by the masker like the one you saw in Chapter 2 (Fig. 2.9).
      4. Since the signal is detected when the excitation it produces is some constant proportion of the excitation produced by the masker in the frequency region of the signal, the masked audiogram should be directly proportional to the excitation pattern of the masker.
      5. This expectation can be complicated by such factors as off-frequency listening and the possible use of combination tones produced by the interaction of signal and masker to detect the signal.
      6. The shapes of excitation patterns for the masker can be derived using the concept of the auditory filter, as illustrated in Fig. 3.13.
        1. The excitation pattern of a sound can be thought of as the output of the auditory filters as a function of their center frequencies.
        2. The masked threshold for the signal at a given frequency is an estimate of the amount of masker excitation at the output of the auditory filter centered at that frequency.
        3. Thus plotting these thresholds against signal frequency gives a crude estimate of the excitation pattern of the masker.
      7. Although the auditory filters were assumed to be symmetrical in this derivation, the derived excitatory pattern is asymmetrical. This happens because the bandwidth of auditory filters increase with their center frequency.
      8. This method can be extended to asymmetric filter shapes, but whereas the lower side of the auditory filter gets less steep with increasing level, the upper side of the masked audiogram (and the excitatory pattern) gets less steep with increasing level.
      9. The latter happens because the upper side of the excitatory pattern is determined by the lower side of the filter and vice versa.
  5. The nature of the critical band and mechanisms of masking.
    1. The origin of the auditory filter.
      1. The physiological basis of the critical band is not completely understood, but it certainly involves the frequency-resolving power of the basilar membrane.
      2. The critical bandwidth or the equivalent rectangular bandwidth of the auditory filter corresponds to a constant distance along the basilar membrane, regardless of center frequency (about 0.9 mm).
      3. The equivalent rectangular bandwidth of auditory filters measured behaviorally in animals corresponds well with the equivalent rectangular bandwidth of tuning curves measured in single auditory neurons in the same species.
    2. Temporal effects.
      1. It is possible that some further sharpening of tuning curves takes place after the basilar membrane due to lateral inhibition at the level of the hair cells.
      2. Data on this point are inconclusive and we shall not consider them until later in this chapter.
    3. The mechanism of masking.
      1. There are two common conceptions of the mechanism by which masking occurs.
        1. The first is that masking involves swamping of the neural activity evoked by the signal.
          1. If the masker produces significant neural activity in the auditory filter being used to detect the signal, the increment in activity produced by the signal may be insufficient to be detected.
          2. On this account masking occurs only if the masker produces excitation in the auditory filter which would otherwise respond to the signal.
        2. A second possibility is that the masker suppresses or inhibits activity which the signal would evoke if presented alone.
      2. There is currently no clear way to distinguish between these two possibilities.
      3. Psychologists have preferred the swamping explanation because suppression is a nonlinear process and masking is well modeled by a system of linear filters.
    4. The neural code used for signal detection.
      1. What aspect of neural activity evoked by the signal might be used for detection?
      2. Similar to the question of how stimulus intensity is neurally encoded which we considered in Chapter 2.
        1. Neural firing rates is certainly part of the answer.
        2. Temporal patterns of neural firing (phase locking) may also be relevant.
          1. There may be little or no phase locking to weak components which are close in frequency to stronger ones.
          2. In the case of noise, neurons fire irregularly. If a tone is added, some neurons may phase lock to it; and hence, fire more regularly.
          3. In either case, a signal may fail to be detected if the subject cannot detect its effect on the time pattern of neural firings.
      3. Addition of a tone to noise also reduces fluctuations in the amplitude envelope of the total sound and this may be a cue in detecting the tone.
  6. Comodulation masking release: Spectro-temporal pattern analysis in hearing.
    1. Initial demonstrations of comodulation masking release.
      1. The power spectrum model of masking assumes that performance is based on the output of the single auditory filter which gives the highest signal-to-noise ratio and that threshold corresponds to a constant signal-to-masker ratio.
        1. This model works well in many situations.
        2. In other situations it is clear that listeners make comparisons across auditory filters and that short-term temporal fluctuations of the masker can have important effects.
      2. Hall et al. (1984) demonstrated that across-filter comparisons could enhance detection of a sinusoidal signal in a fluctuating noise masker if the fluctuations were coherent or correlated across frequency bands.
        1. The threshold for a 1-kHz, 400-ms pure tone was measured as a function of the bandwidth of a noise masker with constant spectrum level (noise density).
        2. The noise was centered at 1 kHz. and was either random or comodulated.
          1. A random noise has irregular fluctuations in amplitude that are independent in different frequency regions.
          2. The comodulated noise was amplitude modulated at a low, irregular rate so that amplitude fluctuations were the same in different frequency regions.
        3. Results of this experiment are shown in Fig. 3.16.
          1. For the random noise, threshold increases with bandwidth up to the critical bandwidth of the auditory filter and then levels off, just as expected from the power spectrum model of masking.
          2. For the modulated noise, threshold increases with bandwidth up to about 100 Hz and then decreases with increasing bandwidth.
          3. The latter result indicates that subjects can compare the outputs of different auditory filters to reduce masking.
      3. Another way to demonstrate comodulation masking release is by using narrow bands of noise which inherently have slow amplitude fluctuations.
        1. One band of noise (on-frequency band) is centered at the signal frequency and the other (flanking band) is placed outside the critical band around the signal.
        2. If amplitude fluctuations in the two bands are uncorrelated, addition of the flanking band either worsens or does not effect detection of the signal.
        3. If amplitude fluctuations in the two bands are correlated, addition of the flanking band improves detection of the signal.
        4. This comodulation masking release occurs even if the signal and on-frequency band are presented to one ear and the flanking band is presented to the other.
        5. This last result indicates that the brain is somehow able to compare the amplitude fluctuations in the two bands and use this information to improve detection of the signal.
    2. The role of within-channel cues.
      1. Modulation of a masker can produce a release from masking even when the masker�s bandwidth is less than the auditory filter bandwidth.
      2. This release cannot arise from comparisons of the outputs of different auditory filters and does not represent comodulation release from masking.
      3. This results from cues available in the output of a single auditory filter.
      4. One example of such a within-channel cue is the decrease in envelope fluctuations when a sinusoidal signal is added to a noise.
      5. Within-channel cues can lead to an overestimate of comodulation release from masking, but are easily avoided in several ways.
        1. Use of brief signals.
        2. Widely separating the on-frequency and flanking bands.
        3. Presenting the two bands of noise to opposite ears.
    3. Factors influencing the magnitude of comodulation release from masking.
      1. Occurs for a wide range of frequencies (500 to 4000 Hz) and does not vary much with signal frequency.
      2. Is largest when the masker is modulated at a low rat and covers a large frequency range.
      3. When within-channel cues are eliminated, comodulation release from masking can be as large as 11 dB.
      4. When flanking band is presented to opposite ear, comodulation release varies little with center frequency, but is larger for flanking bands close to the signal frequency. Suggests a proximity effect separate from the effect of within-channel cues.
      5. Comodulation release tends to increase as the width of on-frequency and flanking bands of noise is decreased, probably because the rate of envelope fluctuations decreases.
      6. If many flanking bands of noise are used, comodulation release can be as large as 16 dB.
    4. Models to explain comodulation release from masking.
      1. Models fall into two general classes.
        1. One possibility is that the auditory system compares envelope modulation patterns at the outputs of auditory filters with different center frequencies and detects disparities.
          1. For a comodulated masker without a signal, the modulation pattern would be similar for all active filters.
          2. Addition of a signal would modify the modulation pattern at the output of a filter tuned to the signal.
          3. If the masker were not comodulated there would be little similarity at the output of different auditory filters in the first place.
        2. Another possibility is the dip-listening model in which the listener monitors fluctuations in auditory filters away from the signal frequency to determine the optimum times to listen for the signal. If the noise is comodulated, an off-frequency minimum is predictive of an optimum signal-to-noise ratio at the signal frequency.
    5. Experimental tests of the models.
      1. Richards (1987) tested the idea that we can detect disparities in envelope modulation patterns at the outputs of different auditory filters by asking subjects to discriminate two comodulated bands of noise from two bands with independent envelopes.
        1. Subjects were successful in this task, indicating that across-filter disparities in modulation pattern can be detected.
        2. Others have shown that this detection improves with increasing bandwidth, which is opposite to what is found in comodulation release experiments and suggests different mechanisms are involved.
      2. A number of experiments attempting to eliminate either across-filter disparities in envelope fluctuations, across-filter disparities in overall level, or dip listening lead to the conclusion that comodulation release from masking does not depend on a single cue or mechanism.
      3. It appears to reflect the operation of flexible mechanisms which can exploit a variety of cues depending on the specific stimuli used and is clearly of considerable value in many real-life situations.
  7. Profile analysis.
    1. Even without distinct envelope fluctuations, subjects are able to compare the outputs of different auditory filters to enhance signal detection.
    2. Green et al. have shown that subjects can detect changes as small at 1-2 dB in the relative level of a sinusoidal signal imbedded in a complex background with a varying overall level.
    3. Such results suggest subjects can detect a change in the shape or profile of the spectrum of the sound.
    4. This should not be surprising since it has been known for years that one of the main factors determining the timbre of a sound is its spectral shape.
      1. We discriminate various vowel sounds from each other on the basis of their spectral shape, regardless of the level of those sounds.
      2. We identify the sound of various musical instruments largely on the basis of the spectral shape of their sound whether they are playing loudly or softly.
  8. Non-simultaneous masking.
    1. Two basic types of non-simultaneous masking.
      1. Backward or pre-stimulatory masking in which the signal precedes the masker.
        1. This phenomenon is poorly understood and seems not to occur in highly practiced subjects.
        2. Could reflect interference of the masker with later processing of the signal at some higher level of the nervous system.
        3. May simply reflect some confusion of the signal with the masker in unpracticed subjects.
      2. Forward or post-stimulatory masking in which the masker precedes the signal.
        1. Must be separated from adaptation or fatigue produced by the masker.
        2. Such separation is possible because forward masking occurs for relatively short masker presentations (usually a few hundred msec) and is limited to signals occurring within about 200 msec of the masker.
    2. Main properties of forward masking.
      1. Forward masking is greater the nearer in time to the masker the signal occurs, as shown in Fig. 3.17.
      2. The rate of recovery from forward masking is greater for higher masker levels and always decays to zero within 100 to 200 msec.
      3. Increments in masker level do not produce equal increments in amount of forward masking as they do in simultaneous masking where threshold corresponds to a constant signal-to-noise ratio. This is shown in the right panel of Fig. 3.17.
      4. Amount of forward masking increases with masker duration up to at least 20 msec, with some studies finding an increase up to 200 msec.
      5. Forward masking is influenced by the relation between the frequencies of the signal and the masker, just as with simultaneous masking.
  9. Evidence for lateral suppression from non-simultaneous masking.
    1. A point of some controversy and considerable neurophysiological complexity is the existence and nature of lateral suppression in the auditory system.
    2. Although your text provides an extended discussion of this topic, it goes well beyond the scope of this course and we shall only make a few basic points in this regard.
    3. It may seem surprising that simultaneous masking is well modeled by linear filters since the behavior of individual auditory neurons is grossly nonlinear and involves lateral suppression processes by which strong activity at one center frequency can inhibit or weaken activity at adjacent center frequencies.
    4. One would not expect to see effects of lateral suppression in simultaneous masking because the masker and the signal are processed simultaneously in the same channel (auditory filter).
    5. If suppression does occur, one might expect to see it in forward masking provided:
      1. The neural level at which suppression occurs is not later than the level at which most of the forward masking effect arises.
      2. Suppression built up by the masker has decayed by the time that the signal is presented.
    6. Psychophysical tuning curves from forward masking are steeper on the high frequency side than those from simultaneous masking as shown in Fig. 3.20.
    7. Several other methods of estimating frequency selectivity also show sharper tuning in forward than in simultaneous masking.
    8. At least one explanation of this difference is that the low frequency side of the excitatory pattern for the masker is sharpened by a suppression process.
    9. Alternative explanations of this difference also involve suppression processes.
    10. It is thus likely that non-simultaneous masking results show effects of a suppression mechanism operating close to the point of transduction of mechanical movements to neural impulses (hair cells or the basilar membrane).
  10. Frequency selectivity in impaired hearing.
    1. A variety of different types of studies show that frequency selectivity is impaired (the critical band is widened) by damage to the cochlea.
    2. Fig. 3.21 shows an example of this by comparing filter shapes determined by the notched-noise method for each ear of subjects with one normal ear and one with a cochlear hearing loss.
    3. The first major consequence of this is a greater susceptibility to masking by interfering sounds which may partly account for the difficulties experienced by those with cochlear impairments in understanding speech in noisy situations.
    4. A second major consequence is difficulty in perceptual analysis of complex sounds such as speech or music with a resulting reduction in timbre discrimination required to discriminate different vowel sounds or different musical instruments.
    5. A hearing aid which simply amplifies sound does nothing to correct impaired frequency selectivity.
  11. General conclusions. Your textbook gives a good summary of the major ideas in this chapter. I shall not attempt to summarize it further here, but suggest that you study it carefully.
  12. Signal detection theory.
    1. Much of Chapters 2 and 3 have been concerned with the measurement of thresholds which were classically conceived as the intensity of a stimulus above which it will be detected and below which it will not.
      1. It has been known for a long time that this view is not satisfactory for a number of reasons.
        1. Psychometric functions are continuous, rather than discrete, and usually have lower limits greater than zero, as shown in Fig. 3.22.
        2. A subject�s performance in such situations is affected by a variety of motivational and response factors such as instructions, payoffs and a priori probabilities.
      2. Signal detection theory provides a way of separating factors not directly associated with the discriminability of the signal (bias, criterion, motivation) from factors relating to purely sensory capabilities.
      3. It also accounts for the fact that responses may vary from trial to trial even when identical stimuli are presented.
    2. Basic assumptions of signal detection theory.
      1. Decisions are based on the internal value of a random variable (x) whose average value is monotonically related to stimulus magnitude and can be thought of in terms of either a sensory or a physiological dimension (e.g., loudness or number of neural firings).
      2. The value of x varies randomly from trial to trial even though the same stimulus is presented.
        1. Some of this variability may be in the stimulus (e.g., a white noise masker has random variations in physical intensity).
        2. Some of this variability is in the nervous system of the listener (e.g., neurons show random firings in the absence of stimulation).
        3. Thus, the average value of x will be greater if a signal is present than if it is not, but the listener cannot be certain that a signal has occurred on a particular occasion because of the inherent variability of x.
      3. On the basis of nonsensory factors the subject adopts some criterion value of x and reports hearing the signal if the value of x on that trial is greater than his/her criterion.
    3. The probability density function for x on trials that do not contain the signal is usually assumed to be normal or Gaussian, as shown in Fig. 3.24.
      1. The probability density function for x on trials that do contain the signal is also usually assumed to be normal and to have the same variance, but a greater mean.
      2. The separation between the means of these two distributions in standard deviation units is called d� and gives a measure of the detectability or discriminability of the signal.
      3. The location of the subject�s criterion gives a measure nonsensory effects.
    4. Hit rate (probability of saying yes given the signal was presented) estimates the area under the signal plus noise distribution to the right of the subjects response criterion.
      1. Similarly, false alarm rate (probability of saying yes given the signal was not presented) estimates the area under the noise alone distribution to the right of the subjects response criterion.
      2. Using these estimates and tables of the normal distribution, d� and criterion location can be computed. These computations are taught in PSYCH 3LL3 if you wish to learn them.
    5. Signal detection theory thus provides a way to separate changes in performance due to purely sensory factors from changes due to motivation or criterion.
      1. This logic is applicable to a wide variety of problems in psychology.
      2. In order to use signal detection theory, one must measure false alarms, as well as the proportion of hits. This information was not obtained in many classical psychophysical studies.
      3. According to this theory there is no such thing as a threshold, only degrees of discriminability.
        1. One can measure the value of the signal that produces a particular level of d�.
        2. Alternatively, one can use a task such as two-alternative forced-choice that attempts to eliminate effects of bias or criterion and arbitrarily define threshold as corresponding to a particular percentage correct (76% correct in two-alternative forced-choice is equivalent to a d� of 1.0).