sign in sign up
<<
Home , Hearing aid, History of hearing aids

Veterans Administration

Journal of Rehabilitation Research and Development Vol. 24 No. 4 Pages 7-20

Digital aids: A tutorial review*

HARRY LEVYITT, PIh,D. GenterJi)r Research in Speech nnd Nearing Sciences, Crud~ruieSchool and Uni~~ersityCenter of tlze Cily Univec~ityof Mew Yo&, New YorA, NY 10036

Abstract-The basic concepts underlying digital signal hearing aids because of the constraints on chip size processing are reviewed briefly, followed by a short and power consumption. But all of the above- historical account of the development of digital hearing mentioned features have already been demonstrated aids. Key problem areas and opportunities are identified, in a master digital (12,14) and it is likely and the various approaches used in attempting to develop that (with the development of more advanced signal- a practical digital hearing aid are discussed. processing chips) an increasing number of these attractive features will be incorporated in the wear- able digital hearing aids of the future. ADVANTAGES OF DIGITAL HEARING AIDS The many advantages offered by digital hearing aids can be subdivided into three broad groups: Digital hearing aids promise many advantages I) Signal-processing capabilities that are analo- over conventional heartng aids. These include: gous to, but superior to, those offered by conven- Programmability; tional analog hearing aids. Much greater precision in adjusting electroacous- 2) Signal-processing capabilities that are ~~nique tic parameters; to digital systems and which cannot be implemented Self-monitoring capabilities; in conventional analog hearing aids. Logical operations for self-testing and self-cali- 3) Methods of processing and controfllng signals "oation ; that change our way of thinking about how hearing Control of acoustic feedback (a serious practical aids should be designed, prescribed, and fitted. problem with high-gain hearing aids); The third type of advantage is the most subtle- The use of advanced signal-processing techniques and the most important. For example, a digital for noise reduction; hearing aid can be programmed not only to amplify, Automatic control of signal levels; and but also to generate, audio signals. As such, the Self-adaptive adjustment to changing acoustic en- instrument can be programmed to serve as an vironments. audiometer in order to facilitate the measurement Only a few of these features are likely to be of audiological characteristics relevant to the pre- included in the first generation of wearable digital scriptive fitting of hearing aids (14,151. Using this approach, it is possible to circumvent the very difficult problem of correcting for the frequency- "Preparation of this paper was supported by Grant No. GO0830251 dependent differences in sound level between the from the National Institute on Disability and Rehabilitation Research and by PHS Grant PO1 NS17764 from the National Institute of traditional audiometer headphone and the patient's Neurological and Communicative Disorders and Stroke. own hearing-aid receiver. The idea of using a hearing Journal of Rehabilitation Research and Development Vol. 24 No. 4 Fall 7987 aid as an audiometer was born out of the realization The middle section of the diagram shows the that the primary differences between the two devices sampled data. Note that the signal is no longer are those of software (i.e., the controlling program) continuous but consists of a series of discrete pulses. rather than fundamental differences in hardware. The height of each pulse is equal to the value of the Another important stimulus to our thinking has continuous waveform at the sampling insturzl. Note been the use of simulation. As noted that, at this stage of the conversion process, the elsewhere in this paper, the development of com- pulse heights are still specified in analog form; i.e., puter-simulation techniques to facilitate the design the value of each sample can be recorded at its and development of vocoders and other - precise numerical level within the range of the oriented speech processing systems led to the re- signal's continuous variation. Only the sampling alization that could also be used to instant is defined discretely. simulate hearing aids, and that eventually digital The bottom section of the diagram shows the processing of audio signals would be possible in a sampled data signal in binary form. For the purposes hearing aid. Further, the first working digital hearing of this example, eight arbitrary levels of quantization aid was achieved using real-time computer simuta- are shown. Each of the analog samples shown in lion (12,14), thereby providing a glimpse of the many the middle section of the diagram has been approx- possible features that could be incorporated in the imated by a sample with a height equal to one of digital hearing aids of the future. these eight levels of quantization (The binary rep- The results currently being obtained using com- resentation of these eight quantization levels appears puter simulation techniques are having a profound on the ordinate.) The continuous waveform has also effect on our thinking with respect to which features been reproduced here in order to show how well should or should not be included in a modern hearing the binary samples approximate the original signal aid. Whereas, in the past, progress in the develop- (at the sampling instants). ment of more effective hearing aids was limited The process of converting an analog sarnple to primarily by what could be achieved technologically, binary form involves a sequence of binary "deci- the present situation, with the help of computer sions." The first decision is whether the signal is sirnufation and the realization of practical digital greater or less than zero. Consider the first sarnple hearing aids, is one in which progress is limited by way of illustration: this sample has a value of primarily by our own lack of understanding of what 0.9 volts and therefore the answer to the first is important in processing signals for hearing im- question is positive (i.e., sample value is greater pairment. than zero) and a I is assigned as the first digit of its binary representation. (The analog-to-digital con- version has been set to cover the range from - I to HOW DOES DIGITAL + I volts. This range has been chosen to exceed WORK? that of the signal being quantized.) After the first binary decision it is evident that the value of the Analog-to Digital Conversion waveform at the first sampling instant must lie The key element in a digital hearing aid is its use between 0 and + 1 volts. of sound signals that have been sampled discretely The next binary decision is whether this sample in time, so that they have become represented by a value lies above or below the mid-value (0.5 volts) series of data poiizts instead of by a continuously- of this range. The answer is again positive (i.e., varying value analogous to the waveform itself. sample value greater than 0.5 volts) and a 1 is once Figure 1 illustrates the process whereby an analog again assigned as the next digit in the binary rep- signal is converted to digital form. The uppermost resentation. It is now known that the waveform section of the diagram shows a continuous wave- must have a value between 0.5 and 1.0 volts at this form; in this example, voltage as a function of time sampling instant. The mid-value of that range is 0.75 is shown. The signal is periodic with a period equal volts and so the third binary decision is whether the to 0.8 msec. It is sampled at regular intervals of 0.1 sample value is greater or less than 0.75 volts. The msec. The value of the waveform at each sampling answer is again positive so once again a 1 is assigned instant is shown by a solid circle. as the next digit in the binary representation. It is LEVITT: Digital Hearing Aids: A Tutorial Review

5 10 SAMPLE NUMBER

Figure 1. Converting a Signal frorn Analog to Digital Form. 'The uppermost section of the diagram shows the waveform of a continuous signal. The solid points show the value of the waveform at each sampling instant. The middle section of the diagram shows the sampled-data version of the analog signal. The lowest section of the diagram shows the sample values after 3-bit quantization. (The original waveform is also shown here in order to illustrate the accuracy of quantization.) now known that the waveform must lie between example is 0.25 volts (e.g., Sample #I lies between 0.75 and 1 .0 volts at this sampling instant. 0.75 and 1 volt, Sample #2 lies between 0 and - 0.25 The above process can be continued iteratively volts, etc.). The largest error that can occur is half until the binary representation reaches the desired a quantization step, which in this case is 0.125 volts. accuracy. (The accuracy of approximation doubles As can be seen from the diagram, all of the binary with each additional binary decision.) In the above samples are within 0.125 volts of the original wave- example, three binary decisions were used leading form. In general, an N-bit analog-to-digital converter to a binary representation consisting of three binary will have 2N quantization levels and the error of digits. The term binary digit is abbreviated as bit quantization will not exceed 112 of 1/2N,or 11(2N+1), and the binary representation derived above is said of the range of the analog-to-digital converter. In to have "an accuracy of 3 bits." the above example, N=3 and the range of quanti- Referring once again to the lowest section of zation was 2 volts; the number of quantization levels Figure 1, it can be seen that each binary sample lies was thus 8 (= 2') and the precision of quantization eitherjust above orjust below the original waveform. was l/(z4)or 1116 of the range, which is 2116 ( = 0.125) The distance between quantization levels in this volts. 10 Journal of Rehabilitation Research and Development Vol. 24 No. 4 Fall 1987

TIME Figure 2. Illustration of Aliasing. Section A of the diagram shows a typical sampled-data sequence. Sections B, C, and D show waveforms that would generate the identical sampled-data sequence if sampled at the same instants in time. The true waveform from which the sampled-data sequence was originally obtained is shown in Section B: note that this waveform is the only one of the three waveforms that is sampled at a rate of more than two samples per period.

The Problem of Aliasing Only one of the waveforms shown in Sections B, In the digitization of analog signals we encounter C and D is the true waveform, i.e., represents the the risk of "aliasing." The problem and meaning of waveform that was originally sampled. The others aliasing is best described in terms of an example. are aliases of the true waveform. (It can be shown Section A of Figure 2 shows a sampled-data se- that there are an infinite number of possible aliases quence representing a continuous waveform. Sec- of the true waveform.) If sampling is to be used as tions B, C and D show three waveforms, each of a means of preparing continuous signals for digital which would yield the same sequence of samples. signal processing, it is essential that there be no Without further information there is no way of ambiguity as to which waveform is represented by knowing which waveform is represented by the a sampled-data sequence. sampled-data sequence shown in Section A. LEVITT: Digital Hearing Aids: A Tutorial Review

A Rule for Avoiding Aliasing also achieved fairly simply, by setting a maximum The waveform in Section 13 of the diagram differs allowable value for the samples contained in the from the other waveforms in one important respect, digital representation. An alternative method of it is represented by more than two samples per output limiting is to adjust the amplification constant, period. The waveforms in Section C and I> are A, in inverse proportion to the short-term energy sampled at a rate of less than two samples per of the signal, thereby reducing the gain as signal period. For example, in Section C there are nine level is increased. The first of these two methods is samples for five periods of the waveform; the rate the digital equivalent of peak-clipping, (a technique of samples is thus less than two samples per period. commonly used in older conventional hearing aids); This is true of any other alias corresponding to the the second method is the digital equivalent of an?- sampled-data sequence shown in Section A. A plitude compression (a technique used increasingly simple rule is thus to assume that the true waveform in modern conventional hearing aids). is the one sampled at more than two samples per In contrast to the above operations, the process period. In order for this rule to hold, it is essential of filtering audio signals represents a particutary that the sampling rate always be greater than twice interesting and innovative application of digital tech- the highest frequency of the waveform being sam- niques. Unfortunately, digital filters are difficult to pled. Stated another way, if it is known that the describe without some use of mathematics. The highest frequency in the signal does not exceed half Appendix provides an introduction to the key con- the sampling rate (i.e., there are at least two samples cepts underlying digital filtering. The non-mathe- per period of the highest frequency component In matical reader may wish to skip the Appendix. the signal) then there can be no aliasing errors. Digital filters, as illustrated in the Appendix, In order to ensure that there are no aliasing errors, operate on sampled-data waveforms in much the it is common practice to use an anti-aliasing filter same way (but not in exactly the same way) as an prior to the sampling operation. An anti-aliasing electronic filter operates on an electrical waveform. filter is typically a lowpass filter with a very high It may be noted that this filtering operation is also rate of attenuation above the cutoff frequency, fc, analagous to taking a running arithmetic average of where fc < 1/2 sampling rate. Similarly, in order to the data sequence. avoid the generation of spurious waveforms when An important advantage of digital filtering over a sampled-data sequence is converted back to analog conventional electronic filtering is the potential for form, it is common practice to use an ar~ti-ima

SAMPLED-DATA SYSTEM

INPUT OUTPUT

ANTI- WAVEFORM ANTI- ALIASING SAMPLER RE- IMAGING FILTER CONSTRUCTION FILTER

ALL DIGI"TL SYSTEM

OUTPUT

ANTI- ANTI- ALIASING FILTER IMAGING FILTER

Figure 3. This figure shows two typical sampled-data systems. The upper The second system is the corresponding all-digital system. It block diagram- revresents an analog-. samoled-data svstem (often consists of an anti-aliasing filter, an analog-to-digital converter referred to as a sampled-data system). It consists of an anti- (this unit contains both a sampler and a circuit for converting aliasing filter, a sampling circuit, a signal processor for operating the samples to binary form), a (this on sampled-data sequences, a circuit for waveform reconstmc- could be a general-purpose digital computer), a digital-to-analog tion, and an anti-imaging filter. converter, and an anti-imaging filter. - LEVITT: Digital Hearing Aids: A Tutorial Review

A BRIEF HISTORY OF DIGITAL introduction of high-level languages for Fdcilitating HEARING AIDS the simulation of audio systems. One such language was BLODI, an acronym for Block Diagram Corn- Early Applications of Digital Techniques piler, developed by Kelly, Lochbaum and Vyssotsky Sampled-data systems were first used in automatic (I 1) in 1961. This language could be used to simulate control work as well as in the transmission of any realizable audio system specified in block dia- information (23,241. Theoretical analyses of these gram form. early rysterns led to the development of several BLODI has been used for a wide range of appli- important theorems, most notably the sampling cations including the computer simulation of a high- theorem described earlier that determines the miin- gain telephone with frequency shaping for hearing- irnurn sampling rate necessary to specify maambig- impaired persons. That simulation, conducted by uously a continuous signal of known bandwidth. the author in 1967, was probably the first use of a Although the theoretical underpinnings of discrete digital computer in simulating a hearing aid. It was digital signal analysis were well developed some recognized at the time that a computer could also time ago (171, it was not until the advent of the be used to adaptively adjust the frequency response digital computer that these techniques started to and other characteristics of a hearing aid SO as to take on a new, practical importance. A concomitant best meet the needs of the user: however, because development of great consequence for the digital of severe practical limitations on the speed of sim- processing of audio signals was the development of ulation and the constraint that all processing be analog-to-digital and digital-to-analog converters fast done off-line this potentially useful approach was enough and with enough precision for the conversion not pursued until well over a decade later. of cotztin1ioi4s audio signals to digital form, and vice The advent of the laboratory cornputer brought versa. A typical system for telephone-quality speech the realization of a digital hearing aid one step would have a sampling rate of 10,000 Hz with &bit closer. An important feature of the laboratory corn- to IO-bit accuracy. A high fidelity system would puler is the relative ease with which 11: can be used require a sampling rate of at least 40,000 Hz with to control laboratory equipment. Computer con- IS-bit accuracy or better. trolled audio systems for research in audition were Much of the early research on the digital proc- developed soon after the introduction of the modern essing of audio signals focussed on speech analysis, laboratory computer. The earliest of these systems speech synthesis, and vocoder design (4). During were configured around a LING-8, the predecessor that early period in the development of digital audio, of the highly successful DEG PDP-8 labratory the time taken by a computer to process audio computer, An early system of this type was devel- signals was extremely long. Typically, a fairly simple oped by Braida and his associates at the Massachu- speech-processing algorithm (by today's standards) setts Institute of Technology (personal communi- would take sc~veru!hundred times reul time. As a cation with author) and has been heavily used in consequence, almost all of the research on the early research in psychoacoustics. A more modern ver- development of digital signal processing for audio sion of the system has been used extensively in applications was done off-line. Even with this major experiments on acoustic amplification and signal Iimitation, it was far more efficient to develop processing for hearing impairment. Although not experimental audio systems using off-line computer designed specifically as such, this early system was simulation rather than by building experimental in essence a computer-controlled master hearing aid prototypes. The latter approach is typically far more used for research in acoustic amplification. costly and time consuming than that of computer simulation. Further, experimental prototypes often Quasi-Digital Hearing Aids do not meet all design requirements. The use of In a "quasi-digital" hearing aid, conventional computer simulation has grown dramatically as a analog and filters are controlled by digital research and development tool and it is now widely means. A simple, practical realization of this ap- used in almost all industrial research laboratories. proach is to use the computer only for programming An important step in the development of com- the hearing aid. Once programmed, the hearing aid puter-simulation techniques in digital audio was the is disconnected from the computer and is then used Journal of Rehabilitation Research and Development Vol. 24 No. 4 Fall 1987

TO EXTERNAL COMPUTER

BlGlTAL CONTROLLER

MICROPHONE PRE- PROGRAMMABLE POWER OUTPUT HEARING AID FILTER AMPLIFIER LlMlTER RECEIVER

Figure 4. A Quasi-Digital Hearing Aid. The analog components of the hearing aid (amplifiers, filter and output limiter) are controlled by the digital controller Note that not ail the analog components need be under digital control. in essentially the same way as a conventional hearing connection. The external programming and control aid. This approach has not had much success com- is performed In the clinic or heanng-aid dispensary mercially with hearing aids, although it has been used when the unit is first prescribed and fitted. effectively with cochlear prostheses (1). The concept of a digitally-controlled analog hear- A closely related application of computers in this ing aid is very attractive from a practical perspective context is that in which a laboratory computer is because of the low power consumption involved. used as an audiometer and prescriptive tool. Popelka The technology of low-power analog amplifiers and and Engebretson (22) have developed a system of filters is well advanced, whereas the present gen- this type in which the computer is used to facilitate eration of chips "for digital signal processing dra~s the measurement, display and interpretation of au- relatively large amounts of power. The possibility diological data in order to prescribe a hearing aid of combining the low power consumption of analog more efficiently. The computer can also be used to components with the greater signal processing ca- search for the best hearing aid for a given patient pability of digital components now appears ts be a from a data base containing detailed electroacoustic viable option. A hearing aid of this type has been specifications of available hearing aids ( L 0). developed by Etymonic Design, Inc., for DahIberg The next step in the development of a practical Electronics (William A. Cole, personal communi- quasi-digital bearing aid is that in which both the cation). This also appears to be the approach taken analog components and the digital controller are by Mangold and Leijon (16) who have developed a combined in a single wearable unit. The basic programmable, multiband hearing aid. Similarly, architecture of such a hearing aid is shown in Figure Graupe. et, al., (7) have developed an adaptive 4. Note that the digital controller not only controls noise-reducing filter on a single chip using a quasi- or programs the operation sf the analog components digital approach. This chip is small enough and of (amplifiers, filter, and limiter), but is itself pro- sufficiently low power consumption to fit in a con- grammed by an external computer using a temporary ventional behind-the- or in-the-ear hearing aid. LEV1Ta: Digital Hearing Aids: A Tutorial Review

Another form of quasi-digital hearing aid (and that the fitter, limiter, and one or more of the which may have also been used in the applications amplifiers have been replaced by equivalent digital. cited above) is that of a sampled-data system in components. Further, because ofthe great Aexibility which the audio signal is sampled at discrete inter- of control afforded by the , it was vals in time, but the samples are kept in analog form possible to program the system to be self-adaptive, during processing. Such a system can be imple- thereby opening the door to the use of advanced mented fairly easily in practice using switched- signal-processing techniques for noise reduction and capacitance techniques. Consider, for example, a intelilglbility enhancement. Although the 8088 mi- sequence of capacitors with each capacitor cont;lln- croprocessor used by Graupe was both slow and ing a charge representing a single sample in a relatively large in size, he clearly anaticipated the sampled-data sequence. At each sampling instant, day when would be fast enough the charge on each capacitor is switched to the next and small enough for use in a practical hearing aid. capacitor in the sequence. The value of the charge At about this tirne (the late 1970s) the concept of on each capacitance is multiplied by a coefficient. an all-digital programmable hearing aid was also The sum of these products is the output of the filter. being developed in Germany (18) as well as at the The filter described here is essentially the same as Institute for Hearing Research in Nottingharn, Eng- that shown in Figure A3 of the Appendix. land (personal communication between M. Haggard The characteristics of this filter are determined and the author). The approach followed by the latter by the choice of the coefficients. In a programmable group was that of using a self-standing digital filter version of the filter, the coefficients are adjusted by controlled by a standard microcomputer, a technique a digital controller. Note that although the incoming that was to be adopted later by several other research audio waveform is sampled discretely in time (hence laboratories (1 5,25). the need for anti-aliasing and anti-imaging filters) Although the concept of a digital hearing aid was the samples themselves remain in analog form anticipated at an early date, two major technical throughout the processing. In an all-digital version problems had to be resolved before anyone could of the above filter, the samples would be converted develop a practicaI all-digital instrument. The first to binary form prior to processing. was the development of a digital signal processor The use of the switched-capacitance technique, fast enough to operate in real tirne. The second, or similar quasi-digital methods, has the important more difficult problem (which has yet to be resolved advantage that digital signal-processing techniques satisfactorily) is that of developing digital circuitry can be used without the need for analog-to-digital that is srnall enough and sufficiently low in power or digital-to-analog converters. The power con- consumption for practical use in a small, wearable sumed by an analog-to-digital converter increases unit. rapidly with degree of quantization. The develop- The first breakthrough came with the development ment of high resolution analog-to-digital converters of the array processor in which an array of numbers of small size and low power consumption, suitable is processed simultaneously, instead of only one for use in a practical hearing aid, is still a difficult number at a time as in a conventional digital com- technical problem. puter. The saving in processing time resulting from the use of this technique is sufficient to allow for All-Digital Hearing Aids real-time processing of audio signals. High-speed In an all-digital hearing aid both the processing array processors were introduced toward the end of the audio signals and the control of the processing of the nineteen seventies and shortly afterwards an are done by digital means. Further, all sampled all-digital hearing aid was developed configured waveforms are converted to binary form for ease of around one of these units (12). Although still too processing and then converted back to analog form large to be wearable, that instrumelat has been used after processing to drive the earphone. Graupe (6) effectively as a master hearing aid in a series of appears to have been the first to implement such a experimental investigations on the prescriptive fit- system using an 8080 microprocessor. The approach ting and evaluation of digital hearing aids (1 3,14,19). used was conceptually similar to that of the digitally Another important development was the intro- controlled analog system shown in Figure 4, except duction of a family of high-speed digital-signal- Journal of Rehabilitation Research and Development Vol. 24 No. 4 Fall 1987 processing (DSP) chips in 1982. Although not as East APPENDIX as an array processor, these chips are fast enough for Limited real-time processing of speech signals. Two Basic of Digital Filters Because of their small size, these chips can be Filtering may be viewed as a forrn of averaging packaged in a unlt small enough to be wearable. or smoothing. A very simple forrn of averaging is Experimental body-worn digital hearing aids were to take the mean of successive pairs of samples. developed soon after high-speed DSP chips became Consider, for example, the sampled-data sequence available (2,3,8,9,20). It is not clear which research representing the waveform shown in Figure 11 of the group was first in the race to develop a body-worn text. The numerical values of this sampled-data digital hearing aid, since in addition to the research sequence are reproduced in the third column of groups identified above, at least three other indus- Table 131. The fourth column of the table shows the trial research laboratories have developed instru- smoothing resulting from the operation: ments of this type but are extremely secretive about U, = 0.5 (X, + X,-,)= 0.5 X, 0.5 X ,,_,, their progress. + [la] The second major problem, that of reducing power where U, is the smoothed sample value at sampling consumption and physical size so that the digital instant i, chips are small enough to fit in a behind-the-ear or X,is the value of the original sample-data in-the-ear hearing aid, has yet to be resolved. Under sequence at sampling instant i, normal operation, the various DSP chips currently on the market (as of March 1987) draw far too much current for a practical behind-the-ear unit. An im- portant recent development is that of the application- Table A1 specific DSP chip dedicated to a specific task and Filtering of sampled-data sequance designed for very low power consumption. Appli- Sample Ui cation-specific analog chips for hearing aid use were Instant (Eqn Ib) developed well over a decade ago with highly suc- cesstut results; the major research groups concerned .85 with the development of a practical, wearable digital .325 - ,268 hearing aid are now actively engaged in developing - ,234 their own low-power, application-specific DSP chips. - ,547 It is likely that a practical solution will be found in - .373 - ,617 the near future. At least one group, Project Phoenix - .408 end of in Madison, Wisconsin, has indicated that it will I st period begin marketing a body-worn all-digital hearing aid before the end of 1987 and that a behind-the-ear unlt will be available within a year. (Personal com- munication between K. Mecox, Project Director, and the author) - .619 - .410 end of ACKNOWLEDGMENTS 2nd period 17 1.7 .9 .695 I would like to thank Linda Ashour and Matt Bakke for their 18 1.8 -.I .247 help in the preparation of this paper. I am also most grateful 19 1.9 - .43 - .306 to Stanley Daly for his editorial input. 20 2.0 - .1 - ,253 21 2.1 -.43 - .557 - .378 - ,619 - .410 end of 3rd period LEVITT: Digital Hearing Aids: A Tutorial Review

and XI_, is the sample value at the preceding sam- pling instant. The effect of the smoothing operation given by Equation [la] is to remove much of the ripple contained in the original sampled-data sequence. The sampled-data sequence represented by the Y, is, in essence, a filtered version of the original sampled-data sequence, X,. The digital filter repre- sented by Equation [la] is shown schematically in the upper half of Figure Al. The symbol D represents a delay of 1 sampling interval. The symbols x and + represent multiplication and addition, respec- tively. Note that the coefficients kl and k2 are both equal to 0.5 in this example. Note also that only four operations are required in order to implement this digital filter, (1 delay, 2 multiplications, and 1 addition). Another very simple digital filter is shown in the lower half of Figure Al. In this case, the output of the digital filter, Y,, is given by

In this case, only three digital operations are required (1 addition, 1 delay, and 1 multiplication). The fourth column of Table A1 shows the output sampled-data sequence Y, for k, = 0.5 in the above filter. This filter is not quite as effective in removing the ripple from the original sampled data sequence, but it has other useful properties. Note that this filter also takes time to settle down. It is only toward the end of the second period of the original data sequence, X,, that the output of the filter, Y,, approximates a Figure Al. Simple Digital Filters. periodic function. The upper section of the diagram shows a simple finite-impulse- It is conceptually convenient to think of the response (FIR) filter. The lower section of the diagram show5 smoothing operations given by Equations [la] and a simple recursive filter having an infinite impulse response [lb] in terms of the frequency responses of the (IIR). associated digital filters. The filter corresponding to Equation [lb] is essentially a lowpass filter. In electrical engineering terms the frequency response Equations [la] and [lb] can be generalized as of this digital filter is analogous (but not identical) follows: to that of a single-pole lowpass network, as com- monly realized by the simple RC network shown in Figure A2. The frequency response of the digital filter given by Equation [la] is more complex and is of the form sinflf, where f represents frequency. In general, if the arithmetic average of n samples is taken, the frequency response of the associated digital filter is where Y, = output sample at time instant i, proportional to sin (nDnf)l(nDnf) where D is the X, = input sample at time instant i, time delay between samples (the sampling interval). and the k, are constants (j = 1,2,3,..., n). Journal of Rehabilitation Research and Development Vol. 24 &o. 4 Fall 1987

(dB1 vin Figure A2. Simple RC-Filter. A simple low-pass filter consisting of a single resistance and capacitance is shown. The form of the frequency response is shown in F"REQL%ENcy the lower half of the diagram.

The digital filter corresponding to Equation [2a] DFT [k,, kZ,. . . , k,] is the discrete Fourier trans- is referred to as a finite-impulse-response (FIR) filter form of the set of coefficients. since the output U, is determined by only the last n Note that as the number of coefficients is in- values of XI. In contrast, the digital filter corre- creased, the precision with which the frequency sponding to Equation [2b] is referred to as a recursive response is specified is increased accordingly; i.e., or infinite impulse response (IIR) filter. In this case, the frequency interval between the discrete corn- the output U, is affected by all previous values of ponents f, is proportional to L/n. Note also that each XI. In order lo illustrate this point, consider the value of F(f,) is a complex number that specifies term U, , in the simplest filter of this type, as given both amplitude and phase of the frequency response by Equation [I b]. The value of U at sampling instant at each discrete frequency. Since the coefficient

1 - 1 is determined from the previous value of U, i.e., array consists of n real numbers, the resulting U(, 1, = X,,-,, + klU(,-z,. The value of V,, ,,, in discrete frequency response consists of n/2 complex turn, is dependent on Y,,-,,, which in turn is de- numbers. For mathematical convenience, n should pendent on U,,-,,, and so on. If the constant k, is be an even number. less than 1, then the earlier values of U become The digital filter corresponding to Equation [2b] progressively less important in deriving U,. is shown in Figure A-4. The frequency response of The digital filter corresponding to Equation [2a] this filter is obtained by finding the roots of an nth is shown in Figure A3. Electrical engineers will order polynomial in f with coefficients k,, k2, k3, recognize this circuit as a transversal filter. The . . . , k,. This is not always easy to do when n frequency response of the filter is determined by exceeds 3. A case of particular interest occurs when the coefficients k,, kz,. . . , k,. Expressed mathe- n = 2; i.e., when the digital filter shown in Figure matically, the frequency response of the filter is A-4 consists of only two delay loops. The frequency response of this filter is relatively easy to derive F(fl) = DFT [k,, kz, k?, . . , k,] . mathematically and is equal to that of a simple where F(f,) is the frequency response of the filter resonant circuit. If f, is the resonant frequency of specified at discrete frequencies f,,(,=,?. and the circuit and a, its bandwidth (i.e., the resonance LEVITT: Digital Nearing Aids: A Tutorial Review

Figure A3. A General Class of Finite-Impulse-Response (FIR) Filters. The digital filter shown in this diagram is a generalization of the very simple filter shown in the upper half of Figure Al.

Figure A4. A General Class of Infinite Impulse Response (IIR) Filters. The digital filter shown in this diagram is a generalization of the very simple filter shown in the lower half of Figure Al. is defined mathematicaliy as consisting of the corn- variations of these basic filter types, the analysis of plex pole-pair o, tj2~rf,),then the coefficient k, which can be quite complex. For further information is given by 2 e-"oD cos(2~rf,D) and k2 is given by on this topic, the reader is referred to the classic - 7-,~ texts of Gold and Rader (5) and Oppenheirn and The examples cited above represent two basic Shafer (21), among others. classes of digital filter. There are, of course, many 20 Journal of Rehabilitation Research and Development Vol. 24 No. 4 Fall 1987

REFERENCES

Clark GM, Crosby PA, Dowell Re, Kuzma JA, Money Levitt H: An array-processor, computer hearing aid. An- DK, Patrick JF, Seligman PM, and Tong YC: The prelim- nual Convention, American. Speech-Language-Wearing inary clinical trial of a multichannel Association, Toronto, 1982. See also ASHA 24:805, 1983. hearing . J Acoltst Soc Atn 74(6): 191 1-191 3, Levitt H, Neuman A, and Toraskar J: Orthogonal poly- 1983. nomial compression amplification for the hearing-impaired. E~~gebretsonAM, Morley RE, and O'Connetl MP: A wear- Proc lOtll Ann Conf' orr Rehnhilitation Tc~chi~oi(RESM able, pocket-sized processor for digital hearing aid and '87), pp 410-412. Washington, DC: Association for the other hearing prosthesis applications. Proceeditzgs of th~ Advancement of Rehabilitation Technology, 1987. Irrtrrnatiort~zlConfirenee on Acoirstics, Speech and Signrri Levitt IP, Neuman A,, Milis R, and Schwander T: A digital P~.ocessing, (ICASSP-19861, Institute of Electrical and master hearing aid. J Rehnbil Rcs Dev 23 (1):79-87, 1986. Electronic Engineering, 1986. Levitt H, Sullivan J, and Hwa~~gJU: A computerized Engebretson AM, Morley RE, and Popelka GR: Develop- hearing aid measurement/sirnulation system. H~rrringIn- ment of an ear-level digital hearing aid and computer- strlrrnents 37(2), 1986. assisted fitting procedure. J. Rehabil Res Dpt, 24(4), 1987 Mangold S and LeGon A: A programmable hearing aid (this issue). with multichannel compression. Scund Aridiol(8):121-126, Flanagan JL: Speeclz Anczlysis synthesis and Perception. 1979. New York: Springer-Verlag, 1965. Milne-Thompson LM: The Caic~irlllsc?fFinite DiJferenr.c>s. Gold B and Rader CM: Digital Processitlg of Sign~ils. London: McMillan and Go.. 1933. New York: McGraw-Hill, 1969. Moser LM: Hearing aid with digital processing for: Cor- Graupe I), Causey GI): Development of ;I hearing aid relation of signals from plural . dynamic range system with independently adjustable subranges of its control, or filtering using an erasable memory. US Patent spectrum using microprocessor hardware. Bui Pro.rtirei #4,187,413, February 5, 1980. Res 12:241-242, 1975. Neurnan A, Levitt W, Mills W, and Schwander T: An Graupe D, Grosspietsch JK, and Basseas SP: A single- evaluation of three adaptive hearing aid selection strate- -based self-adaptive filter of noise from speech gies. J Acoust Soc Atn (In Press) 1987. and its performance evaluation. J Rehabil Res Dev 24(4), Nunley J, Staab W, Steadrnan J, Wechsler P, and Spencer 1987 (this issue). B: A wearable digital hearing aid. The Necirirzg J, pp 29- Hecox K: New frontiers in ; Application of digital 31, 34-35, October, 1983. technology to hearing aids. Cnlif'ornin State Speeclz Lan- Oppenheirn AV and ScRafer RW: Digitcri Sigi~aiProces.sitzg. gltage and Nerrring Convention, Monterey, CA, 1985. Englewood Cliffs, New Jersey: Prentice Mall, 1975. Cummins KL and Hecox K: Ambulatory testing of digital Popelka F and Engebretson AM: A computer-based system hearing aid algorithms. Proc 1Orlz Anrz Conf on Rehabili- for hearing aid assessment. Hecrriizg 1nstrzirnc~nf.r34t7), tation Techno1 (RESNA '87), pp 398-400. Washington, 1983. DG: Assoc. for the Advancement of Rehabilitation Tech- Ragazinni JR and Franklin GF: Sanrpled-Dutcr Corrlrol nology, 1987. Systems. New York: McGraw-Hill 1958. Herlzano T, Levitt H, and Slosberg R: Computer-assisted Shannon CE and Weaver W: The Matlrematicrrl TI?eory of hearing aid selection. Speech Communicution Papers, Cornmlinication. Urbana: University of Illinois Press, 97th Meeting of the Acoltstical Society ofAmerica, Boston, 1949. 1979. Published by Acoustical Society of America, New Sludebaker GA, Sherbecoe RL, and Matesicla JS: Spectrum York, 1979. (See also J Acozrst Soc Am 65S, p 137A, shaping with a hardware digital filter. J Rehabil Res DPIJ 1979.) 24(4), 1987 (this issue). Kelly LJ, Jr., Lochbaum C, and Vyssotsky VA: A block diagram compiler. Bell System Tech J (40):669-676, 1961.