DIGITAL SIGNAL PROCESSING FOR THE AUDITORY SCIENTIST: A TUTORIAL INTRODUCTION

Transcription

1 CHAPTER 3 DIGITAL SIGNAL PROCESSING FOR THE AUDITORY SCIENTIST: A TUTORIAL INTRODUCTION Philip Leong, Tim Tucker and Simon Carlile 1. INTRODUCTION 1.1. OBJECTIVES OF THIS TUTORIAL This chapter is an introduction to digital signal processing (DSP) concepts used in the generation and analysis of auditory signals. It is not a mathematical exposition on DSP, but rather a qualitative description of the DSP techniques commonly used in the processing of audio signals. Readers should refer to the annotated bibliography for a more detailed coverage of specific DSP techniques SIGNAL PROCESSING IN AUDITORY RESEARCH The microprocessor has revolutionized signal processing by shifting the emphasis away from analog techniques. DSP enables the auditory scientist to generate, record and analyze audio signals with greater accuracy, convenience and resolution. Entirely new signal processing techniques such as generating complex stimuli, linear phase filters and Fourier transforms have become invaluable tools in most fields of science and engineering. An advantage of the digital approach is that system control is performed primarily through software, and thus changes to the system involve changes in software rather than hardware. Other advantages include improved tolerances to noise, arbitrarily high signal to noise Virtual Auditory Space: Generation and Applications, edited by Simon Carlile R.G. Landes Company.

2 80 Virtual Auditory Space: Generation and Applications ratios, no changes in performance due to component variation, ambient temperature and aging and ease of data storage. Although general purpose microprocessors such as the Intel Pentium can be used to perform digital signal processing, it is more common to use special purpose digital signal processors for real-time applications. Digital signal processors such as the AT&T DSP32C and the Texas Instruments TMS320C40 are microprocessors which are optimized to perform common digital signal processing operations such as Fourier transforms and filtering at maximum speed. Optimized software libraries for implementing most of the common techniques described in this chapter are available, allowing users to perform DSP without concern for the intricate details of DSP programming. 2. THE NATURE OF SOUND 2.1. SOUND AND MEASUREMENT Sound is created by a source generating a mechanical wave in the surrounding medium. Assuming the usual case of the medium being air, air molecules conduct the vibrations, propagating the sound in all directions away from the source. A listener can detect the sound because air pressure changes cause mechanical vibrations in the listener s eardrums which then pass through the middle ear to be transduced into neural signals in the inner ear. Sound sources create changes in pressure by causing a mechanical particle displacement. For example, striking a drum causes it to vibrate, and the air around it is alternately compressed and rarefied as the skin moves backwards and forwards. Note that the sound waves are longitudinal, and the backwards and forwards particle displacement is in the same direction as the propagation of the sound wave. Other sound sources include vibrating strings (e.g., violin, piano), vibrating air (e.g., organ, flute, human vocal tract) and vibrating solids (drums, loudspeaker, door knocker). Although sounds are longitudinal waves, they can be drawn as transverse waves (displacement being at right angles to the direction of propagation) like the sinusoidal waveform shown in Figure 3.1. Sinusoids are of particular importance since we will later see that all practical waveforms can be constructed from a sum of sinusoids. A sinusoid can be completely characterized by three parameters, frequency (or its reciprocal-period), amplitude and phase. Frequency is related to the pitch of the sound and amplitude is related to its loudness. The phase is only important when we are dealing with more than one sinusoid, and it determines the relative starting times of the sinusoids. The human ear can detect sounds over a wide range of frequency from 20 Hz to about 20 khz, a and can detect pressure ratios of approximately 1 x a In the case of a healthy, young adult.

3 Digital Signal Processing for the Auditory Scientist: A Tutorial Introduction 81 Fig Sine wave showing frequency, amplitude, phase and period. Sound intensity is proportional to the square of the pressure, and is defined as the sound energy transmitted per second through unit area. The magnitude of a sound can be measured as the ratio between two levels with one Bel corresponding to a tenfold difference in the ratio between two intensities. A more convenient unit is the decibel (db) which is one tenth of this amount. Thus we can express the difference in sound intensity between two sounds with intensity I 0 and I 1 (or between pressures of P 0 and P 1 ) as decibels = 10 log 10 (I 1 / I 0 ) = 20 log 10 (P 1 / P 0 ). Since the decibel scale is logarithmic, a positive value means that the ratio difference is greater than one, and this corresponds to amplification (I 1 is greater than I 0 ). When it is negative, I 1 is said to be attenuated from I 0. A +3 db (gain) causes a doubling in the intensity of the sound, whereas -3 db is a halving in intensity. A decibel measure is not an absolute measure of amplitude, but always corresponds to a difference in sound pressure between two signals, i.e., it is based on a ratio. Often, one would like to refer to a sound as an absolute number instead of a difference, and so the sound must be compared with a reference value and levels expressed in this fashion are called sound pressure levels (SPL). The usual reference is the softest 1000 Hz tone the average human can detect, which corresponds to an intensity of about W m 2 (Watt/meter 2 ). Thus

4 82 Virtual Auditory Space: Generation and Applications 20 db SPL refers to an intensity 100 times more than the reference value, namely W m 2. According to Vernon et al, 1 the application of the db scale can be traced back to early developments in telephone engineering. The engineers were interested in measuring how much power drop there was down a telephone line. Because the losses were significant they found a logarithmic ratio system worked well to explain the measurements. Coincidentally this logarithmic system also works well when describing hearing sensitivity. Psychophysically the auditory system has an approximately logarithmic sensitivity quantized to about one db. In a DSP system, the sound pressure is usually converted into an electrical voltage by a microphone. In a dynamic microphone, the change in air pressure created by the sound causes a coil to move in a magnetic field, generating a voltage. The voltage is then amplified and converted to a digital signal using an analog to digital converter (see section 3) NOISE Unwanted noise is present in all practical signals and must also be considered in DSP systems. External sources of noise include electrical interference from nearby electronic equipment and background audio noise. There can also be internal noise sources such as quantization noise (see section 3.3). Noise is often used as a sound stimulus in a DSP system since it contains energy across the whole frequency range of interest. White noise, for instance, contains uniform amounts of energy at all frequencies. The Signal to noise ratio (SNR) is the measurement used to describe the ratio between a signal and the noise present in the system. The SNR is defined as SNR = where v s is the signal voltage and v n is the noise voltage. b For example the minimum SNR for good speech recognition is approximately 6 db and an audio CD player has over 90 db SNR. 3. DISCRETE TIME SYSTEMS log v v 2 s 2 n 3.1. INTRODUCTION In the real world, all signals are analog in value and continuous in time. A DSP system operates in discrete time where signals are repreb Actually, these values are all root mean squared values since we are dealing with AC signals. Also, since noise is a random process, we usually measure it over several readings rather than a single reading.

5 Digital Signal Processing for the Auditory Scientist: A Tutorial Introduction 83 sented by a series of digital numbers. These digital values correspond to the signal s amplitude at uniformly spaced intervals of time. Although the theories corresponding to discrete and continuous time signals are not identical, they have many parallels. Fortunately, with care, these differences do not seriously limit our ability to process signals in the digital domain. A block diagram of a generic DSP system is shown in Figure 3.2. A continuous time signal is first low pass filtered (the necessity of filtering is explained below) by the anti-aliasing filter and converted to a digital signal via an analog to digital converter (ADC). An ADC is an electrical circuit that measures the analog signal c at its input and converts it to a digital number at its output. This is done at discrete, instantaneous points in time. The analog signal has now been converted to a digital representation, so it can be processed in the digital domain. After processing, the digital samples are converted back to an analog signal via a digital to analog converter (DAC) and low pass filter (called the reconstruction filter). Some systems such as a speech synthesizer, start with a digital representation and generate an analog signal so analog to digital conversion is not required. Similarly, some systems designed only to analyze signals do not require digital to analog conversion. All DSP systems have a structure similar to Figure 3.2. Since all of the processing of the system is performed in the digital domain, the functionality of the DSP system is usually derived solely through software. Thus, the same DSP hardware can be used for many different applications. Because software controlled systems are easier to develop and debug than hardware systems, DSP systems can be made to perform ever more complex signal processing tasks SAMPLING OF CONTINUOUS TIME SIGNALS In Figure 3.2, it can be seen that the first operation in a DSP system is to low pass filter and convert the analog signal into a digital one. This conversion is performed by the ADC using a process called discrete time sampling. Discrete time sampling converts a continuous Fig Block diagram of a generic DSP system. c These can be voltages or currents, but in this chapter we will assume that the input of the ADC is a voltage.

6 84 Virtual Auditory Space: Generation and Applications time signal into a discrete time signal. This process is illustrated in Figure 3.3. Voltage measurements are taken at regular intervals determined by the sampling rate of the ADC. In the Figure, our sampling period is T, corresponding to a sampling frequency of f s = 1/T. Note that the digitized version of the waveform is not an exact copy of the signal since it only changes value every sampling period. Two types of artifacts occur in the sampled record as a result of this process and are refereed to as quantization error and aliasing. A number of precautions need to be taken in DSP to ensure that the effects do not distort the final signal. These precautions are discussed in this and the following section. All practical signals contain energy only up to a maximum frequency and thus are said to be bandlimited. The Nyquist sampling theorem states that under certain conditions, a bandlimited continuous time signal can be exactly reconstructed from the discrete samples. This remarkable theorem is the basis of all DSP systems and is the reason why we can process signals using a discrete time representation of a continuous time system. If the maximum frequency contained within a signal is given by f max the Nyquist sampling theorem states that the original signal can be reconstructed from the sampled version if the sampling rate f s is greater than 2f max. This is called the Nyquist rate, i.e., f s f 2 > max If a signal is sampled below its Nyquist rate, the signal will become distorted and this effect is known as aliasing. There are a num- Fig Sampling of a continuous time analog signal to make it into a discrete time signal. The sampling frequency is f s = 1/T Hz.

7 Digital Signal Processing for the Auditory Scientist: A Tutorial Introduction 85 ber of nonintuitive consequences of the digitization process. For instance, if we examine a sampled signal using a frequency representation (see section 4) identical copies of the signal spectra occur at intervals of f s as illustrated in Figure 3.4. If the signal only contains frequency components which are less than f s /2, the copies of the signal at the different frequency bands will not overlap. However, if the maximum frequency is larger than f s /2, the higher frequencies will fold over onto the lower band and corrupt the lower frequency components (Fig. 3.4b). The sampling theorem can be rigorously demonstrated mathematically, but for our purposes is best understood intuitively by considering what happens if we change the frame rate of a movie camera as Fig Frequency domain representation of a signal sampled with a sampling period T. The original signal contains frequencies extending up to f max. Note that copies of the spectra occur at intervals of 2/T and if the sampling rate is less than the Nyquist rate, overlapping will occur causing aliasing [as shown in (b)]. Although only the positive x-axis is shown, copies extend into the negative frequency axis as well [as discussed in Oppenheim and Schafer (1975), see Annotated Bibliography, or any other DSP text].

8 86 Virtual Auditory Space: Generation and Applications we record a person at work. The action is normally sampled quickly so that when it is played back, we can make a faithful reproduction. As we slow the frame rate of the movie camera, a point is reached when we are unable to follow what the person is doing because the movie camera undersamples the signal. The frequency at which this occurs depends on the maximum frequency at which the person moves; after a certain point, we cannot reconstruct the original signal and in fact fast repetitive movements may appear as smooth slow movements (strobing). In a DSP system, an anti-aliasing filter like the one shown in Figure 3.2 is used. The anti-aliasing filter removes high frequencies from the signal, ensuring that it is bandlimited to at least half the sampling frequency. The anti-aliasing filter should not distort the amplitude or phase of the signals of interest, but should remove high frequency noise or unwanted parts of the signal to avoid aliasing. Anti-aliasing filters and reconstruction filters must be analog continuous time filters constructed using analog techniques. Unfortunately, we cannot build a perfect anti-aliasing filter and anti-aliasing filters are subject to tradeoffs between cost, steepness, sensitivity to component variations and phase characteristics. Most real-world DSP systems sample much faster than the Nyquist rate to reduce the requirements of the anti-aliasing filter. This technique, known as oversampling, allows the slope of the low pass filter to be more gradual, making the filter easier to implement. Refer to section 6 for more details on filter design. In practice, the speed at which we can perform our processing (including the time it takes to perform ADC and DAC) will limit the maximum sampling rate of the DSP system. In addition, if the sampled data must be stored, larger amounts of data will be generated for higher sampling rates and extra storage is required QUANTIZATION When a continuous signal is digitized, it must be changed from a value represented by a real number to a value which is represented by a finite number of levels (Fig. 3.3). This process is known as quantization. Quantization is analogous to using a ruler to measure something we only have a resolution of 1/2 of the finest graduation on the ruler, and the total length of the ruler determines the maximum length which we can easily measure. Internally, in a DSP system, an N bit number (which can represent 2 N different values) is used to represent the signal, and N describes the resolution of the converter. Quantizing errors are introduced by the finite resolution of the ADC, which is limited to the maximum value which the ADC can measure divided by 2 N+1. The resolution of an ADC is usually in the range of 8 to 18 bits. Generally speaking, as the resolution increases, the cost of the ADC

9 Digital Signal Processing for the Auditory Scientist: A Tutorial Introduction 87 increases and the speed decreases. To obtain the high conversion rates that are required, video signals are usually processed with 8 bit resolution. For speech signals, 12 bits is normally sufficient; whereas, for high quality audio reproduction, 16 bits or more are used. Converters are usually either bipolar (they can take both positive and negative inputs), or unipolar (they take strictly positive inputs). For example, a unipolar 8 bit converter might take inputs from 0 to 5 V, converting them to values from 0-255; whereas, a bipolar 8 bit converter with the same range would take inputs from -2.5 to +2.5 V and convert them to values of -128 to 127. When implementing analog to digital conversion, care must be taken to match the expected range of the analog input signal to the specified input range for the ADC (signal range and converter range respectively). Ideally, these two ranges will be the same and the conversion will use the full scale of the ADC. If the signal range is smaller than the converter range, we will not be fully utilizing the resolution of the converter. For example, if we use a 12 bit converter with input range 0 to 5 V to convert an analog input signal with range 0 to 1.25 V, only 1/4 of the conversion range is used, with 2 bits of the resolution lost, resulting in 10 bits of effective resolution (see Fig. 3.5). On the other hand, if the signal range is larger than the converter, clipping can occur. This causes large amounts of distortion and is therefore most undesirable. To circumvent this problem, such a signal should be attenuated and perhaps have its DC level shifted so that its range is within that of the converter. Fig Illustration of how resolution is lost if the signal is reduced in range. If a 5V signal is measured, it uses the entire 12 bits of the converter. If the signal is limited to 1.25 V in amplitude, only 10 bits are used.

10 88 Virtual Auditory Space: Generation and Applications 3.4. SIGNAL GENERATION In order to generate an analog signal, discrete time samples are first computed (either mathematically or by sampling a desired analog signal) and stored in the memory of the DSP device. By sending successive samples to a digital-to-analog converter and low pass filtering, a continuous time analog signal can be generated (see Figs. 3.2 and 3.3). The process of signal generation is the exact reverse of sampling analog signals (described in section 3.2). A reconstruction filter is required at the output to remove high frequency copies of the signal caused by sampling. As with sampling of signals, to avoid aliasing, the signal that is being generated must not have frequencies higher than the Nyquist rate. Data generated in this manner can be easily controlled in software and be arbitrarily complicated, thus offering advantages over analog signal generators which are restricted to generating much more simple signals. 4. FREQUENCY DOMAIN DIGITAL SIGNAL PROCESSING 4.1. REVIEW OF COMPLEX ARITHMETIC Since complex arithmetic is required when a frequency domain representation of a signal is used, we need to first provide a brief review of complex arithmetic. For a more complete description of complex numbers the reader is referred to any 1st year undergraduate mathematics text. The problem with real numbers is that polynomial equations such as x 2 +1 = 0 cannot be solved. If we define the imaginary number j = 1, we can solve any polynomial equation and it greatly simplifies a lot of algebra. Note that most scientists and mathematicians use i to represent the imaginary number but electrical engineers like to reserve this symbol for electrical current (we will always use j). In general, a complex number is broken into two parts and expressed in the form a+jb where a is the real part and b is the imagi- Fig A complex number a+jb can be plotted on a realimaginary graph. It can be represented either in cartesian or polar form (r,θ). In polar form, it is expressed as a magnitude and a phase value and in DSP systems, this is the preferred representation.

11 Digital Signal Processing for the Auditory Scientist: A Tutorial Introduction 89 nary part. Since this is two numbers, we can locate a complex number on a graph of the real and imaginary parts as shown in Figure 3.6. As is also shown in Figure 3.6, a complex number a+jb can be described using polar coordinates (r,θ) where r corresponds to the length of the ray extending from the origin of the graph and θ corresponds to the angle the ray makes with the real axis of the graph. These are related to the complex numbers so that the magnitude is given by r = a+jb or r = (a 2 +b 2 ) 1_ 2 and the phase θ = arg(a+jb) or θ = tan 1 (b/a). Note that the second expression in both cases can be simply derived using simple trigonometry from the graph of the real and imaginary parts of the complex number (Fig. 3.6). The complex conjugate of a complex number turns out to have very useful properties for signal processing and is computed by simply changing the sign of the imaginary part of the number, for example, the complex conjugate of 1-8j is 1+8j. As we shall see below, complex exponentials are also very important in signal processing and these are related to sinusoids in the following fashion: jθ e = cosθ + j sinθ 4.2. PERIODIC SIGNALS AND THE FOURIER SERIES A signal is said to be periodic if it repeats after some time. Sinusoids are very important in digital signal processing since any practical periodic waveform can be expressed as a Fourier series an infinite sum d of sinusoids of frequencies which are multiples of a fundamental frequency f o. The Fourier series x(t) is given by: x(t) = Xk e jkω t 0 k= where j = 1, ω 0 = 2πf 0 (frequency in radians per second) and the X k are known as the Fourier coefficients and represent the sinusoidal component of the signal at the frequency corresponding to index k. These coefficients are complex so that they can encode both amplitude and phase relationships of the sinusoids being summed. If only real coefficients were summed, we would not be able to represent arbitrary signals. Just as light can be split into its component wavelengths by using a prism or diffraction grating, electrical signals can also be separated into component frequencies. As an example, Figure 3.7 shows a square wave being constructed from contributions made from its component sinusoids. The overshoot at the corners of the square wave which is d In fact, all practical signals can be represented with a finite sum of sinusoids.

12 90 Virtual Auditory Space: Generation and Applications apparent in the plot with the 11 summed sinusoids shows an example of the Gibbs effect, which will be explained in section FOURIER ANALYSIS AND THE DISCRETE FOURIER TRANSFORM In the previous section, we saw that periodic signals could be represented as a sum of complex exponentials (e raised to the power of a complex number). However, this formulation is not of practical use in a DSP system since the Fourier series represents a continuous time signal as an infinite sum of sinusoids of different frequencies whereas our DSP systems can only deal with discrete time systems and finite sums. Fortunately, for a discrete time signal x[n] of finite duration N, the discrete Fourier transform (DFT) can be used to calculate the Fourier series coefficients X[k] (we normally use capital letters to represent frequency domain descriptions). The DFT, also known as the analysis equation, determines the Fourier coefficients from the sample values and can be expressed mathematically as: X[k] = 1 N N 1 x[n]e jk2πn/ N n=0, k = 0,1 N 1 Using this formulation, the X[k] values that are computed represent the contribution e of each of the component sinusoids required to approximate the input signal x[n]. The number of coefficients that are computed is given by N. If N were infinite, we would be able to exactly reconstruct our signal from the Fourier coefficients. Unfortunately, as we increase N, our computational and storage requirements increase, so a tradeoff must be made between an accurate representation of the input, and our computation time. Thus N determines the resolution (in frequency) of the time domain to frequency domain transform. As can be seen in the formula, the n value loops through the input, summing the contribution of each different frequency (represented by k) to the Fourier coefficient. The inverse DFT (IDFT), or synthesis equation performs the reverse computation, computing the time domain response from the Fourier coefficients. This is immensely useful in signal generation as it means that we can specify any signal in terms of its amplitude and phase characteristics and generate the analog waveform using the IDFT. The IDFT is applied using the following equation: X[n] = 1 N N 1 x[k]e jk2πn/ N k=0, n = 0,1 N 1 e Since it is a complex number, both the magnitude and phase is represented.

13 Digital Signal Processing for the Auditory Scientist: A Tutorial Introduction 91 Fig Plot showing terms of the Fourier series for a square wave being summed. Although the waveform approaches a square wave, the amplitude of the overshoot does not decrease with increasing N (Gibbs effect).

14 92 Virtual Auditory Space: Generation and Applications Thus we have a way to take our N point frequency domain representation, and reconstruct our original input x[n] from it or to synthesize new analog signals. In fact, the IDFT generates a waveform which is exactly equal to x[n] for 0 n < N from the Fourier coefficients X[k]. The Fourier coefficients computed by the DFT describe the frequency spectrum of the signal. We can then think of the DFT as transforming our signal x[n] from a representation in the time domain, to a complex valued frequency representation X[k]. In a DSP context, a polar representation is normally used to separately describe the magnitude and phase of each frequency component that makes up the signal. A very common computation in DSP is to analyze the frequency components of a signal. The procedure is shown in Figure 3.8, where the DFT of the signal is computed and then the amplitude and phase of each of the Fourier coefficients are plotted as a function of frequency. Figure 3.9 shows an example of a time domain input signal composed of two sine waves (1 khz and 3 khz) plus a small amount of noise which has all been digitized at 10 khz. The corresponding amplitude and phase components responses have been computed using the DFT and are also plotted on the figures. Note that as described in section 3.2 there are identical copies of the frequency components of the signal reflected around the frequency corresponding to half the sampling frequency (see also Fig. 3.4). Since the DFT quantizes the frequency components of the signal into N discrete values, there is an issue of tradeoff between resolution and range. After we apply the DFT, if the sampling period is T, each Fourier coefficient a[k] represents the magnitude and phase of the signal at a single frequency: f k = k, n = 0,1, N /2 NT Thus the DFT can only provide information about the signal at frequencies which are multiples of the fundamental frequency 1/NT. The separation in frequencies 1/NT, given by the DFT, is called the binwidth. The range of frequencies covered can be improved by reducing T (increasing the sampling rate) but this increases the binwidth and therefore reduces the frequency resolution. On the other hand resolution can be improved by increasing N (this does not affect the frequency range) which serves to sample more periods of the signal. For instance, if a 1024 point DFT is applied to a signal digitized at 40 khz, the binwidth is 39 Hz and we can analyze signals up to 20 khz. If we wanted better resolution, we could do a 2048 point DFT which would make our Fourier coefficients 19.5 Hz apart. Furthermore, if

15 Digital Signal Processing for the Auditory Scientist: A Tutorial Introduction 93 Fig Frequency domain signal analysis using the DFT (or FFT). Fig The top plot shows a signal composed from two sine waves plus noise. The bottom plots show the magnitude and phase of the 100 point (i.e., N = 100) DFT of this signal. The two peaks in the DFT occur at the frequencies of the sine waves. Note that the magnitude response is symmetric about N/2 and the phase is antisymmetric about N/2. This is one of the effects of sampling and the values of frequency and phase up to N/2 are sufficient to fully describe the system (see also Fig. 3.4).

16 94 Virtual Auditory Space: Generation and Applications we decided we only need to look at frequencies up to 10 khz, the sampling rate could be reduced to 20 khz. f 4.4. FAST FOURIER TRANSFORM For reasons of computational efficiency, the Fourier coefficients are not often calculated directly from the DFT equations given in section 4.3. The fast Fourier transform (FFT), as its name suggests, is an extremely efficient algorithm for calculating the DFT with the restriction that N be a power of 2. Whereas the DFT has a computational complexity of the order of N 2 multiplications, the FFT only requires order Nlog 2 (N) multiplications. In addition the architecture of modern DSP chips are optimized to perform FFTs at maximum speed. 5. TIME DOMAIN ANALYSIS 5.1. IMPULSE RESPONSE An impulse is a short duration signal pulse. It contains equal energy over all frequencies and has zero starting phase, which makes it a valuable test signal. If we wish to analyze how a system transforms a signal (such as the filtering processes of the outer), we can apply an impulse as the input and record the output (using a microphone inside the ear canal for example). The resulting impulse response that we measure enables us to determine the transfer function of the system. This type of measurement is critical in generating high fidelity virtual auditory space. The ear acts as a filter which varies as a function of the location in space from which the input signal arrives. By applying an impulse input from a particular point in space, the head related transfer function (HRTF) for that location in space can be measured (see chapter 2, section 1.4), and the response of the ear to any other input signal (such as music) can be computed from the measured impulse response information. An impulse function δ[n] is defined as: { δ[ n 0 ]= 1 n = 1 otherwise A practical example is striking a bell with a hammer. The hammer blow is an impulse input, which causes the bell to ring at certain frequencies determined by the geometry and materials of the bell. The bell can be considered as our system which filters the input signal to produce an output. This also illustrates a very important property of the impulse response; namely that it contains a very wide range of f Note that in this case we would need to adjust our anti-aliasing filters to ensure that aliasing would not occur if we lowered the sampling rate.

17 Digital Signal Processing for the Auditory Scientist: A Tutorial Introduction 95 frequencies. This can be demonstrated by summing together a large number of harmonically related sine waves all with the same starting phase. At the end of this operation one is left with an impulse whose duration is related to the highest frequency added into the complex. When the bell is struck it is essentially being presented with this very wide range of frequencies but only transmits those frequencies to which it is resonantly tuned. Impulse responses are very important in DSP since they completely characterize a linear time-invariant (LTI) system. g From the impulse response we can determine the system s response to arbitrary inputs. The frequency response of a system is simply the discrete Fourier transform of the impulse response IMPULSE RESPONSE MEASUREMENT AND GOLAY CODES Impulse responses can be measured by simply applying an impulse to the input of a system and measuring the output. Although the impulse response contains a very wide range of frequencies, the overall energy level is quite low. Noise in the measurement process can cause poor signal to noise ratios and therefore is not an entirely satisfactory method of measuring the impulse response. In recent years, Golay codes have become popular for accurately measuring the impulse response of various systems. In particular this technique has been used for measuring HRTFs (e.g., refs. 2-4) and provides a significant improvement in the signal to noise ratio. This process relies on the presentation of a pair of specially constructed codes, and postprocessing the resulting output to obtain the impulse response. The initial pair of Golay codes is a 1 = {+1,+1} and b 1 = {1,-1}. The next Golay code is obtained by appending b 1 to a 1 so that a 2 = {+1,+1,-1} and b 2 is obtained by appending -b 1 to a 1 to get b 2 = {+1,+1,-1,+1}. This is repeated recursively to obtain the Nth code pairs which are of length L = 2 N. g A system is said to be linear if a signal obeys the property of superposition. That is, if the response of a system to signal x 1 (t) is y 1 (t) and the response of the system to signal x 2 (t) is y 2 (t), then for some constants a and b, the response of the system to ax 1 (t)+bx 1 (t)= ay 1 (t)+by 1 (t). Even nonlinear systems can be approximated as linear ones if the excursions of the inputs and outputs are kept small. A system is time-invarient if a time shift in the input causes a time shift in the output. Mathematically, if the response of the system to x(t) is y(t), then the response to x(t-t 0 ) (i.e., x(t) time-shifted by t 0 is y(t-t0)). Linear time invariant (LTI) systems are very important since these conditions represent a large class of systems, and are necessary for much of the theory behind digital signal processing.

18 96 Virtual Auditory Space: Generation and Applications In order to measure the impulse response of a system h(t), we apply the Golay codes a N and b N to the system and measure the frequency response H A [f] and H B [f] (obtained using the DFT of the response to each code a N and b N ). These responses can then be processed by multiplying each response by the complex conjugate of the DFT of each code a N and b N. If we then take the IDFT, the impulse response is obtained. Mathematically, this can be expressed: h()= t IDFT 1 { HA[ f ] DFT ( a )+ HB[ f ] DFT n ( bn) } 2L where DFT* denotes the complex conjugate of the discrete Fourier transform (see section 4.1 and 4.3). This method of computing the impulse response has a signal to noise ratio which is 10log 10 (2L) db higher than if it were computed directly using an impulse signal (since there is much more energy present in the input signal). In practice, typical lengths used are L = 512 (ref. 2) and L = 1024 (ref. 3) which correspond to improvements in signal to noise ratio of 30.1 db and 33.1 db respectively. For a further discussion of this important processing method and some implementation examples, see Zhou et al CONVOLUTION In DSP two equivalent methods, in either the frequency or time domain, can be used to filter an input signal with an impulse response. In the frequency domain, direct multiplication of the DFT of the time domain input signal with the DFT of the transfer function of the system followed by an IDFT will produce the output (see Fig. 3.10). The time domain equivalent to multiplication in the frequency domain is to convolve the input signal x[n] with the impulse response h[n] of the system in order to produce the output y[n] of that system (see Fig. 3.10). Convolution is the infinite sum of time shifted values, and the operation is commutative. h We use the * operator to denote convolution so: y[n] = x[k]h[n k] k = = x * h = h * n A most useful property of the DFT is that convolutions in the time domain correspond to multiplication in the frequency domain. h If * is a commutative operator, x * h = h * x. For example, addition is commutative but division is not.

19 Digital Signal Processing for the Auditory Scientist: A Tutorial Introduction 97 Fig Multiplication in the frequency domain is equivalent to convolution in the time domain. This property can be exploited to make efficient implementations of FIR filters as discussed in section FILTER DESIGN As its name suggests, filtering involves modification of certain signal components. This could involve either amplifying or attenuating portions of the signal. A decibel scale is normally used to describe these factors with positive gain corresponding to amplification (the signal is larger than the original), and negative gain attenuating the signal. Digital filters have significant advantages over their analog counterparts since they can be made more accurately and have steeper edges. Since they are often implemented in software, the same hardware can be used to apply many different filtering functions. The desired characteristics of a filter are usually described in the frequency domain, selecting and/or rejecting different frequencies (standard filters are lowpass, highpass, bandpass and bandstop) as illustrated in Figure For example, a low pass filter might be used to remove high frequency noise from a signal in an antialiasing application; a high pass filter can be used to remove the DC level from a signal and shift it into range for an ADC; a bandpass filter could extract a signal which is known to be in a certain frequency range;

20 98 Virtual Auditory Space: Generation and Applications Fig Ideal filter magnitude response characteristics. and a bandstop filter could be used to remove 50 or 60 Hz power line noise from a recording. Filters are normally designed only with regard to the amplitude of the frequency response H[k] and then the resulting phase response arg(h[k]) is analyzed to see if it is satisfactory (see also section 4.1). A signal is said to be undistorted if the filter only amplifies and/or time delays the signal. Amplification corresponds to a frequency independent scaling of the magnitude. The time delay (or group delay) is given by: τ[k] = d dk arg(h[k]) and for this to be constant, the phase response must be linear with frequency (see also chapter 2, section 1.2). Mathematically, if c is a constant: arg(h[k]) = ck If the phase response is nonlinear, the time delay of the filter will be a function of the frequency and hence the output will be distorted. This is usually not a problem as long as the phase response is reason-

21 Digital Signal Processing for the Auditory Scientist: A Tutorial Introduction 99 ably smooth. However, linear phase is desirable when high fidelity is required. The specification of a filter is the first step in filter design. An ideal lowpass filter has unit gain in the passband, no transition region, and infinite attenuation in the stopband (this terminology is defined in Fig. 3.12). In practice, arbitrarily close approximations can be made to this ideal as long as we have enough computing power to implement our filter. For a given amount of computation, changing any parameter will affect the others. In fact, most of the time spent in filter design involves dealing with tradeoffs between magnitude response, phase response and computational requirements. This is why there is such a wide choice of filter approximations. Linear, time invariant systems can be divided into those with finite impulse response (FIR), and those with infinite impulse response (IIR). FIR filters can have precisely linear phase and guaranteed stability; whereas, IIR filters are implemented recursively, have steeper characteristics than a FIR filter (given the same amount of computing time), but are more sensitive to quantization effects which can make the filters unstable, and hence oscillate. Quantization effects in filters are beyond the scope of this introduction but interested readers can refer to Oppenheim and Schafer 5 for a good description of these effects. Fig Low pass filter specifications. A perfect low pass filter has no passband ripple and infinite attenuation in the stopband. Although practical filters such as the one shown cannot achieve this, they can make arbitrarily close approximations to this ideal.

22 100 Virtual Auditory Space: Generation and Applications 6.1. FIR filters FIR filters are usually implemented by convolving (see section 5.3) the filter s impulse response with the input signal x[k]. Since the impulse response is finite, the sum for the convolution becomes finite. If the impulse response is of length N, the computation is given by: N y[n] = h[k]x[n 1] k = 0 As can be seen from the equation, the computation requires the N previous values of the input x, and then these values can be considered as taps on a delay line (see Fig. 3.13). The inputs x[n] are multiplied and accumulated with the coefficients of the filter represented by h[k] to produce the output signal y[n]. DSP chips are optimized so that they can compute the multiply and accumulate function (known as MAC) at maximum speed. Larger values of N allow steeper filters to be implemented, but they are computationally more expensive. Very complicated transfer functions can be easily implemented using FIR filters. For example, head related transfer functions of the outer ear can be simulated in DSP for virtual reality applications by first measuring the impulse response h of the filter (see section 5.2), and then convolving the input with the impulse response to obtain a filtered output (see Fig. 1.1 and chapter 4). For large values of N, it is possible to reduce the amount of computation required by using the fact that convolution in the time domain is equivalent to multiplication in the frequency domain (see sec- Fig Implementation of a FIR filter. The FIR filter can be modeled as the weighted sum of the outputs of a delay line.

23 Digital Signal Processing for the Auditory Scientist: A Tutorial Introduction 101 tion 5.3). In this case, the FFT of the input signal is computed, multiplied by the FFT of the impulse response, and then the inverse FFT is computed to generate the output Windowing functions In section 4.2, we saw that any signal can be reconstructed from its Fourier series which is an infinite sum. We cannot, in general, produce arbitrary signals using a finite sum, but it is possible to make an arbitrarily close approximation by summing many terms of the Fourier series. In this section, we describe windowing functions which are used to reduce the length of a long (or even infinite) impulse response to produce a finite impulse response of length N. The most obvious technique would be to simply truncate the impulse response. This rectangular windowing function can be described mathematically as: w R (n) = { 0 n N 01 otherwise The impulse function h[n] is then multiplied by the windowing function to give h new [n]: h new [n] = w R (n)h[n] n = 0 N-1 The problem with this technique is that if it is applied to a discontinuous function (such as desired in filters), it leads to overshoot or ringing in the time domain response. This problem arises because we are trying to approximate the discontinuity using a finite number of Fourier series terms (when we really need an infinite number of terms), with the resulting effect known as the Gibbs phenomenon. For any finite value of N, a truncated Fourier series will, in general, exhibit overshoot if the rectangular window is applied. An example of this effect can be seen Figure 3.7. The amplitude of the overshoot does not decrease with increasing N. Although a large value of N will make the energy in the ripples negligible, by applying other windowing functions we can overcome this problem. For this reason, the rectangular window is not used in practice. Windowing functions such as the Hanning, Hamming and Kaiser windows can be used to progressively reduce the Fourier coefficients instead of the discontinuous weighting applied by the rectangular window. The Hamming window is a raised cosine function given by: 2πn w n (n) = cos N 1 As can be seen in Figure 3.14, the Hamming window has a gradual transition to the null at N-1 avoiding the discontinuity which would be caused by a rectangular windowing function.

24 102 Virtual Auditory Space: Generation and Applications Frequency sampling design A FIR filter can be designed simply in the frequency domain by applying the IDFT to N uniformly spaced samples of the desired frequency response, thus obtaining the impulse response. After having made a filter design using frequency sampling, the DFT of the impulse response should be calculated and plotted to check the design. Fig The Hamming window function. The tapered response of the function causes a gradual truncation of the impulse response.

25 Digital Signal Processing for the Auditory Scientist: A Tutorial Introduction 103 The larger the value of N, the closer the filter will be to the desired response. Care must be taken when specifying the amplitude and phase values before performing the IDFT. When using a frequency sampling design the amplitude and phase values must be arranged symmetrically around f 2 /2 as is illustrated in Figure In order to achieve this symmetry the IDFT coefficients must occur in complex conjugate pairs i and obey the following equations: For N odd, H(N-k) = H * (k) k = 1,2, (N-1)/2 Fig Example of a frequency sampled design of a complex filter. Plots (a) and (b) show the desired magnitude and phase response of the filter. Our frequency sampled design with N = 50 is computed by taking an IDFT to get the impulse response shown in (c). The resulting frequency response is shown in (d) which passes through all of the sampled points of (a). i We use the * superscript to denote the complex conjugate of a number.

26 104 Virtual Auditory Space: Generation and Applications and for an even N, ( )= ( ) = ( ) HN k H k k 12,, L N/ 2 1 HN ( / 2)= 0 Figure 3.15 shows an example of a frequency sampling design. The frequency response that we wish to achieve is that of the HRTF shown in the top two plots of the figure and is represented by the 150 point magnitude and phase response. The symmetry of the magnitude and phase response ensure that the above equations are satisfied. For an impulse response of length 50, the figure shows the impulse response, and resulting frequency response, of a filter generated using the frequency sampling approach Optimal FIR filters As an example of the trade-offs in filter design, a filter that is allowed to have ripples in the passband and stopband can be made to have a steeper transition region than one which is constrained to be monotonic. In practice, ripples in the passband and stopband (see Fig. 3.16) are acceptable, with the allowable ripple specified in the filter design. The most widely used FIR design algorithm was developed by Parks and McClellan. 6 This design is optimal in the sense that for a given length (number of taps), it has the smallest maximum error in the stopband and passband. Designing such filters is a computationally expensive task which involves optimization to determine the coefficients. Such filters are designed using computer aided design (CAD) programs which takes the specifications as inputs, and produces the impulse response of the filter as the output. More sophisticated programs can generate DSP specific subroutines which can then be called from an applications program. Figure 3.16 shows an example of a filter design using the MATLAB Signal Processing Toolbox from Mathworks Inc IIR FILTERS An infinite impulse response (IIR) filter has the following formula N y[n] = a[k]x[n k] b[ j]y[n j] k =0 where x[n] are the inputs, y[n] the outputs and a[k], b[j] are the filter coefficients. Compared with the FIR filter which is a function of the previous N inputs the IIR filter is also a function of the previous M outputs, and it is this recursive formulation that makes the impulse response infinite in length. For nearly all applications, the filter coefficients a[k] and b[j] are one of the four standard filter types (Butterworth, Chebyshev, inverse M j=0

27 Digital Signal Processing for the Auditory Scientist: A Tutorial Introduction 105 Fig Example showing the design of a 500 Hz lowpass filter using the MATLAB DSP toolkit. The transition region is set between 500 Hz (Fpass) and 600 Hz (Fstop), the passband ripple is 3 db (Rpass) and stopband attenuation is 50 db (Rstop). The filter design software uses the Parks-McClellan algorithm to determine the filter order (27) and compute the filter coefficients. Only the magnitude response is shown, but this filter has linear phase. Chebyshev or elliptic). Each type is optimal in some fashion, and the different types represent tradeoffs in steepness, ripple and phase response. In order to illustrate the differences in these four types of filters, Figure 3.17 shows an 8th order bandpass filter design using the Butterworth, Chebyshev and inverse Chebyshev methods. Butterworth filters are maximally flat and their transfer function changes monotonically with frequency. Chebyshev filters allow ripples in the passband, enabling them to obtain a steeper filter (at the expense of more nonlinearity in the phase response). The inverse Chebyshev filter is maximally flat in the passband and has ripples in the stopband. The elliptic filter(not shown) has the steepest transfer characteristic, with ripples in both passband and stopband. Unfortunately, the elliptic filter also has the worse phase response.