Digital Signal Processing. The Scientist and Engineers Guide. Steven W. Smith

FIGURE 21-3

Windowed--sinc and Chebyshev step responses. The step responses are shown for a 51 point windowed-sinc filter and a 6 pole, 0.5% ripple Chebyshev recursive filter. Each of these filters has a cutoff frequency of 0.2. The windowed-sinc has a slightly better step response because it has less overshoot and a zero phase.

millivolt signal at 61 hertz that is riding on a 120 volt power line at 60 hertz. Figure 21-4 shows how these two filters compare when you need maximum performance. The recursive filter is a 6 pole Chebyshev with 0.5% ripple. This is the maximum number of poles that can be used at a 0.05 cutoff frequency with single precision. The windowed-sinc uses a 1001 point filter kernel, formed by convolving a 501 point windowed-sinc filter kernel with itself. As shown in Chapter 16, this provides greater stopband attenuation.

How do these two filters compare when maximum performance is needed? The windowed-sinc crushes the Chebyshev! Even if the recursive filter were improved (more poles, multistage implementation, double precision, etc.), it is still no match for the FIR performance. This is especially impressive when you consider that the windowed-sinc has only begun to fight. There are strong limits on the maximum performance that recursive filters can provide. The windowed-sinc, in contrast, can be pushed to incredible levels. This is, of course, provided you are willing to wait for the result. Which brings up the second critical issue: speed.

Comparing these filters for speed is like racing a Ferrari against a go-cart. Figure 21-5 shows how much longer the windowed-sinc takes to execute, compared to a six pole recursive filter. Since the recursive filter has a faster roll-off at low and high frequencies, the length of the windowed-sinc kernel must be made longer to match the performance (i.e., to keep the comparison fair). This accounts for the increased execution time for the windowed-sinc near frequencies 0 and 0.5. The important point is that FIR filters can be expected to be about an order of magnitude slower than comparable IIR filters (go-cart: 15 mph, Ferrari: 150 mph).

Match #3: Moving Average vs. Single Pole

Our third competition will be a battle of the time domain filters. The first fighter will be a nine point moving average filter. Its opponent for today's match will be a single pole recursive filter using the bidirectional technique. To achieve a comparable frequency response, the single pole filter will use a sample-to-sample decay of x ' 0.70 . The battle begins in Fig. 21-6 where the frequency response of each filter is shown. Neither one is very impressive, but of course, frequency separation isn't what these filters are used for. No points for either side.

Figure 21-7 shows the step responses of the filters. In (a), the moving average step response is a straight line, the most rapid way of moving from one level to another. In (b), the recursive filter's step response is smoother, which may be better for some applications. One point for each side.

These filters are quite equally matched in terms of performance and often the choice between the two is made on personal preference. However, there are

two cases where one filter has a slight edge over the other. These are based on the trade-off between development time and execution time. In the first instance, you want to reduce development time and are willing to accept a slower filter. For example, you might have a one time need to filter a few thousand points. Since the entire program runs in only a few seconds, it is pointless to spend time optimizing the algorithm. Floating point will almost certainly be used. The choice is to use the moving average filter carried out by convolution, or a single pole recursive filter. The winner here is the recursive filter. It will be slightly easier to program and modify, and will execute much faster.

The second case is just the opposite; your filter must operate as fast as possible and you are willing to spend the extra development time to get it. For instance, this filter might be a part of a commercial product, with the potential to be run millions of times. You will probably use integers for the highest possible speed. Your choice of filters will be the moving average

FIGURE 21-7

Step responses of the moving average and the bidirectional single pole filter. The moving average step response occurs over a smaller number of samples, while the single pole filter's step response is smoother.

CHAPTER

Audio Processing

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Audio processing covers many diverse fields, all involved in presenting sound to human listeners. Three areas are prominent: (1) high fidelity music reproduction, such as in audio compact discs,

(2) voice telecommunications, another name for telephone networks, and (3) synthetic speech, where computers generate and recognize human voice patterns. While these applications have different goals and problems, they are linked by a common umpire: the human ear. Digital Signal Processing has produced revolutionary changes in these and other areas of audio processing.

Human Hearing

The human ear is an exceedingly complex organ. To make matters even more difficult, the information from two ears is combined in a perplexing neural network, the human brain. Keep in mind that the following is only a brief overview; there are many subtle effects and poorly understood phenomena related to human hearing.

Figure 22-1 illustrates the major structures and processes that comprise the human ear. The outer ear is composed of two parts, the visible flap of skin and cartilage attached to the side of the head, and the ear canal, a tube about 0.5 cm in diameter extending about 3 cm into the head. These structures direct environmental sounds to the sensitive middle and inner ear organs located safely inside of the skull bones. Stretched across the end of the ear canal is a thin sheet of tissue called the tympanic membrane or ear drum. Sound waves striking the tympanic membrane cause it to vibrate. The middle ear is a set of small bones that transfer this vibration to the cochlea (inner ear) where it is converted to neural impulses. The cochlea is a liquid filled tube roughly 2 mm in diameter and 3 cm in length. Although shown straight in Fig. 22-1, the cochlea is curled up and looks like a small snail shell. In fact, cochlea is derived from the Greek word for snail.

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The primary advantage of having two ears is the ability to identify the direction of the sound. Human listeners can detect the difference between two sound sources that are placed as little as three degrees apart, about the width of a person at 10 meters. This directional information is obtained in two separate ways. First, frequencies above about 1 kHz are strongly shadowed by the head. In other words, the ear nearest the sound receives a stronger signal than the ear on the opposite side of the head. The second clue to directionality is that the ear on the far side of the head hears the sound slightly later than the near ear, due to its greater distance from the source. Based on a typical head size (about 22 cm) and the speed of sound (about 340 meters per second), an angular discrimination of three degrees requires a timing precision of about 30 microseconds. Since this timing requires the volley principle, this clue to directionality is predominately used for sounds less than about 1 kHz.

Both these sources of directional information are greatly aided by the ability to turn the head and observe the change in the signals. An interesting sensation occurs when a listener is presented with exactly the same sounds to both ears, such as listening to monaural sound through headphones. The brain concludes that the sound is coming from the center of the listener's head!

While human hearing can determine the direction a sound is from, it does poorly in identifying the distance to the sound source. This is because there are few clues available in a sound wave that can provide this information. Human hearing weakly perceives that high frequency sounds are nearby, while low frequency sounds are distant. This is because sound waves dissipate their higher frequencies as they propagate long distances. Echo content is another weak clue to distance, providing a perception of the room size. For example,

sounds in a large auditorium will contain echoes at about 100 millisecond intervals, while 10 milliseconds is typical for a small office. Some species have solved this ranging problem by using active sonar. For example, bats and dolphins produce clicks and squeaks that reflect from nearby objects. By measuring the interval between transmission and echo, these animals can locate objects with about 1 cm resolution. Experiments have shown that some humans, particularly the blind, can also use active echo localization to a small extent.

Timbre

The perception of a continuous sound, such as a note from a musical instrument, is often divided into three parts: loudness, pitch, and timbre (pronounced "timber"). Loudness is a measure of sound wave intensity, as previously described. Pitch is the frequency of the fundamental component in the sound, that is, the frequency with which the waveform repeats itself. While there are subtle effects in both these perceptions, they are a straightforward match with easily characterized physical quantities.

Timbre is more complicated, being determined by the harmonic content of the signal. Figure 22-2 illustrates two waveforms, each formed by adding a 1 kHz sine wave with an amplitude of one, to a 3 kHz sine wave with an amplitude of one-half. The difference between the two waveforms is that the one shown in (b) has the higher frequency inverted before the addition. Put another way, the third harmonic (3 kHz) is phase shifted by 180 degrees compared to the first harmonic (1 kHz). In spite of the very different time domain waveforms, these two signals sound identical. This is because hearing is based on the amplitude of the frequencies, and is very insensitive to their phase. The shape of the time domain waveform is only indirectly related to hearing, and usually not considered in audio systems.

FIGURE 22-2

Phase detection of the human ear. The human ear is very insensitive to the relative phase of the component sinusoids. For example, these two waveforms would sound identical, because the amplitudes of their components are the same, even though their relative phases are different.