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U. Glaeser

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FIGURE 39.65<br />

straight line drawn between adjacent input samples. Figure 39.65 illustrates the result of linearly interpolating<br />

a sine wave. The quality is quite good if the frequency of the sine wave is low relative to the<br />

Nyquist, but the quality deteriorates significantly as the frequency approaches the Nyquist. The linear<br />

interpolator has a low pass filtering effect that becomes noticeable above one half the Nyquist frequency.<br />

In addition, the alias rejection is not very good for the images of signals above one-half the Nyquist<br />

frequency. Thus, linear interpolation affects the quality in both the frequency response and aliasing<br />

distortion for high frequencies.<br />

Multi-Point Interpolation<br />

Multi-point interpolation can produce much better quality than linear interpolation in both frequency<br />

response and aliasing distortion. The ideal interpolator has a frequency response that is perfectly flat<br />

within the passband and attenuates all other frequencies to zero. Convolving the input waveform with a<br />

sinc function that runs from negative to positive infinite time produces such a frequency response.<br />

Unfortunately, we must work within the limits of finite time to build a real interpolator. In 1984, Gossett<br />

and Smith showed an efficient way to use a finite-length, windowed sinc function as a finite-impulseresponse<br />

(FIR) filter for sample rate conversion over a wide range of pitches [6]. The definition of the<br />

sinc function is sint/<br />

t.<br />

The convolution equation is ∑n=0 anx T−n,<br />

where x is the input waveform and a is a selected set of<br />

coefficients, possibly a windowed sinc function.<br />

The hardware implementation of a Gossett-Smith sample rate converter consists of a read-onlymemory<br />

(ROM) containing the filter coefficients, a linear interpolator to increase the resolution of the<br />

filter coefficient set, and a multiply-accumulate unit to perform the convolution. Figure 39.66 shows a<br />

block diagram of a typical Gossett-Smith interpolation system. Because the sinc function and other lowpass<br />

FIR filters are symmetric about their centers, it is only necessary to store half of the points in the<br />

ROM. Simple address mirroring makes the ROM appear to contain all the points.<br />

Perceived quality is often more important than measured quality. That is, it is more important to sound<br />

good to humans than to measure low distortion on laboratory instruments. Using a perceptual approach,<br />

one can design sample-rate conversion filters that sound better [7]. For example, humans cannot hear<br />

above 20 kHz, yet a 48 kHz sample-rate can represent frequencies up to 24 kHz. A filter that allows<br />

distortion within this guardband between 20 and 24 kHz can achieve better quality within the audible<br />

range of 20 Hz to 20 kHz.<br />

N−1<br />

© 2002 by CRC Press LLC<br />

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Linear interpolation of simple time series.

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