he noise power spectrum. As illustrated, the quantization noise is uniformly distributed in the frequency range of [-ws/2,w/s]. Figure 1.9(a) shows the case where oversampling is not used (i.e., Ws = W2). In this case, the noise is distributed in the same frequency range as the message signal. On the other hand, Figure 1.9(b) illustrates the spectrum of the noise when oversampling is in use (i.e., Ws > 2 an). The area under the shaded curves in both figures is equal because, as is shown by equation 1.13, the quantization noise power depends only on the quantization step size and not the sampling rate. When using oversampling, the lowpass filter rejects the higher-frequency noise components, which results in overall noise reduction. The noise can Pulse Code Modulation Lowpass Filter Lowpass Filter Message Message Spectrum Spectrum Noise Noise Spectrum Spectrum Fig. 1.10 Spectrum of quantization noise (a) without oversampling and (b) with oversampling. Example of two-level quantization: (a) original image, (b) thresholded image, (c) dithered image. be further reduced by employing noise-shaping filters. In fact, a noise-shaping filter con- tours the noise SO that it is reduced in the audible regions and increased in the inaudible regions Dithering is another technique that can be used to improve the perceptual quality of speech signals. The idea of dithering is to randomize quantization error when a high- resolution signal is converted to low resolution. Dithering is widely used in image- processing applications. For example, because most printing facilities use a binary (blacks and whites only) scheme to print a picture, given a gray-scale picture, a conversion is needed before printing. This is a special case of quantization with only two quantization levels. One solution is to use a simple fixed threshold level. However, a more efficient way is to use dithering to spread the quantization noise spatially across the image pixels. In other words, when a pixel is quantized, the quantization error is calculated and is