Imagine you are riding a bicycle and suddenly you are caught in a downpour. You are unable to keep your eyes open in the rain, so you start blinking them, opening them only as frequently as necessary to see the changing scene before you. You are using the sampling technique without really being aware of it. Your brain constructs the full image from the samples obtained when the eyes are open for a brief period. When you watch a cinema you are actually shown image samples projected on the screen (typically 16 to 25 frames per second). Your brain perceives the individual frames as a continuous image without flicker at this rate.

We are increasingly dealing with digital information as in, for example, digital cameras, CDs and DVDs, digital video, digital audio and digital cellular phones, which are inherently sampled. Most of the information generated by natural processes, however, is analog. The human speech, audio signals produced by musical instruments, video signals from cameras, outputs of measuring instruments like seismograph or thermometer, etc. are examples of analog signals. The analog signal is characterized by being continuous in time and amplitude. Such signals, when used with digital systems, need to be converted to digital signals as shown in Figure 1. The output of the digital system is converted back to analog form. The purpose of all this is to take advantage of the digital systems over their analog counterparts. It is convenient and efficient to process, store and transmit signals in the digital format than in their natural analog format. However, we cannot perceive the signals directly in digital domain. Try feeding a digital audio stream directly to a speaker system or a digital video stream to the video input of a TV set. What you hear or see is just noise. So it is essential to convert the digitally stored, received or processed signal back to the original analog format. A fundamental question is “Do we loose any information in the signal in the process of these conversions from analog to digital and back to analog?” We loose nothing if we do the job in the right way, if not we loose almost everything. Our first task is to understand the right way of doing these conversions so that we do not loose any information in the end-to-end process.

Before understanding the conversion to digital let's see what digital really is. A digital signal has two basic characteristics. The signal changes at fixed time intervals, i.e., discretely in time. Second, the amplitude of the signal changes in discrete steps, i.e., the signal can have only certain finite values. In contrast, an analog signal can change continuously in time and amplitude. The amplitude values are infinite even within a finite range of the analog signal. Hence, converting an analog signal to digital involves two steps: first, making it discrete in time, which is carried out by the sampling process and second, making it discrete in amplitude, which is carried out by a process called quantization. Figure 2 shows how a signal flows through an analog to digital converter. It goes through a bandlimiting low-pass filter, the significance of which will be made clear shortly, a sampler, a quantizer and an encoder.

Although the signal at the output of the quantizer could be called digital it is still a multilevel signal changing at fixed intervals. The levels are assigned binary numbers in the encoder. The output of the encoder has multiple bits each with two levels — high and low, or logic ‘one’ and logic ‘zero.’ This is the true format of the digital signal that we find in all our applications. Figure 3 shows the signal waveforms at different stages in Figure 2. The analog input (a) contains high-frequency components that are removed by the bandlimiting filter. The smoothened output (b) is sampled at intervals of T seconds (c). The samples are then quantized (d) and encoded (e). An eight-level quantizer shown requires a three-bit encoder. It is clear that the quantization is an approximation process where the samples are approximated by the closest fixed level, introducing an error in the quantized samples. This is an irrecoverable process in the sense that the original signal can never be exactly recovered from the quantized signal. The error, however, can be made as small as desired by increasing the number of quantizer levels. For example, if the quantizer levels in Figure 3 are increased to 16, the error is halved, and we need four bits to encode the levels (2⁴ = 16). In general, n bits can encode 2ⁿ levels. The resulting error power, measured relative to the signal power and expressed as a ratio of signal to quantization noise (S/N), is given by

(S/N) = 6n + 1.8 dB (1)

Clearly, doubling the number of quantizer levels and using one extra bit in the encoder improves the performance by 6 dB.

Typical values of n, in practice, are eight bits for speech and video, and 16 bits to 20 bits for music. Once the number n is chosen for an application we have decided the distortion level that we can tolerate. No matter what we do, we can never remove this distortion from the signal. When we process, store or transmit the digitized signal the reconstructed analog signal quality cannot be better than that given by Equation 1. Because there is always a loss of quality, no matter how small, the quantization is a lossy process. (See the sidebar “Is digital better than analog?”)

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