[8.3 ANALOG-TO-DIGITAL](#page-13-0) (A/D) CONVERSION

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8.3 ANALOG-TO-DIGITAL (A/D) CONVERSION

The amplitude of an analog signal can take on any value over a continuous range. Hence, analog signal amplitude can take on an infinite number of values. In contrast, a digital signal amplitude can take on only a finite number of values. An analog signal can be converted into a digital signal by means of sampling and quantizing (rounding off). Sampling an analog signal alone will not yield a digital signal because a sample of analog signal can still take on any value in a continuous range. It is digitized by rounding off its value to one of the closest permissible numbers (or quantized levels), as illustrated in Fig. 8.14a, which represents one possible quantizing scheme. The amplitudes of the analog signal x(t) lie in the range (−V,V). This range is partitioned into L subintervals, each of magnitude = 2V/L. Next, each sample amplitude is approximated by the midpoint value of the subinterval in which the sample falls (see Fig. 8.14a for L = 16). It is clear that each sample is approximated to one of the L numbers. Thus, the signal is digitized with quantized samples taking on any one of the L values. This is an L-ary digital signal (see Sec. 1.3-2). Each sample can now be represented by one of L distinct pulses.

From a practical viewpoint, dealing with a large number of distinct pulses is difficult. We prefer to use the smallest possible number of distinct pulses, the very smallest number being 2. A digital signal using only two symbols or values is the binary signal. A binary digital signal (a signal that can take on only two values) is very desirable because of its simplicity, economy, and ease of engineering. We can convert an L-ary signal into a binary signal by using pulse coding. Figure 8.14b shows one such code for the case of L = 16. This code, formed by binary representation of the 16 decimal digits from 0 to 15, is known as the natural binary code (NBC). For L quantization levels, we need a minimum of b binary code digits, where 2*^b* = L or b = log2 L.

Each of the 16 levels is assigned one binary code word of four digits. Thus, each sample in this example is encoded by four binary digits. To transmit or digitally process the binary data, we need to assign a distinct electrical pulse to each of the two binary states. One possible way is to assign a negative pulse to a binary 0 and a positive pulse to a binary 1 so that each sample is now represented by a group of four binary pulses (pulse code), as depicted in Fig. 8.14b. The resulting binary signal is a digital signal obtained from the analog signal x(t) through A/D conversion. In communications jargon, such a signal is known as a pulse-code-modulated (PCM) signal.

^† The error in pulse detection can be made negligible.

(a)

Digit	Binary equivalent	Pulse code waveform
0	0000
1	0001
2	0010
3	0011
4	0100
5	0101
6	0110
7	0111
8	1000
9	1001
10	1010
11	1011
12	1100
13	1101
14	1110
15	1111

(b)

Figure 8.14 Analog-to-digital (A/D) conversion of a signal: (a) quantizing and (b) pulse coding.

The convenient contraction of “binary digit” to bit has become an industry standard abbreviation.

The audio signal bandwidth is about 15 kHz, but subjective tests show that signal articulation (intelligibility) is not affected if all the components above 3400 Hz are suppressed [3]. Since the objective in telephone communication is intelligibility rather than high fidelity, the components above 3400 Hz are eliminated by a lowpass filter.† The resulting signal is then sampled at a rate of 8000 samples/s (8 kHz). This rate is intentionally kept higher than the Nyquist sampling rate of 6.8 kHz to avoid unrealizable filters required for signal reconstruction. Each sample is finally quantized into 256 levels (L = 256), which requires a group of eight binary pulses to encode each sample (28 = 256). Thus, a digitized telephone signal consists of data amounting to 8 × 8000 = 64,000 or 64 kbit/s, requiring 64,000 binary pulses per second for its transmission.

The compact disc (CD), a high-fidelity application of A/D conversion, requires the audio signal bandwidth of 20 kHz. Although the Nyquist sampling rate is only 40 kHz, an actual sampling rate of 44.1 kHz is used for the reason mentioned earlier. The signal is quantized into a rather large number of levels (L = 65,536) to reduce quantizing error. The binary-coded samples are now recorded on the CD.

A HISTORICAL NOTE

The binary system of representing any number by using 1s and 0s was invented in India by Pingala (ca. 200 BCE). It was again worked out independently in the West by Gottfried Wilhelm Leibniz (1646–1716). He felt a spiritual significance in this discovery, reasoning that 1 representing unity was clearly a symbol for God, while 0 represented the nothingness. He reasoned that if all numbers can be represented merely by the use of 1 and 0, this surely proves that God created the universe out of nothing!

EXAMPLE 8.5 ADC Bit Number and Bit Rate

A signal x(t) bandlimited to 3 kHz is sampled at a rate 33^{1 3}% higher than the Nyquist rate. The maximum acceptable error in the sample amplitude (the maximum error due to quantization) is 0.5% of the peak amplitude V. The quantized samples are binary-coded. Find the required sampling rate, the number of bits required to encode each sample, and the bit rate of the resulting PCM signal.

The Nyquist sampling rate is fNyq = 2×3000 = 6000 Hz (samples/s). The actual sampling rate is fA ^{= 6000}×(1^{1 3} ) = 8000 Hz.

The quantization step is , and the maximum quantization error is ±/2, where = 2V/L. The maximum error due to quantization, /2, should be no greater than 0.5% of the

^†Components below 300 Hz may also be suppressed without affecting the articulation.

802 CHAPTER 8 SAMPLING: THE BRIDGE FROM CONTINUOUS TO DISCRETE

signal peak amplitude V. Therefore,

$$\frac{\Delta}{2} = \frac{V}{L} = \frac{0.5}{100}V \implies L = 200$$

For binary coding, L must be a power of 2. Hence, the next higher value of L that is a power of 2 is L = 256. Because log2 256 = 8, we need 8 bits to encode each sample. Therefore the bit rate of the PCM signal is

8000×8 = 64,000 bits/s

DR ILL 8.5 Bit Number and Bit Rate for ASCII

The American Standard Code for Information Interchange (ASCII) has 128 characters, which are binary-coded. A certain computer generates 100,000 characters per second. Show that

• (a) 7 bits (binary digits) are required to encode each character
• (b) 700,000 bits/s are required to transmit the computer output.