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Clearly, the output of the sensor is an analog signal; however, the display can show only a nite number of readouts (101, to be precise) Because the display itself can only take a value out of a discrete set of states the integers from 0 to 100 we call it a digital display, indicating that the variable displayed is expressed in digital form Now, each temperature on the display corresponds to a range of voltages: each digit on the display represents one hundredth of the 5-volt range of the sensor, or 005 V = 50 mV Thus, the display will read 0 if the sensor voltage is between 0 and 49 mV, 1 if it is between 50 and 99 mV, and so on Figure 132 depicts the staircase function relationship between the analog voltage and the digital readout This quantization of the sensor output voltage is in effect an approximation If one wished to know the temperature with greater precision, a greater number of display digits could be employed
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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Display readout
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Sensor output voltage (mV)
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Figure 132 Digital representation of an analog signal
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The most common digital signals are binary signals A binary signal is a signal that can take only one of two discrete values and is therefore characterized by transitions between two states Figure 133 displays a typical binary signal In binary arithmetic (which we discuss in the next section), the two discrete values f1 and f0 are represented by the numbers 1 and 0 In binary voltage waveforms, these values are represented by two voltage levels For example, in the TTL convention (see 10), these values are (nominally) 5 V and 0 V, respectively; in CMOS circuits, these values can vary substantially Other conventions are also used, including reversing the assignment for example, by letting a 0-V level represent a logic 1 and a 5-V level represent a logic 0 Note that in a binary waveform, knowledge of the transition between one state and another (eg, from f0 to f1 at
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Temperature ( F)
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f (t) f1
t1 t 2 t3
t6 t
Figure 133 A binary signal
13
Digital Logic Circuits
t = t2 ) is equivalent to knowledge of the state Thus, digital logic circuits can operate by detecting transitions between voltage levels The transitions are often called edges and can be positive (f0 to f1 ) or negative (f1 to f0 ) Virtually all of the signals handled by a computer are binary From here on, whenever we speak of digital signals, you may assume that the text is referring to signals of the binary type, unless otherwise indicated
THE BINARY NUMBER SYSTEM
Table 131 Conversion from decimal to binary Decimal number, n10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Binary number, n2 0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111 10000
The binary number system is a natural choice for representing the behavior of circuits that operate in one of two states (on or off, 1 or 0, or the like) The diode and transistor gates and switches studied in 10 fall in this category Table 131 shows the correspondence between decimal and binary number systems for decimal numbers up to 16 Binary numbers are based on powers of 2, whereas the decimal system is based on powers of 10 For example, the number 372 in the decimal system can be expressed as 372 = (3 102 ) + (7 101 ) + (2 100 ) while the binary number 10110 corresponds to the following combination of powers of 2: 10110 = (1 24 ) + (0 23 ) + (1 22 ) + (1 21 ) + (0 20 ) It is relatively simple to see the correspondence between the two number systems if we add the terms on the right-hand side of the previous expression Let n2 represent the number n base 2 (ie, in the binary system) and n10 the same number base 10 Then, our notation will be as follows: 101102 = 16 + 0 + 4 + 2 + 0 = 2210 Note that a fractional number can also be similarly represented For example, the number 325 in the decimal system may be represented as 32510 = 3 100 + 2 10 1 + 5 10 2 while in the binary system the number 10011 corresponds to 100112 = 1 21 + 0 20 + 0 2 1 + 1 2 2 + 1 2 3 =2+0+0+
= 237510
Table 132 Rules for addition 0+0=0 0+1=1 1+0=1 1 + 1 = 0 (with a carry of 1)
Table 131 shows that it takes four binary digits, also called bits, to represent the decimal numbers up to 15 Usually, the rightmost bit is called the least signi cant bit, or LSB, and the leftmost bit is called the most signi cant bit, or MSB Since binary numbers clearly require a larger number of digits than decimal numbers, the digits are usually grouped in sets of four, eight, or sixteen Four bits are usually termed a nibble, eight bits are called a byte, and sixteen bits (or two bytes) form a word Addition and Subtraction The operations of addition and subtraction are based on the simple rules shown in Table 132 Note that, just as is done in the decimal system, a carry is generated
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