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Introduction to Digital:Combinational Logic and Systems DesignNumbering SystemsBinary Code.Binary to Decimal Conversion.Decimal to Binary Conversion.Binary Coded Decimal. BCD CodeNumbers with other bases.Fundamental Digital Devices: The inverter.Introduction to Digital: Combinational Logic and Systems Design So far we have been discussing the generation, transmission and processing of signals whose amplitude (voltage, current) varies continuously in time and can in principle take any value. At a certain instant of time we may represent a signal by displaying its amplitude in an analog form or in a digital format. The graphics below demonstrate the familiar representation of the two forms. Both displays are asked to display the number 4.7. In this case the digital display on the left has the required resolution to represent the number exactly. If we try to display the number 4.76, this digital display, with its ability to display only 2 digits will have to round off the number, representing it either as 4.7 or 4.8, depending on how the system processes the numerics. Our reading of the number off the analog display requires some interpolation of the value but in principle the resolution is only limited by our ability to identify the position of the measuring needle. In general the characteristics of the digital display correspond to the digital signal which is required to generate the digital display in the first place. If we now consider an analog signal which varies continuously in time, see Figure 2a, then if we sample the signal at discrete times (τ, 2τ, 3τ, κλπ) we will obtain the values indicated by the solid circles on Figure 2b. Furthermore, if we consider the quantization of the signal at these discrete sampling times we obtain the signal indicated on Figure 2c which is a digital signal. The analog signal is also shown on Figure 2c to emphasize the relationship between it and its digital representation. 6.071/22.071 Spring 2006, Chaniotakis and Cory 12468Signal (V) Timeτ2τ 3τ 4τ 5τ 6τ 7τ0 (a) 2468Signal (V)Timeτ2τ 3τ 4τ 5τ 6τ 7τ0 (b) Signal (V)Timeτ2τ 3τ 4τ 5τ 6τ 7τ0Level 1Level 2Level 3Level 4Level 5Level 6Level 7 (c) Figure 2. Schematic representation of analog and digital signals. This review should have motivated us to ask a few questions about these signals and in particular about the digital signal shown on Figure 2c. Some of these questions are: 1. How is the information embodied by the digital signal represented? 2. How is the signal generated? a. How is the sampling frequency selected and how is it related to the “quality” of signal representation? b. How is the amplitude quantization achieved? 3. What are the advantages and disadvantages of generating the digital signal? For example, how does it perform in a. Accuracy b. Transmission c. Noise immunity d. Information storage e. Computation In the next few classes we will answer these questions and explore the fundamental issues associated with the design of digital circuits. 6.071/22.071 Spring 2006, Chaniotakis and Cory 2Numbering Systems Binary Code. In digital electronics the signals are formed with only two voltage values, HI and LOW, or level 1 and level 0 and it is called binary digital signal.1 Therefore, the information contained in the digital signal is represented by the numbers 1 and 0. In most digital systems the state 1 corresponds to a voltage range from 2V to 5V while the state 0 corresponds to a voltage range from a fraction of a volt to 1 volts. Digital operations are performed by creating and operating on binary numbers. Binary numbers are comprised of the digits 0 and 1 and are based on powers of 2. Each digit of a binary number, 0 or 1, is called a bit, an abbreviation for binary digit. Four bits together is a nibble, 8 bits is called a byte. (8, 16, 32, 64 bit arrangements are also called words) The rightmost bit is called the Least Significant Bit (LSB) while the leftmost bit is called the Most Significant Bit (MSB). The schematic below illustrates the general structure of a binary number and the associated labels. N1010 1101 0110 1010nibblebyteword  MSB LSB 1 In addition to binary digital systems and its associated binary logic, multivalued logic also exists but we will not consider it in our discussion. 6.071/22.071 Spring 2006, Chaniotakis and Cory 3Binary to Decimal Conversion. The conversion of a binary number to a decimal number may be accomplished by taking the successive powers of 2 and summing for the result. For example let’s consider the four bit binary number 0101. The conversion to a decimal number (base 10) is illustrated below. NNNN32101001010x2 1x2 0x2 1x20401=⇓⇓⇓⇓↓↓↓↓++++++5 For this four bit binary number the range of powers of 2 goes from 0, corresponding to the LSB, to 3, corresponding to the MSB. The number 5 is shown as to indicate that it is a decimal number (power of 10). 105 The signal represented on Figure 2c has a value of 5 V at time=6τ. The binary representation of that value is 0101 and it is shown on Figure 3 replacing Level 4. We will see more of this later when we consider the fundamentals of the device which converts the analog signal to a digital signal. Signal (V)Timeτ2τ 3τ 4τ 5τ 6τ 7τ0Level 1Level 2Level 30101Level 5Level 6Level 7 Figure 3. In the next few examples we will use the subscript 2 to indicate a binary number but the subscripts will be omitted after that. 6.071/22.071 Spring 2006, Chaniotakis and Cory 4Examples: Verify the Binary to Decimal conversion 21021021021021211111 = 151111 0000 = 2401111 1111 = 2551101 1011 = 2190001 0101 1011 = 3471001 0101 1011 = 239500 Decimal to Binary Conversion. The conversion of a decimal number to a binary number is accomplished by successively dividing the decimal number by 2 and recording the remainder as 0 or 1. Here is an example of the conversion of decimal number 125 to binary. 12562 126231 023115 121571 0111 11012731231121012⎫=+⎪⎪⎪=+⎪⎪⎪=+⎪⎪=+⎪⎪⇒⎬⎪=+⎪⎪⎪=+⎪⎪⎪=+⎪⎪⎪⎭ LSB MSB Practice number conversion by verifying the conversions from decimal to binary: Decimal Binary 69 0100 0101 299 0001 0010 1011 756 0010 1111 0100 6.071/22.071 Spring 2006, Chaniotakis and Cory 5Representation of fractions and signed numbers. A fractional number may be represented as a


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MIT 6 071J - Combinational Logic and Systems Design

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