Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 261Reading Assignment: Chapter 3 in Logic and Computer Design Fundamentals, 4th Edition by Mano Combinational Functions and CircuitsFunctions and fundamental circuits are introduced in this chapter which are useful in designing larger digital circuits.These circuits are presented as functional blocks – fundamental, reusable circuits. The focus is to develop the functional blocks such that they are reusable, are easily expanded for use with larger functions, and are efficiently designed for implementing via VHDL programming.Examples of functional blocks to be introduced:• Decoders• Encoders• Priority encoders• Multiplexers• De-multiplexers• Magnitude Comparators• Programmable Logic (Chapter 6)Lecture #7 EGR 270 – Fundamentals of Computer Engineering2Combinational Logic Using MSI and LSI devicesAlthough our focus will be more on defining functional blocks that are designed to be reusable and to be implemented using VHDL, many of these fundamental circuits are also available as commercially available IC’s.Commercial devices can perform complex functions using perhaps a single IC, thus saving space. They are typically faster that equivalent circuits that we might build using discrete logic gates. It might be a good idea to browse through a Logic Data Book to see what is available. A few devices are listed below.Assortment of commercially available combinational logic devicesLecture #7 EGR 270 – Fundamentals of Computer Engineering3DecodersAn N-bit decoder has 2N outputs, only one of which may be activated at a given time.If the device is active-HIGH , then only one output may be HIGH at any time.If the device is active-LOW , then only one output may be LOW at any time.Example: A 3-bit decoder might also be called a 3-line-to-8-line decoder or a 3x8 decoder. The block diagram is shown below:D0 D1 D2 D3 D4 D5 D6 D7 Input Code Only one output is activated (HIGH) 3 x 8 Decoder 2 2 2 1 2 0 x y z Discuss:•basic operation•the truth tableLecture #7 EGR 270 – Fundamentals of Computer Engineering4Active-LOW versus Active-HIGH decodersEnable lines – essentially act as ON/OFF switchesExample: Show the truth table and block diagram for an active-LOW 2x4 decoder with an enable line, E.Lecture #7 EGR 270 – Fundamentals of Computer Engineering5Circuit DesignShow that decoder outputs are essentially minterms and draw a circuit for:• 1x2 decoder (no enable, active-HIGH outputs) • 2x4 decoder (no enable, active-HIGH outputs)• 3x8 decoder (no enable, active-HIGH outputs)• Note the gate input count on the 2x4 and 3x8 decoder circuitsLecture #7 EGR 270 – Fundamentals of Computer Engineering6Decoder expansion using hierarchyThe text introduces a procedure for forming any (n x 2n) decoder by expanding smaller decoders. The result requires only 2-input AND gates (rather than n-input) and inverters. This technique is especially useful for building large decoders using reusable fundamental blocks. Figure 3-19 below illustrates a 3x8 decoder constructed using this method:Lecture #7 EGR 270 – Fundamentals of Computer Engineering7Implementing Boolean functions using decodersNote that the decoder outputs for active-HIGH decoders are simply minterms, soF = (minterms) = (active-HIGH decoder outputs) Example: Implement f(A,B,C) = (0, 3, 5, 6) using a 3 x 8 decoder with active-HIGH outputsNote that the decoder outputs for active-LOW decoders are simply maxterms, soF = (maxterms) =  (active-LOW decoder outputs) Example: Implement f(A,B,C) = (0, 3, 5, 6) using a 3 x 8 decoder with active-LOW outputsLecture #7 EGR 270 – Fundamentals of Computer Engineering8Decoder IC’s74155 Data Sheet (dual 2x4 decoder/single 3x8 decoder) – see next page•The 74156 is similar to the 74155 except that it has open-collector outputs instead of totem-pole outputs (discuss the advantage of this).•Show how to connect the 74155 as a 2x4 decoder and also as a 3x8 decoder.•Show how to use two 74155’s to form a 4x16 decoder.Lecture #7 EGR 270 – Fundamentals of Computer Engineering9Lecture #7 EGR 270 – Fundamentals of Computer Engineering10EncoderAn encoder is essentially the opposite of a decoder.An N-bit encoder has 2N inputs lines, one of which is active, and N output lines that carry the binary code corresponding to the active input.The 8 x 3 encoder shown below might also be called an octal-to-binary encoder.Example: Show an encoder with sample inputs and outputs.D0 D1 D2 D3 D4 D5 D6 D7 Output Code Only one input is activated 8 x 3 Encoder 2 2 2 1 2 0 x y z (active-HIGH inputs shown) Lecture #7 EGR 270 – Fundamentals of Computer Engineering11Example: Show an 8x3 encoder followed by a 3x8 decoder with some sample inputs and outputs.Note that encoders and decoders perform the opposite functions.Example: Show a 3x8 decoder followed by an 8x3 encoder with some sample inputs and outputs.Lecture #7 EGR 270 – Fundamentals of Computer Engineering12Basic Encoder DesignDraw the truth table for an 8x3 encoder. From the truth table, determine expressions for the outputs x, y, and z.Valid Output? One problem with the encoder design above is that there is no way to indicate that an invalid input occurred. This problem can be resolved by using an additional output called a valid line, V.D0 D1 D2 D3 D4 D5 D6 D7 Output Code V = 1 for a valid code (0 if invalid)8 x 3 Encoder 2 2 2 1 2 0 x y z VLecture #7 EGR 270 – Fundamentals of Computer Engineering13Decoders and Encoders - ApplicationsDecoders can be used to reduce the number of wires needed to control multiple outputs.Encoders can be used to reduce the number of wires needed to read multiple inputs.These wires might be to external devices or might be within a digital circuit.Encoder Example 1 – Reading 256 external sensors with a computer•Suppose that a computer was used to read the status of 256 sensors in a special application where only one sensor would ever be HIGH at a given time. •One option would be to find a 256-pin connector to work with the computer (good luck). •A better option would be to use an 256x8 encoder and use an 8-bit connector on the computer. The computer could then simply read the code to determine which