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algebsimp - CS2204 DIGITAL LOGIC STATE MACHINE DESIGN FALL...

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CS2204 DIGITAL LOGIC & STATE MACHINE DESIGN FALL 2009 DIGITAL CIRCUIT DESIGN & ALGEBRAIC SIMPLIFICATION EXAMPLES Polytechnic Institute of NYU Page 1 of 8 Handout No : 5 September 22, 2009 Example 1 1-bit 2-to-1 Multiplexer A 1-bit 2-to-1 Multiplexer (MUX) is a selector which selects one of the two inputs based on a select signal. As seen below, it has three inputs and one output. Two inputs (b and c) are data inputs .. The third input (a) is the control input , the select input. The single output is always equal to either b or c at any time. The input/output relationship (the operation, purpose) is given below in two different ways : A textual description and a truth table . a b y(a, b, c) If a = 0 then y = b 1-bit c a = 1 then y = c 2-to- 1 MUX a b c y(a, b, c) 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 1 2 3 1 0 0 0 1 1 1 1 4 5 6 7 1 0 1 1 1 1 0 0 The MUX is a 1-bit MUX since when an input is selected, there is only one data line selected. The minterm list, canonical SOP and minimal SOP extressions are shown below. y(a,b, c) m(2, 3, 5, 7) = a b c + a b c + a b c + a b c 0 1 0 0 1 1 1 0 1 1 1 1 2 3 5 7 y(a, b, c) = a b ( c + c) + a c ( b + b) k(m+p) = km + kp = a b 1 + a c 1 k + k = 1 = a b + a c k1 = k As the gate network shows, the MUX has three gate delays : a a b c y a a One can develop k-bit 2-to-1 MUXes from 1-bit 2-to-1MUXes. For example, 2-bit 2-to-1 MUXes, 4-bit 2-to-1 MUXes, etc. are implemented by using 1-bit 2-to-1 MUXes. Example 2 shows how to implement a 4-bit 2-to-1 MUX by using four 1-bit 2-to-1 MUXes. Note also that there are k-bit 4-to-1 MUXes, k-bit 8-to-1 MUXes, etc.
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Polytechnic Institute of NYU Page 2 of 8 CS2204 Handout No : 5 September 22, 2009 Example 2 4-bit 2-to-1 MUX A 4-bit 2-to-1 MUX selects between two 4-bit inputs. As a black box, it has 9 inputs and 4 outputs as shown below. The single control input is Select and the 4-bit inputs, K and M, are data inputs. The 4-bit output is Y which is K if Select is 0 and M if Select is 1 : 4-bit K M 4 4 If Select = 0 then Y = K Since there are 9 inputs, we need to partition it into simpler pieces ! (K3,K2,K1,K0) (M3,M2,M1,M0) 2-to-1 MUX Select 4 Y (Y3,Y2,Y1,Y0) else Y = M We have to obtain the operation table of the 4-bit 2-to-1 MUX : Select Operation 0 Y = K 1 Y = M The major operations are not clear on this operation table. We need to get a different , more detailed operation table : Select Operation 0 Y3 = K3 ; Y2 = K2 ; Y1 = K1 ; Y0 = K0 1 Y3 = M3 ; Y2 = M2 ; Y1 = M1 ; Y0 = M0 There are four identical major operations : 1-bit 2-to-1 MUXing !
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This note was uploaded on 02/02/2011 for the course CS 2204 taught by Professor Hadimioglu during the Spring '10 term at NYU Poly.

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algebsimp - CS2204 DIGITAL LOGIC STATE MACHINE DESIGN FALL...

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