BT0064 – Logic Design, BSC IT 1st SEM, Assignment

1. Convert the following hexadecimal numbers to base 2.

a)    (D73)16

Ans: The number 2*(D73)*16 written under base sixteen is written 24FC4 in basis sixteen.

b)    (A21)16

Ans: The number 2*(A21)*16 written under base sixteen is written 1BDAC in basis sixteen.

2. Convert the following binary numbers to base 16:

a. (11001011)2

Ans: The binary number 11001011₂ is equivalent to the hexadecimal number CB₁₆ (0xCB or CBh).

b. (11100011)2

Ans: The binary number 11100011₂ is equivalent to the hexadecimal number E3₁₆ (0xE3 or E3h).

3. Explain ring counter with diagrams.

Ans: A ring counter is a type of counter composed of a type circular shift register. The output of the last shift register is fed to the input of the first register. It has a hamming distance of 1.

There are two types of ring counters:

·         A straight ring counter or Overbeck counter connects the output of the last shift register to the first shift register input and circulates a single one (or zero) bit around the ring. For example, in a 4-register one-hot counter, with initial register values of 1000, the repeating pattern is: 1000, 0100, 0010, 0001, 1000... . Note that one of the registers must be pre-loaded with a 1 (or 0) in order to operate properly.

·         A twisted ring counter, also called Johnson counter or Möbius counter (also Moebius), connects the complement of the output of the last shift register to the input of the first register and circulates a stream of ones followed by zeros around the ring. For example, in a 4-register counter, with initial register values of 0000, the repeating pattern is: 0000, 1000, 1100, 1110, 1111, 0111, 0011, 0001, 0000... .

By looping the output back to the input, we can convert a standard shift register into a ring counter. Consider the circuit below.

The synchronous Ring Counter example above is preset so that exactly one data bit in the register is set to logic "1" with all the other bits reset to "0". To achieve this, a "CLEAR" signal is firstly applied to all the flip-flops together in order to "RESET" their outputs to a logic "0" level and then a "PRESET" pulse is applied to the input of the first flip-flop ( FFA ) before the clock pulses are applied. This then places a single logic "1" value into the circuit of the ring counter.

So on each successive clock pulse, the counter circulates the same data bit between the four flip-flops over and over again around the "ring" every fourth clock cycle. But in order to cycle the data correctly around the counter we must first "load" the counter with a suitable data pattern as all logic "0's" or all logic "1's" outputted at each clock cycle would make the ring counter invalid.

This type of data movement is called "rotation", and like the previous shift register, the effect of the movement of the data bit from left to right through a ring counter can be presented graphically as follows along with its timing diagram:

4. Simplify the following three-variable Boolean functions algebraically:

a.    f1 = Σ1, 2, 5, 6

b.    f2 = Σ0, 1, 2, 3, 7

Ans: f1 = A'B'C + A'BC' + AB'C + ABC'
B'C (A' + A) + BC' (A' + A) <-- T10 (Theorem 10 ) A' + A = 1
B'C + BC' <-- Exclusive OR (sort of sits outside theorems - something you must know + XNOR)
f1 = B ⊕ C

f1 = A'B'+ A'B + AB' + AB
A' (B'+ B) + A (B'+ B) <-- T10
A' + A <-- T10
f2 = 1

 5. Minimize the following functions using Quine-McCluskey tabular method:

a.	f ( A, B, C, D)  0,1,3,6,9,10,11,12,14,15
b.	f ( A, B, C, D, E)  0,1,5,8,11,12,14,16,20,21,25,27,28,30,31
(with don’t care terms 2,7,13,22,23)

Ans:

A.       F =  A'B'C' + B'D + BCD' + AC + ABD'

B.       F =  ACD + B'CE + A'B'D'E + A'C'D'E' + AB'D'E' + BC'DE + BCE' + ABC'E

6. Explain operation and application of digital to analog convertor.

Ans: In electronics, a digital-to-analog converter (DAC or D-to-A) is a device that converts a digital (usually binary) code to an analog signal (current, voltage, or electric charge). An analog-to-digital converter (ADC) performs the reverse operation. Signals are easily stored and transmitted in digital form, but a DAC is needed for the signal to be recognized by human senses or other non-digital systems.A common use of digital-to-analog converters is generation of audio signals from digital information in music players. Digital video signals are converted to analog in televisions and cell phones to display colors and shades. Digital-to-analog conversion can degrade a signal, so conversion details are normally chosen so that the errors are negligible. Due to cost and the need for matched components, DACs are almost exclusively manufactured on integrated circuits (ICs). There are many DAC architectures which have different advantages and disadvantages. The suitability of a particular DAC for an application is determined by a variety of measurements including speed and resolution.

Digital-to-analog conversion is a process in which signals having a few (usually two) defined levels or states (digital) are converted into signals having a theoretically infinite number of states (analog). A common example is the processing, by a modem, of computer data into audio-frequency (AF) tones that can be transmitted over a twisted pair telephone line. The circuit that performs this function is a digital-to-analog converter (DAC).Basically; digital-to-analog conversion is the opposite of analog-to-digital conversion. In most cases, if an analog-to-digital converter (ADC) is placed in a communications circuit after a DAC, the digital signal output is identical to the digital signal input. Also, in most instances when a DAC is placed after an ADC, the analog signal output is identical to the analog signal input. Binary digital impulses, all by themselves, appear as long strings of ones and zeros, and have no apparent meaning to a human observer. But when a DAC is used to decode the binary digital signals, meaningful output appears. This might be a voice, a picture, a musical tune, or mechanical motion.

Both the DAC and the ADC are of significance in some applications of digital signal processing. The intelligibility or fidelity of an analog signal can often be improved by converting the analog input to digital form using an ADC, then clarifying the digital signal, and finally converting the "cleaned-up" digital impulses back to analog form using an DAC.