Digital Logic & Design Instructor: Aneel Ahmed Lecture #1

Text Books Digital Logic and Computer Design – M. Morris Mano Lecture Slides. Every thing discussed in class is part of the course material.

Lecture Analog values & Digital Values. Analog & Digital Signals. Representing continuous signals in the form of discrete values. Representing Digital Values. Merits of a Digital System. Number Systems.

Analog and Digital Both data and the signals that represent them can be either analog or digital in form. Analog and Digital Data Data can be analog or digital. The term Analog data refers to information that is continuous. Digital data refers to information that has discrete states. For example, an analog clock that has hour, minute, and second hands gives information in a continuous form; the movements of the hands are continuous. On the other hand, a digital clock that reports the hours and the minutes will change suddenly from 8:05 to 8:06.

Analog and Digital Data Analog data, such as the sounds made by a human voice, take on continuous values. When someone speaks, an analog wave is created in the air. This can be captured by a microphone and converted to an analog signal or sampled and converted to a digital signal.

Analog and Digital Data Digital data take on discrete values. For example, data are stored in computer memory in the form of 0s and 1s. They can be converted to a digital signal or modulated into an analog signal for transmission across a medium.

Analog and Digital Signals Like the data they represent, signals can be either analog or digital. An analog signal has infinitely many levels of intensity over a period of time. As the wave moves from value A to value B, it passes through and includes an infinite number of values along its path. A digital signal, on the other hand, can have only a limited number of defined values. Although each value can be any number, it is often as simple as 1 and O.

Analog and Digital Signals The simplest way to show signals is by plotting them on a pair of perpendicular axes. The vertical axis represents the value or strength of a signal. The horizontal axis represents time. The Figure illustrates an analog signal and a digital signal. The curve representing the analog signal passes through an infinite number of points. The vertical lines of the digital signal, however, demonstrate the sudden jump that the signal makes from value to value.

Analog and Digital Signals

Representing continuous signal in the form of discrete values This is a continuous signal.

Representing continuous signal in the form of discrete values A continuous signal can be represented digitally by taking samples at regular and fixed intervals.

Representing continuous signal in the form of discrete values Digital Representation.

Representing continuous signal in the form of discrete values In the diagram we took 10 samples at time intervals. The digital representation of the continuous signal only approximates the original signal. and cannot truly represent the original signal as can be seen by plotting the values. The reconstructed continuous signal does not give the exact replica of the original signal. The reconstructed signal has sharp edges and corners in contrast to the original signal which has smooth curves.

Representing continuous signal in the form of discrete values If the number of samples collected are reduced by half, the resulting reconstructed signal is very different from reconstructed signal.

Representing continuous signal in the form of discrete values If the number of samples collected are reduced by half, the resulting reconstructed signal is very different from reconstructed signal. The peak in the continuous signal at 38 and the depth at -22 are all together missing from the reconstructed signal. This is due to the small number of samples taken. A better approximation of the original signals can be obtained by increasing the number of samples. An infinite number of samples very accurately represents the original continuous signal.

Representing digital values We saw a continuous signal and its digital representation. These digital values have to be processed electronically by a digital system. Generally there are two type of electronic systems : analog systems and digital systems. Analog systems : process continuous signals. So a continuous quantity has to be converted into electrical voltage terms. For example, a continuous signal of 42 deg C would be represented by perhaps 42 mV, a continuous temperature signal of deg C will be represented by mV. Digital systems: as mentioned before, use digital or discrete values. So are we going to be representing these discrete values in terms of voltages? Let us see.

Representing digital values Consider a calculator which is an example of digital system. Let us assume that the calculator has been internally calibrated to represent the number 1 by 1 mV x 10 ^15 volts which is a very large voltage value and cannot be practically represented by any circuit.

Representing digital values We saw that it is not practical to represent discrete digital values in terms of voltages in the digital system. Basically digital systems are based in two voltage values, they work with two voltage values. +5 volts which represents the logic high state or logic 1 state. 0 volts which represent the logic low state or logic 0 state. Using these two voltage values or these two states, we can represent any quantity or value which has two states. For example numbers 0 and 1, the color black and white, the temperature hot and cold, an object might be moving or stationary, so just two values.

Representing digital values Now how can we represent multiple values or more than 2 values in a digital system? Digital systems are based on binary number systems. A single digit or a bit of binary number system can represent only 2 values, a zero and a one. To represent large values, we combine these bits. So a combination of 2 bits would allow us to use four different values or four quantities. Normally we have been doing this in decimal number system. A single digit in decimal number system can represent up to 10 values, from 0 to 9. Now how do u represent more than 10 values. Well u use a combination of 2 decimal digits.so 2 digits would allow u to use 100 values, from 0 to 99.

Representing digital values Similarly in a binary number system, we combine a number of binary bits to represent multiple values. The number 39 can be represented by a combination of six bits. So in terms of binary, 39 is equal to As mentioned before, in a digital system, the binary numbers are represented in terms of voltages. So the number 39 will be represented in terms of voltages as 5V 0V 0V 5V 5V 5V.

Merits of a digital system Digital systems are extensively being used. They offer a number of advantages compared to the analog system. Efficient Processing & Data Storage. ( Computers for example are very efficient at processing information that is in digital binary form, infact computers work with digital information. Another example a CD can store a large number of digitized audio and video clips storing the same number of audio and video clips in an analog form would require a large number of audio or video cassettes.) Efficient & Reliable Transmission. Detection and Correction of Errors. (and less prone to errors. Even if error occurs detection and correction of errors in digital data is easier. We will be looking at the simple example of detecting error using the parity bit method).

Merits of a digital system Precise & Accurate Reproduction.( For example, the picture quality and sound quality of digitized video or audio stored on CDs can be reproduced with a far superior quality as compared to the analog audio and video) Easy Design and Implementation. Occupy minimum space. ( Digital circuits in the form of IC occupy a very small space. For Example, the PC has a motherboard which has an area less than one square foot. This mother board has all the important circuitry of the computer. Digital memory on the hand is implemented as an integrated circuit. It is small enough to fit in the palm of your hand but it can store an entire collection of books. )

Number Systems Decimal Number System Binary Number System Octal Number System Hexadecimal Number System

Number Systems

Decimal Number System

Example Decimal Number System

Binary Number System

Example Binary Number System

Representing Numbers in Different Number Systems

BIT

Octal Number System

Hexadecimal Number System

Converting a Number of another Base to Decimal Number

Converting a Decimal Number to a Number of Another Base

Converting a Number of some Base to a Number of Another Base