Applied Mathematics V

Probability and topics in Statistics.

Statistical experiments with random outcomes, Sample space, probability defined on the basis of sample space and on the basis of events and their combinations.

Theorem on probabilities, conditional probability. Bayes theorem. Random variable, probability distribution for discrete and continuous random variables. Density function and distribution functions. Expected values, variance , moments, moment generating functions, Bernoullis trials, Binomial , Poisson, normal distributions for detailed study with proof, Other common distributions,

T , F, Beta, Gamma, X with indication of the applications(without proof)

Central limit theorem., Bivariate probability and frequency distributions, Correlations, regression, lines of regression.

Introduction to random samples, use of random numbers, stochastic processes, Time series , queuing theory.

Introduction to Discrete Structures.

Mathematical logic, prepositions, statement and negation, combinations of statements, their truth tables, logical equivalence.

Operations on sets, relations and their functions, partial order and equivalence relations, Peanos axioms and mathematical induction, Injective, Surjective, Bijective functions, Pigeonhole principle and its applications.

Formal mathematical systems, elements of theory of some algebras such as rings, integral domains, group, fields Boolean algebra, semigroup.

Digital Integrated Circuits.

Introduction:

Review of binary and hexadecimal number system, Boolean logic & minimisation techniques.

Basic digital circuits:

Basic operation of gates, noise margin, transfer characteristics, propagation delay & fan out, propagation delay of following: MOS inverter, MOS logic gates, CMOS inverter, CMOS logic gates, TTL gates, TTL logic LS, S families, ECL circuits, comparison 

Combinational digital circuits:

Functional logic diagram of following ICs & their working: Standard gate ICs , binary adder, subtractor & 4-bit ALU, digital comparator, Parity checker-generator decoder-demultiplexer, multiplexer, encoder.

Applications of combinational digital circuits:

Multiplexers, demultiplexers-decoders, 4-bit adder, digital comparator, parity generator-checker, BCD- binary code converter, binary- BCD code converter, priority encoder,BCD-7 segment decoder & driver.

Sequential circuits & systems:

Latch, clocked SR flip-flop, shift registers, asynchronous counters including design, synchronous counters including design.

Large scale digital integrated systems:

Dynamic MOS devices, read write memory cell of dynamic memory, static RAM cell, RAM organisation 

4 digit up/down counter 7217, LCD display systems.

Electromagnetic Fields and Waves.

Static electric fields in vacuum:

Coulombs law, electrostatic potential gradient, fields due to point, line surface & volume charges, electric dipoles & its field.

Dielectric media:

Polarisation, surface & volume charge densities due to polarisation, Displacement vector, boundary conditions

Gauss law & its applications

Laplace & Poissons equations

Electrostatic energy & force:

Energy of a charge distribution, energy density, pressure & forces.

Method of images:

Point charge & a conducting plane.

Steady electric current:

Current density, equation of continuity.

Static magnetic fields in vacuum:

Lorentz force equation, Biot-Savarts law & its applications, Amperes circuital law & its applications. Divergence and curl of B, magnetic scalar & vector potentials.

Electromagnetic induction:

Faradays laws in integral & point form, simple problems.

Summary of the results in the form of Maxwells equations, modification of equations for time varying fields.

Uniform plane waves in free space:

Derivation of wave equation & its solution in its simplest form, concept of space & time variations, E,H, & the direction of wave propagation, Intrinsic impedance, sinusoidally time varying uniform plane waves in free space, phase constant, wavelength, phase velocity, classification of waves for communication.

Principles of Control Systems.

Introduction

Open loop and closed loop systems, regulating systems, servo mechanisms, Transfer function & impulse response of system.

Study of components:

OP-Amp as differential amplifier or error detector, stepper motor, synchro control transformer, synchro control transmitter, potentiometer & two phase servo motor.

Modelling of systems:

RLC circuits, armature & field controlled dc motors, importance of a mathematical model.

Block diagrams:

Various techniques of block diagram reduction, Masons gain formula & its application to block diagram reduction.

Transient response of systems:

Importance of second order system models, time domain specifications of system & analysis of transient response using second order model.

Steady state error analysis:

Types of systems, steady state error analysis of different types of systems using step, ramp, parabolic input signals.

Stability analysis:

Introduction to the concept of stability, stability analysis using Routh -

Hurwitz criterion

Frequency domain analysis of systems

Frequency domain specifications of systems , Resonance peak & peak resonance frequency regarding complex poles and zeroes, Relationship between time & frequency domain specification of systems

Stability analysis of systems using Bode plots

Stability analysis of systems using root locus & the concepts of dominant, closed loop pole pair

Introduction to polar plots.

Principle of argument & its application to the study of stability of systems.

Constant gain & phase loci & their use in the stability study of systems.

Microprocessors I.

Introduction to microprocessors:

Features , Programmers model, external & internal organisation.

8085 Architecture:

8085 Architecture & organisation, Instruction cycles, machine cycles and T-states, address decoding techniques, minimum system design, Memory interfacing with timing considerations, clock, reset & buffering circuits

8085 Instruction set

Instruction format, addressing modes, classification of instruction set.

8085 Programming:

Assembly language programming:- basic structure, data transfer, arithmetical, logical, transfer of control & miscellaneous instruction types.

Stack & subroutines:

Stack operations, limitations, subroutine concepts, parameter passing

techniques, subroutine design, delay subroutine design & applications, Re-entrant & recursive subroutines, concept of counters and timers.

I/O data transfer techniques:

I/O interface concepts, speed considerations, program controlled I/O,

asynchronous & synchronous I/O techniques interrupt driven program controlled I/O, direct memory access data control techniques, handshake signals, concepts of serial communication, matrix keyboard & multiplexed display interface.

Interrupts:

Requirements, single level interrupt, multilevel interrupt & vector interrupt system, 8085 interrupt structure and its operation,8259A interrupt controller.

I/O controllers:

Features, organisation & operating modes of 8155 multifunction device,8255 Programmable peripheral interface, 8254 programmable timer,8237 programmable DMA controller.

Signals and Systems.

Introduction:

Basic signals in continuous and discrete time domain. Classification of signals: periodic / aperiodic, even/odd, deterministic/stochastic, energy/power

Singular functions: ramp, step & impulse, Representing a system as a mapping of I/O signals

Representing a system in terms of differential and difference equation respectively.

Classification of systems: causal/non causal, time varying/time invariant, stable/unstable invertible / non invertible & lumped & distributed parameters.

Fourier analysis of continuous time signals

Orthogonal functions, Fourier series representation in terms of sine , cosine, exponential.

Complex Fourier series, Properties of Fourier series Convergence of Fourier series

Gibbs phenomenon, Fourier transform & its properties. Fourier transform of singular functions

Energy density spectrum

Continuous time systems:

Linear differential equations, Representation by impulses, systems impulse response & convolution integral. Evaluation & Interpretation of Convolution integral. System stability in time and frequency domain. Transient & steady state response of linear systems. Frequency response of linear system.

Response of systems to causal periodic inputs.

Laplace transform:

Convergence, properties of Laplace transform, double sided transform, application of Laplace transform to solutions of differential equations, relationship between Fourier & Laplace transform.

Z-transform:

Definition, convergence, properties & inversion of Z-properties, single & double sided transform. Analysis of discrete time systems using Z-transform Relationship between Laplace & Z-transform.

Random signals: