EG7016 Design of Discrete Systems: Discrete-time Systems

EG7016 Design of Discrete Systems: Discrete-time Systems

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EG7016 Design of Discrete Systems

Question:

Discrete-time systems

Introduction – the importance of discrete-time systems and DSP Signals and systems – definitions

Continuous and discrete signals and systems

Sinusoidal signals and complex exponential signals – definition of amplitude, angular frequency, frequency and phase

Linear systems

Time invariant systems

Linear Time Invariant systems

Impulse response of a digital LTI systems

The output of a LTI system is the convolution of the input signal and the impulse response

Difference equations and how to apply them to filter a signal

The importance of discrete-time systems and DSP

With the ubiquity and the speed of digital computers and microcomputers, mainly since the late 1970s and early 1980s (with the popularisation of the Apple II microcomputer and then the IBM-PC), followed, in the late 1990s and early 2000s by the proliferation of fast modern ‘smart’ mobile phones with their ‘Apps’, the understanding of the importance of discrete-time systems and digital signal processing (DSP) is important not only to engineers, but, dare I say, to every person who makes use of these devices. As engineers and programmers, of course that is much more important for us, who design, perfect and program those devices. These devices use a discrete version of signals from the real world and, through the black magic of ‘digital signal processing’, produce useful results. This EG7016 module deals with these ‘discrete-time systems’ and the associated Digital Signal Processing that they perform. We shall start with some definitions, we shall then cover various ways to describe and analyse discrete-time systems and then move into 2 important chapters of DSP, namely digital filtering and spectral analysis. We shall cover both analysis (given a system, understand what it does and how) and synthesis (given requirements, design a system that satisfies those requirements).

Signal – a record of the pattern of variations of a physical quantity that carries information (generally along time). Signals can be stored, manipulated or transmitted.

System – a device or instrument that can store, manipulate or transmit signals.

The elementary operations for the manipulation of signals are: storage, delay, addition, subtraction, and multiplication by constants.

A signal can be continuous, x(t), or discrete, x(nT). A continuous signal is described by a dense function, i.e., it has values for all t, while a discrete signal only has values at the sampled points. These are normally equally spaced in time at nT, where n is the counter of samples and T is the sampling period.

EG7016 Design of Discrete Systems

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