DSP for Engineering Aplications DSP for Engineering Aplications ECI Semester /2010 Department of Engineering and Design London South Bank University

Lecturer (Theoretical Part) Dr. Z. Zhao Room: Room: T409 Tel: Tel:

Textbook Alan V. Oppenheim, Ronald W. Schafer, Discrete-time Signal Processing, 2ed, Prentice Hall, ISBN: Monson H. Hayes, Digital Signal Processing, McGraw-Hill, ISBN

Unit Structure ( Theoretical Part ) Introduction to DSP Discrete-time signals and Systems the Fourier transforms of discrete-time signals (DTFT) The z-transform The discrete Fourier transform (DFT) and its efficient computation (FFT)

Teaching and Learning Methods Lecture: 2 hour each week Tutorial: 1 hour in even weeks Laboratory work (Matlab exercises):2 hour in odd weeks Self learning: 102 hours

Assessment 3-hour written examination: 70% Phase test (Week 7) 10% Workshop assignment: 20% 1. log book 2. formal written reports

Introduction to DSP 1.1 What is DSP? DSP, or Digital Signal Processing, is concerned with the use of programmable digital software and/or hardware (digital systems) to perform mathematical operations on a sequence of discrete numbers (a digital signal).

Introduction to DSP 1.2 A General (Engineering) DSP System Anti-aliasing filter A/D DSP D/A Reconstruction filter Analog signal Analog signal Analog signal Analog signal Digital signal Digital signal

An Example

Introduction to DSP 1.3 Advantages: Programmable Well-defined, stable, and repeatable Manipulating data in the digital domain provides high immunity from noise Use of computer algorithms allows implementation of functions and features that are impossible with analog methods

Introduction to DSP 1.4 Disadvantages: Relatively low bandwidths Signal resolution is limited by the D/A and A/D converters.

Introduction to DSP 1.5 Applications: digital sound recording such as CD and DAT speech and compression for telecommunications and storage implementation of wire-line and radio modems image enhancement and compression speech synthesis and speech recognition Stock Market information processing

What is DSP Used For? …And much more!

Speech Coding – Vo-coder Pulse Train Random Noise Vocal Tract Model V/U Synthesized Speech Decoder Original Speech Analysis: Voiced/Unvoiced decision Pitch Period (voiced only) Signal power (Gain) Signal Power Pitch Period Encoder LPC-10:

JPEG Example Original JPEG (100:1)JPEG (4:1)

Discrete time Signals and Systems Discrete-time signal and its classification What is discrete-time signal? Special sequences used in DSP Signal properties and and basic operations Discrete-time systems and properties Properties of discrete-time systems Convolution sum and methods for performing convolution LCCDE Linear Constant Coefficient Difference Equation.

Discrete time Signals A discrete-time signal is an indexed sequence of real or complex numbers. It is a functions of an integer-valued variable, n, that is, often, denoted by x(n). Complex Sequences z(n) = a(n)+jb(n) = Re{z(n)}+jIm{z(n)} = |z(n)|exp[jarg{z(n)}] Where |z(n)| is the magnitude and arg{z(n)} is the phase angle The conjugate of z(n) is z*(n) = a(n)-jb(n) = Re{z(n)}-jIm{z(n)} = |z(n)|exp[-jarg{z(n)}]

Some fundamental sequences Unit sample Unit step The exponential sequences

Signal Duration Finite length sequence Left-sided sequence Right-sided sequence Two side sequence

Periodic and Aperiodic Sequences A signal x(n) is said to be periodic if, for some positive real integer N, x(n) = x(n+N) Fundamental period – N is smallest integer of the last equation. Examples:

Symmetric Sequences A real valued signal is said to be even if, for all n: x(n) = x(-n) Whereas a signal is said to be odd if, for all n: x(n) =- x(-n) Any signal can be decomposed as a combination of even and odd signal: x(n) = xe(n) + xo(n) xe(n) = ½ [(x(n) + x(-n) ] xo(n) = ½ [(x(n) - x(-n) ] Complex value sequence: It is said to be conjugate symmetric if, for all n x(n) = x*(-n) It is said to be conjugate asymmetric if, for all n x(n) = - x*(-n)

Signal Manipulations Shifting Reversal Scaling Addition Multiplication Time-scaling y(n) = x(mn) y(n)=x(n/N) Shifting, reversal and time-scaling operation are order dependent.

Signal Decomposition: The unit sample may be used to decompose an arbitrary signal x(n) into a sum of weighted and shifted unit sample as follows

Discrete-time Systems and properties A discrete-time system is a mathematical operator or mapping that transforms one signal ( the input) into another signal ( the output) by means of a fixed set of rules or operation.

System Properties Memory-less system Definition: A system is said to be memoryless if the output at any time n=n0 depends only on the input at time n=n0. Ex:y(n) = x2(n) Y(n) = x(n)+x(n-1) Additive systems: T[x1(n) + x2(n)] = T[x1(n)] + T[x2(n)] Homogeneity: T[cx(n)] =c T[x(n)] Linear system: T[a1x1(n) + a2x2(n)] =a1 T[x1(n)] + a2T[x2(n)] h(n) = T[δ(n)] h k (n) = T[δ(n-k)] Examples

System Properties (Cont’d) Shift Invariant System: For y(n)=T[x(n), the system is said to be shift invariant if, for any delay n 0, the response to x(n-n 0 ) is y(n-n 0 ). LSI ( Linear Shift Invariant) System: For LSI : h k (n) = h(n-k) For LSI system, any input x(n) will have output: = x(n)*h(n) Causality A system is said to be causal if, for any n0 the response of the system at time n0 depends only on the input to time n= n0. Stability A sytem is said to be stable in the bounded input-bounded output sense if, for any input that is bounded, the output will be bounded,

Convolution Sums

Difference Equations Difference equation provide a method for computing the response of a system, y(n), to an arbitrary input x(n). Approaches to solve LCCDE: Classical approach of finding homogeneous and particular solution. Using z-transform.