Document Sample

Digital Signal Processing Spring 2012 Lecture 2: Discrete-Time Signals & Systems Instructor: Engr. Asif Iqbal Department of Electrical Engineering NUCES-FAST, Peshawar asif.iqbal@nu.edu.pk Part 1: Discrete-Time Signals • Sequence of numbers , ∞ ∞ 10-Feb-12 where n is an integer • Periodic sampling of an analog Digital Signal Processing, Lecture 2, Spring 2012 signal , ∞ ∞ where T is the sampling period 2 Sequence Operations • The product and sum of two sequences x[n] and y[n]: sample- by-sample product and sum, respectively. 10-Feb-12 • Multiplication of a sequence x[n] by a number : Digital Signal Processing, Lecture 2, Spring 2012 multiplication of each sample value by . • Delay or shift of a sequence x[n] , where is an integer 3 Basic Sequences • Unit Sample Sequence (discrete-time Impulse or Impulse) 0, 0, Unit Sample 1, 0. 10-Feb-12 ... ... Digital Signal Processing, Lecture 2, Spring 2012 • Any sequence can be represented as a sum of scaled, delayed impulses 3 1 3, • More generally, ∑!" 4 Unit Step Sequence • Defined as Unit Step 1, $ 0, # 0, 0. 10-Feb-12 ... ... Digital Signal Processing, Lecture 2, Spring 2012 • Related to the Impluse by # ∑% !" , # 1 2 ∙∙∙ or # ∑!" • Conversely # # 1 5 Exponential Sequences • These are very important in representing and analyzing LTI discrete-time systems. Real 10-Feb-12 • Defined as Exponential ( % ... ... Digital Signal Processing, Lecture 2, Spring 2012 • If A and are real numbers, the sequence is real. • If 0 1 and A is positive, the sequence values are positive and decrease with increasing n. • If 0 ) ) 1 , the sequence values alternate in sign, but again decrease in magnitude with increasing n. • If ) 1 , the sequence values increase with increasing n. 6 Combining Basic Sequences • An exponential sequence that is zero for 0 ( % , ≥ 0 = 0, < 0 10-Feb-12 Digital Signal Processing, Lecture 2, Spring 2012 • A much simpler expression = ( % #[ ] 7 Sinusoidal Sequences • A general sinusoidal function ( cos - . for all 10-Feb-12 With ( and . are real constants • The ( % with complex has real and imaginary parts that are Digital Signal Processing, Lecture 2, Spring 2012 exponentially weighted sinusoids. If / 012 and ( ( / 03 , then % =( = ( / 03 / 012 % , = ( /0 12 %43 , = ( cos - + . + j sin - +. 8 Complex Exponential Sequence • When 1, = ( /0 12 %43 10-Feb-12 = ( cos - + . + j sin - +. • By analogy with the continuous-time case, the quantity - is Digital Signal Processing, Lecture 2, Spring 2012 called the frequency of the complex sinusoid or complex exponential and . is called the phase. • is always an integer (differences between discrete-time and continuous-time) 9 An important difference: Frequency Range • Consider a frequency - 28 (/ 0 12 49: % (/ 012 % / 09:% (/ 012 % , 10-Feb-12 • More generally, - + 28; , ; being an integer = (/ 0 12 49:< % = (/ 012 % / 09:<% = (/ 012 % , Digital Signal Processing, Lecture 2, Spring 2012 • Same for sinusoidal sequences = ( cos - + 28; ( + . = ( cos - + .) • So, only consider frequencies in an interval of 28, such as −8 < - ≤ 8 or 0 ≤ - < 28 10 Another important difference: Periodicity • In the continuous-time case, a sinusoidal signal and a complex exponential signal are both periodic. 10-Feb-12 • In the discrete-time case, a periodic sequence is defined as >, for all Digital Signal Processing, Lecture 2, Spring 2012 where the period N is necessarily an integer. • For sinusoid, ( cos - ( + . = ( cos - + - > + .) 9:! which requires that - > = 28 ?; > = 12 where is an integer 11 Another important difference: Periodicity • Same for complex exponential sequence / 012 %4@ / 012 % 10-Feb-12 which is true for - 28 • So, complex exponential and sinusoidal sequences Digital Signal Processing, Lecture 2, Spring 2012 • are not necessarily periodic in with period 28/- • and, depending on the value of - , may not be periodic at all. • Consider :% • cos , with a period of N=8 B :% • 9 = cos , with a period of N=16 C • Increase in frequency => Increase in period! 12 Another important difference: Frequency • For a continuous-time sinusoidal signal D ( cos Ω D . , 10-Feb-12 as Ω increases, D oscillates more and more rapidly Digital Signal Processing, Lecture 2, Spring 2012 • For the discrete-time sinusoidal signal ( cos - . • As - increases from 0 towards 8, oscillates more and more rapidly • As - increases from 8 towards 28, the oscillations become slower 13 Part 2: Discrete-Time Systems • A transformation or operator that maps input into output F G 10-Feb-12 Digital Signal Processing, Lecture 2, Spring 2012 • Examples: • The ideal delay system H , ∞ ∞ • A memory less system 9, ∞ ∞ 14 Linear Systems Digital Signal Processing, Lecture 2, 10-Feb-12 15 Spring 2012 • a Time-invariant Systems Digital Signal Processing, Lecture 2, 10-Feb-12 16 Spring 2012 Causality • The output sequence value at the index depends only on the input sequence values for ≤ 10-Feb-12 • Example Digital Signal Processing, Lecture 2, Spring 2012 H , −∞ < <∞ • Causal for H >0 • Non-causal for H <0 17 Stability • A system is stable in the BIBO sense if and only if every bounded input sequence produces a bounded output 10-Feb-12 sequence. Digital Signal Processing, Lecture 2, Spring 2012 • Example 9, −∞ < <∞ 18

DOCUMENT INFO

Description:
DSP Lecture

OTHER DOCS BY NumanSaeed

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.