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