ELEG 305: Digital Signal Processing

ELEG 305: Digital Signal Processing Lecture 1: Course Overview; Discrete-Time Signals & Systems Kenneth E. Barner Department of Electrical and Computer Engineering University of Delaware Fall 2008 K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 1 / 28 Outline 1 Course Information 2 Lecture Objectives 3 Discrete-Time Signals and Systems Discrete-Time Signals Discrete-Time Systems Analysis of DT LTI Systems Implementation of Discrete-Time Systems K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 2 / 28

Course Information Course Information Course Info. All information is on My courses (Web CT); Syllabus; TA information; Homework assignments; Prior homework solutions; Grades Grading (2) exams (40%), final (30%), homework (25%), unannounced quizzes (5%) Homework Due Wednesdays; Assignments turned in at 310 Evans (by 5 p.m.); Engineering paper mandatory; no late assignments accepted Objective Successful students will be proficient at characterizing, analyzing, and manipulating discrete-time signals and systems K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 3 / 28 Lecture Objectives Lecture Objectives Objective Define and review basic discrete-time signal and system notation and properties; Analysis of such systems (impulse representations of signals, LTI systems, causality, stability); Difference equations Reading Chapter 1 (not covered in class) and Chapter 2 (2.1-2.5); Next lecture, Chapter 3 K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 4 / 28

Discrete-Time Signals Elementary Discrete-Time Signals Definition (unit sample sequence) { 1, for n = 0 δ(n) = 0, for n 0 Definition (unit step signal) { 1, for n 0 u(n) = 0, for n < 0 Other important signals: unit ramp signal, u r (n) =nu(n), and exponential signal, x(n) =a n K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 5 / 28 Discrete-Time Signals and Systems Discrete-Time Signals Classification of Signals Definitions Let x(n) R, n = 0, ±1, ±2,..., be a discrete signal signal energy E = x(n) 2 n= signal power P = 1 N lim x(n) 2 N 2N + 1 n= N 0 < E < Energy Signal 0 < P < Power Signal Other signal classifications: Periodic vs. aperiodic; symmetric (even), x( n) =x(n), and antisymmetric (odd), x( n) = x(n) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 6 / 28

Discrete Time Systems Discrete-Time Systems System input and output x(n) T y(n) x(n) T y(n) T : x(n) y(n) or x(n) T y(n) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 7 / 28 Discrete-Time Signals and Systems Discrete-Time Systems Block Diagram Representations x1(n) Adder + y(n) = x1(n) + x2(n) x2(n) Scaler x(n) a y(n) = a x(n) x1(n) Multiplier y(n) = x1(n) x2(n) x2(n) x(n) Delay z -k y(n) = x(n-k) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 8 / 28

Discrete-Time Systems Example (I) x(n) T y(n) y(n) = 1 y(n 1)+x(n) x(n 1) 2 K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 9 / 28 Example (II) Discrete-Time Signals and Systems Discrete-Time Systems x(n) T y(n) y(n) = a 1 y(n 1) a 2 y(n 2)+b 0 x(n) +b 1 x(n 1)+b 2 x(n 2)+b 3 x(n 3) 2 3 = a i y(n i)+ b i x(n i) i=1 i=0 K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 10 / 28

Discrete-Time Systems Classification of Discrete-Time Systems Definition (Static) T is static, or memoryless, if y(n) depends only on x(n), i.e., no future/ past samples. Other cases are dynamic systems. If y(n) depends only on x(n k), k = 0, 1,...,N, the system has memory of duration N. Definition (Time Invariant) T is time invariant iff Definition (Linearity) T is linear iff x(n) T y(n) x(n k) T y(n k) T [a 1 x 1 (n)+a 2 x 2 (n)] = a 1 T [x 1 (n)] + a 2 T [x 2 (n)] for arbitrary a 1, x 1, a 2, x 2 K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 11 / 28 Discrete-Time Signals and Systems Discrete-Time Systems Linear systems satisfy the superposition principle, which has two components: Scaling Property T [ax(n)] = at [x(n)] Additive Property T [x 1 (n)+x 2 (n)] = T [x 1 (n)] + T [x 2 (n)] Taken further x(n) = a k x k (n) T y(n) = a k y k (n) where y k (n) =T [x k (n)] K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 12 / 28

Discrete-Time Systems Definition (Causal) Fiscausal if y(n) =F[x(n), x(n 1), x(n 2),...] The output depends on present and past inputs. y(n) = 1 [x(n)+x(n 1)+x(n 2)] 3 CAUSAL? y(n) = 1 [x(n + 1)+x(n)+x(n 1)] 3 CAUSAL? Definition (Bounded Input Bounded Output) A system is BIBO iff x(n) M 1 < T [x(n)] M 2 < n and bounded x(n) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 13 / 28 Discrete-Time Signals and Systems Analysis of DT LTI Systems Analysis of DT LTI Systems Any DT sequence can be written as a weighted sum of impulses. x(n) = k x(k)δ(n k) δ(n k) = { 1, for n = k 0, for n k Example (III) x(n) = {3, 5, 1, 7} x(n) = 3δ(n + 1)+5δ(n) δ(n 1)+7δ(n 2) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 14 / 28

For T Linear Time Invariant (LTI), Analysis of DT LTI Systems h(n) T [δ(n)] h(n k) =T [δ(n k)] Hence, T x(n) y(n) y(n) = T [x(n)] [ ] = T x(k)δ(n k) = k k T [x(k)δ(n k)] [Linearity] = k x(k)h(n k) = x(n) h(n) [Convolution] Note: T [x(k)δ(n k)] = x(k)t [δ(n k)] as x(k) is not a funtion of n. K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 15 / 28 Discrete-Time Signals and Systems Analysis of DT LTI Systems Convolution Steps Letting n = n 0, y(n 0 )= k x(k)h(n 0 k) Folding Fold h(k) about k = 0 to obtain a h( k) Shifting Shift h( k) by n 0 to the right (left) if n 0 is positive (negative), to obtain h(n 0 k) Multiplication Multiply x(k) by h(n 0 k) to obtain the product sequence v n0 (k) =x(k)h(n 0 k) Summation Sum the terms in the product sequence v n0 (k) =to obtain the value of the output at time n = n 0 K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 16 / 28

Analysis of DT LTI Systems K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 17 / 28 Discrete-Time Signals and Systems Analysis of DT LTI Systems K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 18 / 28

Convolution Properties Analysis of DT LTI Systems Shifting x(n) δ(n k) =x(n k) Communative x(n) h(n) =h(n) x(n) Associative [x(n) h 1 (n)] h 2 (n) =x(n) [h 1 (n) h 2 (n)] Distributive x(n) [h 1 (n)+h 2 (n)] = x(n) h 1 (n)+x(n) h 2 (n) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 19 / 28 Discrete-Time Signals and Systems Analysis of DT LTI Systems Causal LTI Systems y(n) = k h(k)x(n k) 1 = h(k)x(n k)+ h(k)x(n k) k= Causality 1 k= h(k)x(n k) =0 [not a function of past input samples] h(k) = 0, for k < 0. y(n) = h(k)x(n k) = n x(m)h(n m) [m = n k] K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 20 / 28

Analysis of DT LTI Systems Theorem (LTI System Stability) A LTI system is BIBO stable if it s impulse response is absolutely summable, S= k h(k) <. Proof. Let x(n) M x < and y(n) = k h(k)x(n k). y(n) = k h(k)x(n k) h(k) x(n k) }{{} k Mx M x h(k) M x S M y k Note: The condition is sufficient and can be shown to be necessary. K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 21 / 28 Discrete-Time Signals and Systems Analysis of DT LTI Systems Impulse Duration (Causal Systems) IIR Infinite Impulse Response y(n) = h(k)x(n k) FIR Finite Impulse Response h(n) = 0 for n < 0 and n M y(n) = M h(k)x(n k) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 22 / 28

Implementation of Discrete-Time Systems Question: How can these systems be efficiently implemented? Suppose 1 n y(n) = x(k) n + 1 y(n 1) = 1 n 1 x(k) n n (n + 1)y(n) = x(k) = n 1 x(k)+x(n) [running average] = ny(n 1)+x(n) y(n) = n 1 y(n 1)+ n + 1 n + 1 x(n) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 23 / 28 Discrete-Time Signals and Systems Implementation of Discrete-Time Systems Result: The system can be implemented Recursively y(n) = F[y(n 1),...,y(n N), x(n),...,x(n M)] Nonrecursively Question: Which is more efficient? y(n) = F[x(n),...,x(n M)] K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 24 / 28

Implementation of Discrete-Time Systems Direct Form I realization Let T be LTI and described by a constant coefficient difference equation y(n) = N M a k y(n k)+ b k x(n k) k=1 Let N = M = 1, y(n) = a 1 y(n 1)+b 0 x(n)+b 1 x(n 1) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 25 / 28 Discrete-Time Signals and Systems Implementation of Discrete-Time Systems Direct Form II realization Direct Form I By linearity; combining delays Direct Form II K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 26 / 28

Implementation of Discrete-Time Systems In the general case, N M y(n) = a k y(n k)+ b k x(n k) k=1 K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 27 / 28 Lecture Summary Lecture Summary DT Systems input output descriptions, block diagram representations, classification of systems (linear, time invariant, static, etc.) Analysis of DT LTI Systems resolving signals as impulses, convolution, convolution properties, causal systems, BIBO stability, finite and infinite impulse responses Implementation of DT Systems recursive and nonrecursive systems, difference equations, Direct Form I and II Next Lecture z Transforms K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 2008 28 / 28