READING ASSIGNMENTS LECTURE OBJECTIVES OVERVIEW. ELEG-212 Signal Processing and Communications. This Lecture:

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1 ELEG-212 Signal Processing and Communications Lecture 11 Linearity & Time-Invariance Convolution READING ASSIGNENTS This Lecture: Chapter 5, Sections 5-5 and 5-6 Section 5-4 will be covered, but not in depth Other Reading: Recitation: Ch. 5, Sects 5-6, 5-7 & 5-8 CONVOLUTION Next Lecture: start Chapter 6 ECE-212 Signal Processing First 2 LECTURE OBJECTIVES GENERAL PROPERTIES of S LINEARITY LTI SYSTES TIE-INVARIANCE > CONVOLUTION BLOCK DIAGRA REPRESENTATION Components for Hardware Connect Simple Filters Together to Build ore Complicated Systems ECE-212 Signal Processing First 3 OVERVIEW IPULSE RESPONSE, h[ FIR case: same as { b } CONVOLUTION GENERAL: h[ GENERAL CLASS of SYSTES LINEAR and TIE-INVARIANT ALL LTI systems have h[ & use convolution ECE-212 Signal Processing First 4

2 DIGITAL ING BUILDING BLOCKS x(t) A-to-D D-to-A y(t) OUTPUT INPUT + + CONCENTRATE on the (DSP) DISCRETE-TIE SIGNALS FUNCTIONS of n, the time index INPUT OUTPUT BUILD UP COPLICATED S FRO SIPLE ODULES Ex: ODULE IGHT BE 3-pt FIR ECE-212 Signal Processing First 5 ECE-212 Signal Processing First 6 GENERAL FIR COEFFICIENTS {b } DEFINE THE For example, 3 b b n ] { 3, 1, 2,1} ECE-212 Signal Processing First 7 b n ] 3 n 1] + 2 n 2] + n 3] ATLAB for FIR yy conv(bb,xx) VECTOR bb contains Filter Coefficients DSP-First: yy firfilt(bb,xx) COEFFICIENTS {b } b n ] conv2() for images ECE-212 Signal Processing First 8

3 SPECIAL INPUT SIGNALS FIR IPULSE RESPONSE SINUSOID FREQUENCY RESPONSE has only one NON-ZERO VALUE 1 n δ [ n UNIT-IPULSE 1 n ECE-212 Signal Processing First 9 Convolution Filter Definition Filter Coeffs Impulse Response hy [ n ] b x δ [ n ] ECE-212 Signal Processing First 1 ATH FORULA for h[ LTI: Convolution Sum Use SHIFTED IPULSES to write h[ h[ δ [ δ [ n 1] + 2δ [ n 2] δ [ n 3] + δ [ n 4] b h[ { 1, 1, 2, 1, 1} n 1 ECE-212 Signal Processing First 11 Output Convolution of & h[ NOTATION: h[ Here is the FIR case: Same as b FINITE LIITS h[ ] n ] FINITE LIITS ECE-212 Signal Processing First 12

4 CONVOLUTION Example h[ δ [ δ [ n 1] + 2δ [ n 2] δ [ n 3] + δ [ n 4] u[ n h[ h[] h[1] n 1] h[2] n 2] h[3] n 3] h[4] n 4] ECE-212 Signal Processing First 13 GENERAL FIR SLIDE a Length-L WINDOW over n-] ECE-212 Signal Processing First 14 DCONVDEO: ATLAB GUI HARDWARE STRUCTURES b n ] INTERNAL STRUCTURE of WHAT COPONENTS ARE NEEDED? HOW DO WE HOOK THE TOGETHER? SIGNAL FLOW GRAPH NOTATION ECE-212 Signal Processing First 15 ECE-212 Signal Processing First 17

5 HARDWARE ATOS FIR STRUCTURE Add, ultiply & Store b n ] Direct Form SIGNAL FLOW GRAPH b n ] β y [ x1[ + x2[ n 1] ECE-212 Signal Processing First 18 ECE-212 Signal Processing First 19 oore s Law for TI DSPs SYSTE PROPERTIES LOG SCALE SYSTE Double every 18 months? ATHEATICAL DESCRIPTION TIE-INVARIANCEINVARIANCE LINEARITY CAUSALITY No output prior to input ECE-212 Signal Processing First 2 ECE-212 Signal Processing First 21

6 TIE-INVARIANCE TESTING Time-Invariance IDEA: Time-Shifting the input will cause the same time-shift in the output EQUIVALENTLY, We can prove that The time origin (n) is piced arbitrary ECE-212 Signal Processing First 22 ECE-212 Signal Processing First 23 LINEAR SYSTE TESTING LINEARITY LINEARITY Two Properties SCALING Doubling will double SUPERPOSITION: Adding two inputs gives an output that is the sum of the individual outputs ECE-212 Signal Processing First 24 ECE-212 Signal Processing First 25

7 LTI SYSTES CASCADE SYSTES LTI: Linear & Time-Invariant COPLETELY CHARACTERIZED by: IPULSE RESPONSE h[ CONVOLUTION: *h[ The rule defining the system can ALWAYS be rewritten as convolution FIR Example: h[ is same as b Does the order of S 1 & S 2 matter? NO, LTI SYSTES can be rearranged!!! WHAT ARE THE COEFFS? {b } S 1 S 2 ECE-212 Signal Processing First 26 ECE-212 Signal Processing First 28 CASCADE EQUIVALENT Find overall h[ for a cascade? S 1 S 2 S 2 S 1 ECE-212 Signal Processing First 29