Lecture Outline ESE 531: Digital Signal Processing ! Discrete Time Systems ! LTI Systems ! LTI System Properties Lec 3: January 17, 2017 ! Difference Equations Discrete Time Signals and Systems Penn ESE 531 Spring 2017 - Khanna Penn ESE 531 Spring 2017 - Khanna 2 Discrete Time Systems Discrete-Time Systems Penn ESE 531 Spring 2017 - Khanna 4 System Properties Examples ! Causality ! Causal? Linear? Time-invariant? Memoryless? " y[n] only depends on x[m] for m<=n BIBO Stable? ! Linearity ! Time Shift: " Scaled sum of arbitrary inputs results in output that is a scaled sum of corresponding outputs " x[n]y[n =] x[n-m]= x[n − m] " Ax [n]+Bx [n] # Ay [n]+By [n] 1 2 1 2 ! Accumulator: ! Memoryless " y[n] n " y[n] depends only on x[n] y[n] = x[k] ! Time Invariance ∑ k=−∞ " Shifted input results in shifted output " x[n-q] # y[n-q] ! Compressor (M>1): ! BIBO Stability y[n] x[Mn] " A bounded input results in a bounded output (ie. max signal value = exists for output if max ) Penn ESE 531 Spring 2017 - Khanna 5 Penn ESE 531 Spring 2017 - Khanna 6 1 Non-Linear System Example Spectrum of Speech ! Median Filter " y[n]=MED{x[n-k], …x[n+k]} Speech " Let k=1 " y[n]=MED{x[n-1], x[n], x[n+1]} Corrupted Speech Penn ESE 531 Spring 2017 - Khanna 7 Penn ESE 531 Spring 2017 - Khanna 8 Low Pass Filtering Low Pass Filtering Corrupted Speech LP-Filtered Speech Penn ESE 531 Spring 2017 - Khanna 9 Penn ESE 531 Spring 2017 - Khanna 10 Median Filtering LTI Systems Corrupted Speech Med-Filter Speech Penn ESE 531 Spring 2017 - Khanna 11 Penn ESE 531 Spring 2017 - Khanna 12 2 LTI Systems Convolution ! LTI system can be completely characterized by its impulse response ! Then the output for an arbitrary input is a sum of weighted, delay impulse responses y[n] = x[n]∗h[n] Penn ESE 531 Spring 2017 - Khanna 13 Convolution Example Convolution Example Penn ESE 531 Spring 2017 - Khanna 15 Penn ESE 531 Spring 2017 - Khanna 16 Convolution is Commutative LTI Systems in Series Penn ESE 531 Spring 2017 - Khanna 17 Penn ESE 531 Spring 2017 - Khanna 18 3 LTI Systems in Parallel Example Penn ESE 531 Spring 2017 - Khanna 19 Penn ESE 531 Spring 2017 - Khanna 20 Causal System Revisited Duration of Impulse Penn ESE 531 Spring 2017 - Khanna 21 Penn ESE 531 Spring 2017 - Khanna 22 Duration of Impulse BIBO Stability Revisited Penn ESE 531 Spring 2017 - Khanna 23 Penn ESE 531 Spring 2017 - Khanna 24 4 BIBO Stability Revisited BIBO Stability – Sufficient Condition Penn ESE 531 Spring 2017 - Khanna 25 Penn ESE 531 Spring 2017 - Khanna 26 BIBO Stability – Sufficient Condition BIBO Stability – Sufficient Condition Penn ESE 531 Spring 2017 - Khanna 27 Penn ESE 531 Spring 2017 - Khanna 28 BIBO Stability – Necessary Condition BIBO Stability – Necessary Condition Penn ESE 531 Spring 2017 - Khanna 29 Penn ESE 531 Spring 2017 - Khanna 30 5 BIBO Stability – Necessary Condition BIBO Stability – Necessary Condition Penn ESE 531 Spring 2017 - Khanna 31 Penn ESE 531 Spring 2017 - Khanna 32 Examples Example Penn ESE 531 Spring 2017 - Khanna 33 Penn ESE 531 Spring 2017 - Khanna 34 Difference Equations Difference Equations ! Accumulator example ! Accumulator example n n y[n] = ∑ x[k] y[n] = ∑ x[k] k=−∞ k=−∞ n−1 n−1 y[n] = x[n]+ ∑ x[k] y[n] = x[n]+ ∑ x[k] k=−∞ k=−∞ y[n] = x[n]+ y[n −1] y[n] = x[n]+ y[n −1] y[n]− y[n −1] = x[n] y[n]− y[n −1] = x[n] N M a y[n k] b y[n m] ∑ k − = ∑ m − k=0 m=0 Penn ESE 531 Spring 2017 - Khanna 35 Penn ESE 531 Spring 2017 - Khanna 36 6 Difference Equations Big Ideas ! Accumulator example ! LTI Systems are a special class of systems with n significant signal processing applications y[n] = ∑ x[k] " Can be characterized by the impulse response k=−∞ ! LTI System Properties n−1 y[n] = x[n]+ ∑ x[k] " Causality and stability can be determined from impulse k=−∞ response y[n] = x[n]+ y[n −1] ! Difference equations suggest implementation of y[n]− y[n −1] = x[n] systems " Give insight into complexity of system N M " More on this next time… a y[n k] b y[n m] ∑ k − = ∑ m − k=0 m=0 Penn ESE 531 Spring 2017 - Khanna 37 Penn ESE 531 Spring 2017 - Khanna 38 Admin ! Homework schedule changed " Due on Fridays at midnight instead of Thursday " Course calendar updated ! HW 1 out now " Due 1/27 at midnight " Submit in Canvas Penn ESE 531 Spring 2017 - Khanna 39 7 .