Lecture Outline

ESE 531: Digital ! Discrete Time Systems ! LTI Systems ! LTI System Properties Lec 3: January 17, 2017 ! Difference Equations Discrete Time Signals and Systems

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Discrete Time Systems

Discrete-Time Systems

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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- 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

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Low Pass Filtering Low Pass Filtering

Corrupted Speech

LP-Filtered Speech

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Median Filtering

LTI Systems Corrupted Speech

Med-Filter Speech

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2 LTI Systems

! LTI system can be completely characterized by its

! Then the output for an arbitrary input is a sum of weighted, delay impulse responses

y[n] = x[n]∗h[n]

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Convolution Example Convolution Example

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Convolution is Commutative LTI Systems in Series

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3 LTI Systems in Parallel Example

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Causal System Revisited Duration of Impulse

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Duration of Impulse BIBO Stability Revisited

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4 BIBO Stability Revisited BIBO Stability – Sufficient Condition

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BIBO Stability – Sufficient Condition BIBO Stability – Sufficient Condition

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BIBO Stability – Necessary Condition BIBO Stability – Necessary Condition

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5 BIBO Stability – Necessary Condition BIBO Stability – Necessary Condition

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Examples Example

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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

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