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ESE 531: Digital Signal Processing
Lec 3: January 24, 2019 Discrete Time Signals and Systems
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Lecture Outline
! Discrete Time Systems ! LTI Systems ! LTI System Properties ! Difference Equations
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Discrete-Time Systems
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Discrete Time Systems
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! Systems manipulate the information in signals ! Examples
" Speech recognition system that converts acoustic waves into text " Radar system transforms radar pulse into position and velocity " fMRI system transform frequency into images of brain activity " Moving average system smooths out the day-to-day variability in a
stock price
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System Properties
! Causality " y[n] only depends on x[m] for m<=n
! Linearity " Scaled sum of arbitrary inputs results in output that is a scaled sum of
corresponding outputs " Ax1[n]+Bx2[n] # Ay1[n]+By2[n]
! Memoryless " y[n] depends only on x[n]
! Time Invariance " Shifted input results in shifted output
" x[n-q] # y[n-q]
! BIBO Stability " A bounded input results in a bounded output (ie. max signal value
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Stability
! BIBO Stability " Bounded-input bounded-output Stability
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Examples
! Causal? Linear? Time-invariant? Memoryless? BIBO Stable?
! Time Shift: " x[n] = x[n-m]
! Accumulator: " y[n]
! Compressor (M>1):
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y[n]= x[n−m]
y[n]= x[k]k=−∞
n
∑
y[n]= x[Mn]
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Non-Linear System Example
! Median Filter " y[n]=MED{x[n-k], …x[n+k]}
" Let k=1 " y[n]=MED{x[n-1], x[n], x[n+1]}
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Spectrum of Speech
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Speech
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Spectrum of Speech
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Speech
Corrupted Speech
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Low Pass Filtering
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Low Pass Filtering
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Corrupted Speech
LP-Filtered Speech
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Median Filtering
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Corrupted Speech
Med-Filter Speech
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LTI Systems
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LTI Systems
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! 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]
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Convolution
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! Convolution formula:
! Convolution method: " 1) Time reverse the impulse response and shift it n time steps to the
right " 2) Compute the inner product between the shifted impulse response
and the input vector " Repeat for evey n
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Convolution Example
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! Convolve a unit pulse with itself
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Convolution is Commutative
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! Convolution is commutative:
! These block diagrams are equivalent
! Implication: pick either h or x to flip and shift when convolving
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LTI Systems in Series
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! Impulse response of the cascade of two LTI systems:
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LTI Systems in Series
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! Impulse response of the cascade of two LTI systems:
! Proof by picture
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LTI Systems in Parallel
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! Impulse response of the parallel connection of two LTI systems:
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Causal System Revisited
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! An LTI system is causal if its impulse response is causal:
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Causal System Revisited
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! An LTI system is causal if its impulse response is causal:
! To prove, note that the convolution does not look into the future if the impulse response is causal
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Duration of Impulse
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Duration of Impulse
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! Example: Moving average
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Duration of Impulse
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! Example: Moving average
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Duration of Impulse
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Duration of Impulse
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! Example: Recursive average
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Duration of Impulse
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! Example: Recursive average
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BIBO Stability Revisited
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BIBO Stability Revisited
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! Bounded input and output:
! Where x
∞=max x[n]
x∞<∞ and y
∞<∞
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BIBO Stability Revisited
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! Bounded input and output:
! An LTI system is BIBO stable if and only if
x∞<∞ and y
∞<∞
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BIBO Stability – Sufficient Condition
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BIBO Stability – Sufficient Condition
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BIBO Stability – Necessary Condition
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BIBO Stability – Necessary Condition
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BIBO Stability – Necessary Condition
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Examples
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Examples
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Example
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Example
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y[n]= x[k]k=−∞
n
∑
y[n]= x[n]+ x[k]k=−∞
n−1
∑
y[n]= x[n]+ y[n−1]y[n]− y[n−1]= x[n]
Difference Equations
! Accumulator example
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Difference Equations
! Accumulator example
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ak y[n− k]k=0
N
∑ = bmx[n−m]m=0
M
∑
y[n]= x[k]k=−∞
n
∑
y[n]= x[n]+ x[k]k=−∞
n−1
∑
y[n]= x[n]+ y[n−1]y[n]− y[n−1]= x[n]
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Difference Equations
! Accumulator example
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ak y[n− k]k=0
N
∑ = bmx[n−m]m=0
M
∑
y[n]= x[k]k=−∞
n
∑
y[n]= x[n]+ x[k]k=−∞
n−1
∑
y[n]= x[n]+ y[n−1]y[n]− y[n−1]= x[n]
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Example: Difference Equation
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! Moving Average System
! Let M1=0 (i.e. system is causal)
y[n]= 1M1 +M 2 +1
x[n− k]k=−M1
M2
∑
y[n]= 1M 2 +1
x[n− k]k=0
M2
∑
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Big Ideas
! LTI Systems are a special class of systems with significant signal processing applications " Can be characterized by the impulse response
! LTI System Properties " Causality and stability can be determined from impulse
response
! Difference equations suggest implementation of systems " Give insight into complexity of system
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Admin
! HW 1 posted tomorrow " Due 2/3 at midnight " Submit in Canvas
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