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EE3054 Signals and Systems
Lecture 2Linear and Time Invariant
Systems
Yao WangPolytechnic University
Most of the slides included are extracted from lecture presentations prepared by McClellan and Schafer
1/30/2008 © 2003, JH McClellan & RW Schafer 2
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1/30/2008 © 2003, JH McClellan & RW Schafer 3
� General FIR System� IMPULSE RESPONSE
� FIR case: h[n]=b_n
� CONVOLUTION� For any FIR system: y[n] = x[n] * h[n]
Review of Last Lecture
][][][ nxnhny ∗=
][nh
}{ kb
1/30/2008 © 2003, JH McClellan & RW Schafer 4
GENERAL FIR FILTER
� FILTER COEFFICIENTS {bk}� DEFINE THE FILTER
� For example,
∑=
−=M
kk knxbny
0][][
]3[]2[2]1[][3
][][3
0
−+−+−−=
−=∑=
nxnxnxnx
knxbnyk
k
}1,2,1,3{ −=kb
Feedforward Difference Equation
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GENERAL FIR FILTER� SLIDE a Length-L WINDOW over x[n]� When h[n] is not symmetric, needs to flip h(n) first!
x[n]x[n-M]
1/30/2008 © 2003, JH McClellan & RW Schafer 6
7-pt AVG EXAMPLE
CAUSAL: Use Previous
LONGER OUTPUT
400for)4/8/2cos()02.1(][ :Input ≤≤++= nnnx n ππ
1/30/2008 © 2003, JH McClellan & RW Schafer 7
≠
==
00
01][
n
nnδ
Unit Impulse Signal
� x[n] has only one NON-ZERO VALUE
1
n
UNIT-IMPULSE
1/30/2008 © 2003, JH McClellan & RW Schafer 8
},0,0,,,,,0,0,{][ 41
41
41
41
��=nh
4-pt Avg Impulse Response
δδδδ[n] “READS OUT” the FILTER COEFFICIENTS
n=0“h” in h[n] denotes Impulse Response
1
n
n=0n=1
n=4n=5
n=–1
NON-ZERO When window overlaps δδδδ[n]
])3[]2[]1[][(][ 41 −+−+−+= nxnxnxnxny
1/30/2008 © 2003, JH McClellan & RW Schafer 9
What is Impulse Response?
� Impulse response is the output signal when the input is an impulse
� Finite Impulse Response (FIR) system:� Systems for which the impulse response has finite
duration� For FIR system, impulse response = Filter
coefficients� h[k] = b_k� Output = h[k]* input
1/30/2008 © 2003, JH McClellan & RW Schafer 10
FIR IMPULSE RESPONSE
� Convolution = Filter Definition� Filter Coeffs = Impulse Response
∑=
−=M
kknxkhny
0][][][
CONVOLUTION
∑=
−=M
kk knxbny
0][][
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Convolution Operation� Flip h[n]� SLIDE a Length-L WINDOW over x[n]
x[n]x[n-M]
∑=
−=M
kknxkhny
0][][][
CONVOLUTION
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More on signal ranges and lengths after filtering
� Input signal from 0 to N-1, length=L1=N� Filter from 0 to M, length = L2= M+1� Output signal?
� From 0 to N+M-1, length L3=N+M=L1+L2-1
1/30/2008 © 2003, JH McClellan & RW Schafer 13
DCONVDEMO: MATLAB GUI
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� Go through the demo program for different types of signals
� Do an example by hand� Rectangular * rectangular� Step function * rectangular
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CONVOLUTION via Synthetic Polynomial Multiplication
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Convolution via Synthetic Polynomial Multiplication
� More example
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MATLAB for FIR FILTER
� yy = conv(bb,xx)
� VECTOR bb contains Filter Coefficients
� DSP-First: yy = firfilt(bb,xx)
� FILTER COEFFICIENTS {bk} conv2()for images
∑=
−=M
kk knxbny
0][][
1/30/2008 © 2003, JH McClellan & RW Schafer 18
]1[][][ −−= nununy
POP QUIZ
� FIR Filter is “FIRST DIFFERENCE”� y[n] = x[n] - x[n-1]
� INPUT is “UNIT STEP”
� Find y[n]
<
≥=
00
01][
n
nnu
][]1[][][ nnununy δ=−−=
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SYSTEM PROPERTIES
�� MATHEMATICAL DESCRIPTIONMATHEMATICAL DESCRIPTION�� TIMETIME--INVARIANCEINVARIANCE�� LINEARITYLINEARITY� CAUSALITY
� “No output prior to input”
SYSTEMy[n]x[n]
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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=0) is picked arbitrary
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TESTING Time-Invariance
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� Examples of systems that are time invariant and non-time invariant
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LINEAR SYSTEM
� LINEARITY = Two Properties� SCALING
� “Doubling x[n] will double y[n]”
� SUPERPOSITION:� “Adding two inputs gives an output that is the
sum of the individual outputs”
1/30/2008 © 2003, JH McClellan & RW Schafer 24
TESTING LINEARITY
1/30/2008 © 2003, JH McClellan & RW Schafer 25
� Examples systems that are linear and non-linear
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LTI SYSTEMS
� LTI: Linear & Time-Invariant� Any FIR system is LTI� Proof!
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LTI SYSTEMS
� COMPLETELY CHARACTERIZED by:� IMPULSE RESPONSE h[n]� CONVOLUTION: y[n] = x[n]*h[n]
� The “rule”defining the system can ALWAYS be re-written as convolution
� FIR Example: h[n] is same as bk
� Proof of the convolution sum relation� by representing x(n) as sum of delta(n-k), and
use LTI property!
Properties of Convolution
� Convolution with an Impulse� Commutative Property� Associative Property
Convolution with Impulse
� x[n]*δ[n]=x[n]� x[n]*δ[n-k]=x[n-k]� Proof
Commutative Property
� x[n]*h[n]=h[n]*x[n]� Proof
Associative Property
� (x[n]* y[n])* z[n]=x[n]*(y[n]*z[n])� Proof
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HARDWARE STRUCTURES
� INTERNAL STRUCTURE of “FILTER”� WHAT COMPONENTS ARE NEEDED?� HOW DO WE “HOOK” THEM TOGETHER?
� SIGNAL FLOW GRAPH NOTATION
FILTER y[n]x[n]∑=
−=M
kk knxbny
0][][
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HARDWARE ATOMS
� Add, Multiply & Store∑=
−=M
kk knxbny
0][][
][][ nxny β=
]1[][ −= nxny
][][][ 21 nxnxny +=
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FIR STRUCTURE
� Direct Form
SIGNALFLOW GRAPH
∑=
−=M
kk knxbny
0][][
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FILTER AS BUILDING BLOCKS
� BUILD UP COMPLICATED FILTERS� FROM SIMPLE MODULES� Ex: FILTER MODULE MIGHT BE 3-pt FIR
� Is the overall system still LTI? What is its impulse response?
FILTERy[n]x[n]
FILTER
FILTER
++
INPUT
OUTPUT
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CASCADE SYSTEMS
� Does the order of S1 & S2 matter?� NO, LTI SYSTEMS can be rearranged !!!� WHAT ARE THE FILTER COEFFS? {bk}
S1 S2
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� proof
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CASCADE SYSTEMS
� Is the cascaded system LTI?� What is the impulse response of the
overall system?
h1[n] h2[n]x[n] y[n]
CASCADE SYSTEMS
� h[n] = h1[n]*h2[n]!� Proof on board
1/30/2008 © 2003, JH McClellan & RW Schafer 41
Does the order matter?
S1 S2
S1S2
� proof
Example
� Given impulse responses of two systems, determine the overall impulse response
Parallel Connections
h1[n]
h2[n]
x[n] y[n]+
h[n]=?
Parallel and Cascade
h[n]=?
h1[n]
h3[n]
x[n] y[n]+
h2[n]
Summary of This Lecture
� Properties of linear and time invariant systems� Any LTI system can be characterized by its impulse
response h[n], and output is related to input by convolution sum: convolution sum: y[ny[n]=]=x[nx[n]*]*h[nh[n]]
� Properties of convolution� Computation of convolution revisited
� Sliding window� Synthetic polynomial multiplication
� Block diagram representation � Hardware implementation of one FIR� Connection of multiple FIR
• Know how to compute overall impulse response
READING ASSIGNMENTS
� This Lecture:� Chapter 5, Sections 5-5 --- 5-9