lti systems
TRANSCRIPT
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7/29/2019 LTI Systems
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4: Linear Time Invariant Systems
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 1 / 15
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7/29/2019 LTI Systems
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LTI Systems
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 2 / 15
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7/29/2019 LTI Systems
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LTI Systems
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 2 / 15
Linear Time-invariant (LTI) systems have two properties:
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7/29/2019 LTI Systems
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LTI Systems
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 2 / 15
Linear Time-invariant (LTI) systems have two properties:
Linear: H (u[n] + v[n]) = H (u[n]) + H (v[n])
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7/29/2019 LTI Systems
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LTI Systems
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 2 / 15
Linear Time-invariant (LTI) systems have two properties:
Linear: H (u[n] + v[n]) = H (u[n]) + H (v[n])Time Invariant: y[n] = H (x[n]) y[n r] = H (x[n r]) r
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7/29/2019 LTI Systems
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LTI Systems
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 2 / 15
Linear Time-invariant (LTI) systems have two properties:
Linear: H (u[n] + v[n]) = H (u[n]) + H (v[n])Time Invariant: y[n] = H (x[n]) y[n r] = H (x[n r]) r
The behaviour of an LTI system is completely defined by its impulse
response: h[n] = H ([n])
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7/29/2019 LTI Systems
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LTI Systems
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 2 / 15
Linear Time-invariant (LTI) systems have two properties:
Linear: H (u[n] + v[n]) = H (u[n]) + H (v[n])Time Invariant: y[n] = H (x[n]) y[n r] = H (x[n r]) r
The behaviour of an LTI system is completely defined by its impulse
response: h[n] = H ([n])
Proof:
We can always write x[n] =
r= x[r][n r]
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7/29/2019 LTI Systems
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LTI Systems
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 2 / 15
Linear Time-invariant (LTI) systems have two properties:
Linear: H (u[n] + v[n]) = H (u[n]) + H (v[n])Time Invariant: y[n] = H (x[n]) y[n r] = H (x[n r]) r
The behaviour of an LTI system is completely defined by its impulse
response: h[n] = H ([n])
Proof:
We can always write x[n] =
r= x[r][n r]
Hence H (x[n]) = H
r= x[r][n r]
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7/29/2019 LTI Systems
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LTI Systems
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 2 / 15
Linear Time-invariant (LTI) systems have two properties:
Linear: H (u[n] + v[n]) = H (u[n]) + H (v[n])Time Invariant: y[n] = H (x[n]) y[n r] = H (x[n r]) r
The behaviour of an LTI system is completely defined by its impulse
response: h[n] = H ([n])
Proof:
We can always write x[n] =
r= x[r][n r]
Hence H (x[n]) = H
r= x[r][n r]
=
r= x[r]H ([n r])
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7/29/2019 LTI Systems
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LTI Systems
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 2 / 15
Linear Time-invariant (LTI) systems have two properties:
Linear: H (u[n] + v[n]) = H (u[n]) + H (v[n])Time Invariant: y[n] = H (x[n]) y[n r] = H (x[n r]) r
The behaviour of an LTI system is completely defined by its impulse
response: h[n] = H ([n])
Proof:
We can always write x[n] =
r= x[r][n r]
Hence H (x[n]) = H
r= x[r][n r]
=
r= x[r]H ([n r])
=
r= x[r]h[n r]
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7/29/2019 LTI Systems
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LTI Systems
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 2 / 15
Linear Time-invariant (LTI) systems have two properties:
Linear: H (u[n] + v[n]) = H (u[n]) + H (v[n])Time Invariant: y[n] = H (x[n]) y[n r] = H (x[n r]) r
The behaviour of an LTI system is completely defined by its impulse
response: h[n] = H ([n])
Proof:
We can always write x[n] =
r= x[r][n r]
Hence H (x[n]) = H
r= x[r][n r]
=
r= x[r]H ([n r])
=
r= x[r]h[n r]
= x[n] h[n]
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7/29/2019 LTI Systems
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Convolution Properties
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 3 / 15
Convolution: x[n] v[n] =
r= x[r]v[n r]
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7/29/2019 LTI Systems
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Convolution Properties
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 3 / 15
Convolution: x[n] v[n] =
r= x[r]v[n r]
Convolution obeys normal arithmetic rules for multiplication:
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7/29/2019 LTI Systems
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Convolution Properties
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 3 / 15
Convolution: x[n] v[n] =
r= x[r]v[n r]
Convolution obeys normal arithmetic rules for multiplication:
Commutative: x[n] v[n] = v[n] x[n]
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7/29/2019 LTI Systems
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Convolution Properties
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 3 / 15
Convolution: x[n] v[n] =
r= x[r]v[n r]
Convolution obeys normal arithmetic rules for multiplication:
Commutative: x[n] v[n] = v[n] x[n]
Proof:
r x[r]v[n r](i)=
p x[n p]v[p](i) substitute p = n r
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7/29/2019 LTI Systems
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Convolution Properties
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 3 / 15
Convolution: x[n] v[n] =
r= x[r]v[n r]
Convolution obeys normal arithmetic rules for multiplication:
Commutative: x[n] v[n] = v[n] x[n]
Proof:
r x[r]v[n r](i)=
p x[n p]v[p](i) substitute p = n r
Associative: x[n] (v[n] w[n]) = (x[n] v[n]) w[n]
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7/29/2019 LTI Systems
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Convolution Properties
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 3 / 15
Convolution: x[n] v[n] =
r= x[r]v[n r]
Convolution obeys normal arithmetic rules for multiplication:
Commutative: x[n] v[n] = v[n] x[n]
Proof:
r x[r]v[n r](i)=
p x[n p]v[p](i) substitute p = n r
Associative: x[n] (v[n] w[n]) = (x[n] v[n]) w[n] x[n] v[n] w[n] is unambiguous
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7/29/2019 LTI Systems
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Convolution Properties
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 3 / 15
Convolution: x[n] v[n] =
r= x[r]v[n r]
Convolution obeys normal arithmetic rules for multiplication:
Commutative: x[n] v[n] = v[n] x[n]
Proof:
r x[r]v[n r](i)=
p x[n p]v[p](i) substitute p = n r
Associative: x[n] (v[n] w[n]) = (x[n] v[n]) w[n] x[n] v[n] w[n] is unambiguous
Proof:
r,s x[n r]v[r s]w[s](i)=
p,q x[p]v[qp]w[n q](i) substitute p = n r, q = n s
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7/29/2019 LTI Systems
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Convolution Properties
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 3 / 15
Convolution: x[n] v[n] =
r= x[r]v[n r]
Convolution obeys normal arithmetic rules for multiplication:
Commutative: x[n] v[n] = v[n] x[n]
Proof:
r x[r]v[n r](i)=
p x[n p]v[p](i) substitute p = n r
Associative: x[n] (v[n] w[n]) = (x[n] v[n]) w[n] x[n] v[n] w[n] is unambiguous
Proof:
r,s x[n r]v[r s]w[s](i)=
p,q x[p]v[qp]w[n q](i) substitute p = n r, q = n s
Distributive over +:
x[n] (v[n] + w[n]) = (x[n] v[n]) + (x[n] w[n])
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7/29/2019 LTI Systems
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Convolution Properties
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 3 / 15
Convolution: x[n] v[n] =
r= x[r]v[n r]
Convolution obeys normal arithmetic rules for multiplication:
Commutative: x[n] v[n] = v[n] x[n]
Proof:
r x[r]v[n r](i)=
p x[n p]v[p](i) substitute p = n r
Associative: x[n] (v[n] w[n]) = (x[n] v[n]) w[n] x[n] v[n] w[n] is unambiguous
Proof:
r,s x[n r]v[r s]w[s](i)=
p,q x[p]v[qp]w[n q](i) substitute p = n r, q = n s
Distributive over +:
x[n] (v[n] + w[n]) = (x[n] v[n]) + (x[n] w[n])Proof:
r x[n r] (v[r] + w[r]) =
r x[n r]v[r] +
r x[n r]w[r]
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7/29/2019 LTI Systems
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Convolution Properties
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 3 / 15
Convolution: x[n] v[n] =
r= x[r]v[n r]
Convolution obeys normal arithmetic rules for multiplication:
Commutative: x[n] v[n] = v[n] x[n]
Proof:
r x[r]v[n r](i)=
p x[n p]v[p](i) substitute p = n r
Associative: x[n] (v[n] w[n]) = (x[n] v[n]) w[n] x[n] v[n] w[n] is unambiguous
Proof:
r,s x[n r]v[r s]w[s](i)=
p,q x[p]v[qp]w[n q](i) substitute p = n r, q = n s
Distributive over +:
x[n] (v[n] + w[n]) = (x[n] v[n]) + (x[n] w[n])Proof:
r x[n r] (v[r] + w[r]) =
r x[n r]v[r] +
r x[n r]w[r]
Identity: x[n] [n] = x[n]
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7/29/2019 LTI Systems
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Convolution Properties
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 3 / 15
Convolution: x[n] v[n] =
r= x[r]v[n r]
Convolution obeys normal arithmetic rules for multiplication:
Commutative: x[n] v[n] = v[n] x[n]
Proof:
r x[r]v[n r](i)=
p x[n p]v[p](i) substitute p = n r
Associative: x[n] (v[n] w[n]) = (x[n] v[n]) w[n] x[n] v[n] w[n] is unambiguous
Proof:
r,s x[n r]v[r s]w[s](i)=
p,q x[p]v[qp]w[n q](i) substitute p = n r, q = n s
Distributive over +:
x[n] (v[n] + w[n]) = (x[n] v[n]) + (x[n] w[n])Proof:
r x[n r] (v[r] + w[r]) =
r x[n r]v[r] +
r x[n r]w[r]
Identity: x[n] [n] = x[n]
Proof:
r [r]x[n r] (i)= x[n] (i) all terms zero except r = 0.
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7/29/2019 LTI Systems
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BIBO Stability
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 4 / 15
BIBO Stability: Bounded Input, x[n] Bounded Output, y[n]
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7/29/2019 LTI Systems
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BIBO Stability
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 4 / 15
BIBO Stability: Bounded Input, x[n] Bounded Output, y[n]
The following are equivalent:
(1) An LTI system is BIBO stable
-
7/29/2019 LTI Systems
25/121
BIBO Stability
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 4 / 15
BIBO Stability: Bounded Input, x[n] Bounded Output, y[n]
The following are equivalent:
(1) An LTI system is BIBO stable
(2) h[n] is absolutely summable, i.e. n=
|h[n]| <
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7/29/2019 LTI Systems
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BIBO Stability
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 4 / 15
BIBO Stability: Bounded Input, x[n] Bounded Output, y[n]
The following are equivalent:
(1) An LTI system is BIBO stable
(2) h[n] is absolutely summable, i.e. n=
|h[n]| < (3) H(z) region of convergence includes |z| = 1.
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7/29/2019 LTI Systems
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BIBO Stability
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 4 / 15
BIBO Stability: Bounded Input, x[n] Bounded Output, y[n]
The following are equivalent:
(1) An LTI system is BIBO stable
(2) h[n] is absolutely summable, i.e. n=
|h[n]| < (3) H(z) region of convergence includes |z| = 1.
Proof (1) (2):
Define x[n] = 1 h[n] 01 h[n] < 0
then y[0] =
x[0 n]h[n] =
|h[n]|.
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BIBO Stability
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 4 / 15
BIBO Stability: Bounded Input, x[n] Bounded Output, y[n]
The following are equivalent:
(1) An LTI system is BIBO stable
(2) h[n] is absolutely summable, i.e. n=
|h[n]| < (3) H(z) region of convergence includes |z| = 1.
Proof (1) (2):
Define x[n] = 1 h[n] 01 h[n] < 0
then y[0] =
x[0 n]h[n] =
|h[n]|.But |x[n]| 1n so BIBO y[0] =
|h[n]| < .
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7/29/2019 LTI Systems
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BIBO Stability
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 4 / 15
BIBO Stability: Bounded Input, x[n] Bounded Output, y[n]
The following are equivalent:
(1) An LTI system is BIBO stable
(2) h[n] is absolutely summable, i.e. n=
|h[n]| < (3) H(z) region of convergence includes |z| = 1.
Proof (1) (2):
Define x[n] = 1 h[n] 01 h[n] < 0
then y[0] =
x[0 n]h[n] =
|h[n]|.But |x[n]| 1n so BIBO y[0] =
|h[n]| < .
Proof (2) (1):Suppose
|h[n]| = S < and |x[n]| B is bounded.
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7/29/2019 LTI Systems
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BIBO Stability
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 4 / 15
BIBO Stability: Bounded Input, x[n] Bounded Output, y[n]
The following are equivalent:
(1) An LTI system is BIBO stable
(2) h[n] is absolutely summable, i.e. n=
|h[n]| < (3) H(z) region of convergence includes |z| = 1.
Proof (1) (2):
Define x[n] = 1 h[n] 01 h[n] < 0
then y[0] =
x[0 n]h[n] =
|h[n]|.But |x[n]| 1n so BIBO y[0] =
|h[n]| < .
Proof (2) (1):Suppose
|h[n]| = S < and |x[n]| B is bounded.
Then |y[n]| =
r= x[n r]h[r]
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BIBO Stability
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 4 / 15
BIBO Stability: Bounded Input, x[n] Bounded Output, y[n]
The following are equivalent:
(1) An LTI system is BIBO stable
(2) h[n] is absolutely summable, i.e. n=
|h[n]| < (3) H(z) region of convergence includes |z| = 1.
Proof (1) (2):
Define x[n] = 1 h[n] 01 h[n] < 0
then y[0] =
x[0 n]h[n] =
|h[n]|.But |x[n]| 1n so BIBO y[0] =
|h[n]| < .
Proof (2) (1):Suppose
|h[n]| = S < and |x[n]| B is bounded.
Then |y[n]| =
r= x[n r]h[r]
r= |x[n r]| |h[r]|
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BIBO Stability
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 4 / 15
BIBO Stability: Bounded Input, x[n] Bounded Output, y[n]
The following are equivalent:
(1) An LTI system is BIBO stable
(2) h[n] is absolutely summable, i.e. n=
|h[n]| < (3) H(z) region of convergence includes |z| = 1.
Proof (1) (2):
Define x[n] = 1 h[n] 01 h[n] < 0
then y[0] =
x[0 n]h[n] =
|h[n]|.But |x[n]| 1n so BIBO y[0] =
|h[n]| < .
Proof (2) (1):
Suppose
|h[n]| = S < and |x[n]| B is bounded.
Then |y[n]| =
r= x[n r]h[r]
r= |x[n r]| |h[r]|
B
r= |h[r]| BS <
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Frequency Response
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 5 / 15
For a BIBO stable system Y(ej) = X(ej)H(ej)where H(ej)is the DTFT of h[n] i.e. H(z) evaluated at z = ej.
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Frequency Response
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 5 / 15
For a BIBO stable system Y(ej) = X(ej)H(ej)where H(ej)is the DTFT of h[n] i.e. H(z) evaluated at z = ej.
Example: h[n] =
1 1 1
0
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Frequency Response
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 5 / 15
For a BIBO stable system Y(ej) = X(ej)H(ej)where H(ej)is the DTFT of h[n] i.e. H(z) evaluated at z = ej.
Example: h[n] =
1 1 1
H(ej) = 1 + e
j + e
j20
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Frequency Response
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 5 / 15
For a BIBO stable system Y(ej) = X(ej)H(ej)where H(ej)is the DTFT of h[n] i.e. H(z) evaluated at z = ej.
Example: h[n] =
1 1 1
H(ej) = 1 + e
j + e
j2
= ej (1 + 2 cos )
0
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Frequency Response
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 5 / 15
For a BIBO stable system Y(ej) = X(ej)H(ej)where H(ej)is the DTFT of h[n] i.e. H(z) evaluated at z = ej.
Example: h[n] =
1 1 1
H(ej) = 1 + e
j + e
j2
= ej (1 + 2 cos )
H(ej)
= |1 + 2 cos |
0
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Frequency Response
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 5 / 15
For a BIBO stable system Y(ej) = X(ej)H(ej)where H(ej)is the DTFT of h[n] i.e. H(z) evaluated at z = ej.
Example: h[n] =
1 1 1
H(ej) = 1 + e
j + e
j2
= ej (1 + 2 cos )
H(ej)
= |1 + 2 cos |
0
0 1 2 30
1
2
3
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Frequency Response
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 5 / 15
For a BIBO stable system Y(ej) = X(ej)H(ej)where H(ej)is the DTFT of h[n] i.e. H(z) evaluated at z = ej.
Example: h[n] =
1 1 1
H(ej) = 1 + e
j + e
j2
= ej (1 + 2 cos )
H(ej)
= |1 + 2 cos |
0
0 1 2 30
1
2
3
0 1 2 3-10
-5
0
5
10
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Frequency Response
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 5 / 15
For a BIBO stable system Y(ej) = X(ej)H(ej)where H(ej)is the DTFT of h[n] i.e. H(z) evaluated at z = ej.
Example: h[n] =
1 1 1
H(ej) = 1 + e
j + e
j2
= ej (1 + 2 cos )
H(ej)
= |1 + 2 cos |
H(ej) = + 1sgn(1+2 cos)
2
0
0 1 2 30
1
2
3
0 1 2 3-10
-5
0
5
10
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Frequency Response
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 5 / 15
For a BIBO stable system Y(ej) = X(ej)H(ej)where H(ej)is the DTFT of h[n] i.e. H(z) evaluated at z = ej.
Example: h[n] =
1 1 1
H(ej) = 1 + e
j + e
j2
= ej (1 + 2 cos )
H(ej)
= |1 + 2 cos |
H(ej) = + 1sgn(1+2 cos)
2
0
0 1 2 30
1
2
3
0 1 2 3-10
-5
0
5
10
0 1 2 3-2
-1
0
1
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Frequency Response
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 5 / 15
For a BIBO stable system Y(ej) = X(ej)H(ej)where H(ej)is the DTFT of h[n] i.e. H(z) evaluated at z = ej.
Example: h[n] =
1 1 1
H(ej) = 1 + e
j + e
j2
= ej (1 + 2 cos )
H(ej)
= |1 + 2 cos |
H(ej) = + 1sgn(1+2 cos)
2Sign change in (1 + 2 cos ) at = 2.1 gives
(a) gradient discontinuity in |H(ej)|(b) an abrupt phase change of .
0
0 1 2 30
1
2
3
0 1 2 3-10
-5
0
5
10
0 1 2 3-2
-1
0
1
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Frequency Response
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 5 / 15
For a BIBO stable system Y(ej) = X(ej)H(ej)where H(ej)is the DTFT of h[n] i.e. H(z) evaluated at z = ej.
Example: h[n] =
1 1 1
H(ej) = 1 + e
j + e
j2
= ej (1 + 2 cos )
H(ej)
= |1 + 2 cos |
H(ej) = + 1sgn(1+2 cos)2
Sign change in (1 + 2 cos ) at = 2.1 gives(a) gradient discontinuity in |H(ej)|(b) an abrupt phase change of .
Group delay is ddH(ej) : gives delay of the
modulation envelope at each .
0
0 1 2 30
1
2
3
0 1 2 3-10
-5
0
5
10
0 1 2 3-2
-1
0
1
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Frequency Response
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 5 / 15
For a BIBO stable system Y(ej) = X(ej)H(ej)where H(ej)is the DTFT of h[n] i.e. H(z) evaluated at z = ej.
Example: h[n] =
1 1 1
H(ej) = 1 + e
j + e
j2
= ej (1 + 2 cos )
H(ej)
= |1 + 2 cos |
H(ej) = + 1sgn(1+2 cos)2
Sign change in (1 + 2 cos ) at = 2.1 gives(a) gradient discontinuity in |H(ej)|(b) an abrupt phase change of .
Group delay is ddH(ej) : gives delay of the
modulation envelope at each .
Normally varies with but for a symmetric filter it is
constant: in this case +1 samples.
0
0 1 2 30
1
2
3
0 1 2 3-10
-5
0
5
10
0 1 2 3-2
-1
0
1
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Frequency Response
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 5 / 15
For a BIBO stable system Y(ej) = X(ej)H(ej)where H(ej)is the DTFT of h[n] i.e. H(z) evaluated at z = ej.
Example: h[n] =
1 1 1
H(ej) = 1 + e
j + e
j2
= ej (1 + 2 cos )
H(ej)
= |1 + 2 cos |
H(ej) = + 1sgn(1+2 cos)2
Sign change in (1 + 2 cos ) at = 2.1 gives(a) gradient discontinuity in |H(ej)|(b) an abrupt phase change of .
Group delay is ddH(ej) : gives delay of the
modulation envelope at each .
Normally varies with but for a symmetric filter it is
constant: in this case +1 samples.Discontinuities of
k do not affect group delay.
0
0 1 2 30
1
2
3
0 1 2 3-10
-5
0
5
10
0 1 2 3-2
-1
0
1
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Causality
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 6 / 15
Causal System: cannot see into the futurei.e. output at time n depends only on inputs up to time n.
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Causality
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 6 / 15
Causal System: cannot see into the futurei.e. output at time n depends only on inputs up to time n.
Formal definition:
If v[n] = x[n] for n n0 then H (v[n]) = H (x[n]) for n n0.
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Causality
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 6 / 15
Causal System: cannot see into the futurei.e. output at time n depends only on inputs up to time n.
Formal definition:
If v[n] = x[n] for n n0 then H (v[n]) = H (x[n]) for n n0.
The following are equivalent:
(1) An LTI system is causal
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Causality
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 6 / 15
Causal System: cannot see into the futurei.e. output at time n depends only on inputs up to time n.
Formal definition:
If v[n] = x[n] for n n0 then H (v[n]) = H (x[n]) for n n0.
The following are equivalent:
(1) An LTI system is causal
(2) h[n] is causal h[n] = 0 for n < 0
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Causality
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 6 / 15
Causal System: cannot see into the futurei.e. output at time n depends only on inputs up to time n.
Formal definition:
If v[n] = x[n] for n n0 then H (v[n]) = H (x[n]) for n n0.
The following are equivalent:
(1) An LTI system is causal
(2) h[n] is causal h[n] = 0 for n < 0(3) H(z) converges for z =
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Causality
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 6 / 15
Causal System: cannot see into the futurei.e. output at time n depends only on inputs up to time n.
Formal definition:
If v[n] = x[n] for n n0 then H (v[n]) = H (x[n]) for n n0.
The following are equivalent:
(1) An LTI system is causal
(2) h[n] is causal h[n] = 0 for n < 0(3) H(z) converges for z =
Any right-sided sequence can be made causal by adding a delay.
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Causality
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 6 / 15
Causal System: cannot see into the futurei.e. output at time n depends only on inputs up to time n.
Formal definition:
If v[n] = x[n] for n n0 then H (v[n]) = H (x[n]) for n n0.
The following are equivalent:
(1) An LTI system is causal
(2) h[n] is causal h[n] = 0 for n < 0(3) H(z) converges for z =
Any right-sided sequence can be made causal by adding a delay.
All the systems we will deal with are causal.
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Passive and Lossless
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 7 / 15
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Passive and Lossless
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 7 / 15
A passive system can never gain energy:
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Passive and Lossless
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 7 / 15
A passive system can never gain energy:
n= |y[n]|2
n= |x[n]|2 for any finite energy input x[n].
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Passive and Lossless
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 7 / 15
A passive system can never gain energy:
n= |y[n]|2
n= |x[n]|2 for any finite energy input x[n].
A passive LTI system must haveH(ej) 1
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Passive and Lossless
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 7 / 15
A passive system can never gain energy:
n= |y[n]|2
n= |x[n]|2 for any finite energy input x[n].
A passive LTI system must haveH(ej) 1
Somewhat analogous to a circuit consisting of R, L and C only.
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Passive and Lossless
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 7 / 15
A passive system can never gain energy:
n= |y[n]|2
n= |x[n]|2 for any finite energy input x[n].
A passive LTI system must haveH(ej) 1
Somewhat analogous to a circuit consisting of R, L and C only.
A lossless system always preserves energy exactly:
-
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Passive and Lossless
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 7 / 15
A passive system can never gain energy:
n= |y[n]|2
n= |x[n]|2 for any finite energy input x[n].
A passive LTI system must haveH(ej) 1
Somewhat analogous to a circuit consisting of R, L and C only.
A lossless system always preserves energy exactly:
n= |y[n]|2
=
n= |x[n]|2
for any finite energy input x[n].
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Passive and Lossless
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 7 / 15
A passive system can never gain energy:n= |y[n]|2
n= |x[n]|2 for any finite energy input x[n].
A passive LTI system must haveH(ej) 1
Somewhat analogous to a circuit consisting of R, L and C only.
A lossless system always preserves energy exactly:
n= |y[n]|2
=
n= |x[n]|2
for any finite energy input x[n].
A lossless LTI system must have
H(ej)
= 1
called an allpass system.
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Passive and Lossless
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 7 / 15
A passive system can never gain energy:n= |y[n]|2
n= |x[n]|2 for any finite energy input x[n].
A passive LTI system must haveH(ej) 1
Somewhat analogous to a circuit consisting of R, L and C only.
A lossless system always preserves energy exactly:
n= |y[n]|2
=
n= |x[n]|2
for any finite energy input x[n].
A lossless LTI system must have
H(ej)
= 1
called an allpass system.
Somewhat analogous to a circuit consisting of L and C only.
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Passive and Lossless
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 7 / 15
A passive system can never gain energy:n= |y[n]|2
n= |x[n]|2 for any finite energy input x[n].
A passive LTI system must haveH(ej) 1
Somewhat analogous to a circuit consisting of R, L and C only.
A lossless system always preserves energy exactly:
n= |y[n]|2
=
n= |x[n]|2
for any finite energy input x[n].
A lossless LTI system must have
H(ej)
= 1
called an allpass system.
Somewhat analogous to a circuit consisting of L and C only.
Passive and lossless building blocks can be used to design systems which
are insensitive to coefficient changes
P ll l d S i
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Parallel and Series
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 8 / 15
Two LTI systems in parallel:
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Parallel and Series
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 8 / 15
Two LTI systems in parallel:
h[n] = h1[n] + h2[n]
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Parallel and Series
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 8 / 15
Two LTI systems in parallel:
h[n] = h1[n] + h2[n] H(z) = H1(z) + H2(z)
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Parallel and Series
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 8 / 15
Two LTI systems in parallel:
h[n] = h1[n] + h2[n] H(z) = H1(z) + H2(z)
Two LTI systems in series (a.k.a. cascade):
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Parallel and Series
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 8 / 15
Two LTI systems in parallel:
h[n] = h1[n] + h2[n] H(z) = H1(z) + H2(z)
Two LTI systems in series (a.k.a. cascade):h[n] = h1[n] h2[n]
Parallel and Series
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Parallel and Series
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 8 / 15
Two LTI systems in parallel:
h[n] = h1[n] + h2[n] H(z) = H1(z) + H2(z)
Two LTI systems in series (a.k.a. cascade):h[n] = h1[n] h2[n] H(z) = H1(z)H2(z)
Parallel and Series
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Parallel and Series
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 8 / 15
Two LTI systems in parallel:
h[n] = h1[n] + h2[n] H(z) = H1(z) + H2(z)
Two LTI systems in series (a.k.a. cascade):h[n] = h1[n] h2[n] H(z) = H1(z)H2(z)
Proof: y[n] = h2[n] (h1[n] x[n]) = (h2[n] h1[n]) x[n]associativity of convolution
Convolution Complexity
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Convolution Complexity
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 9 / 15
y[n] = x[n] h[n]: convolve x[0 : N 1] with h[0 : M 1]
x
Convolution Complexity
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Convolution Complexity
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 9 / 15
y[n] = x[n] h[n]: convolve x[0 : N 1] with h[0 : M 1]
x
Convolution sum:
y[n] =
M1r=0 h[r]x[n r]
Convolution Complexity
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Convolution Complexity
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 9 / 15
y[n] = x[n] h[n]: convolve x[0 : N 1] with h[0 : M 1]
x
Convolution sum:
y[n] =
M1r=0 h[r]x[n r]
y[0] y[9]
N = 8, M = 3M + N 1 = 10
Convolution Complexity
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Convolution Complexity
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 9 / 15
y[n] = x[n] h[n]: convolve x[0 : N 1] with h[0 : M 1]
x
Convolution sum:
y[n] =
M1r=0 h[r]x[n r]
y[n] is only non-zero in the range0 n M + N 2
y[0] y[9]
N = 8, M = 3M + N 1 = 10
Convolution Complexity
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Convolution Complexity
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 9 / 15
y[n] = x[n] h[n]: convolve x[0 : N 1] with h[0 : M 1]
x
Convolution sum:
y[n] =
M1r=0 h[r]x[n r]
y[n] is only non-zero in the range0 n M + N 2
Thus y[n] has onlyM + N 1 non-zero values
y[0] y[9]
N = 8, M = 3M + N 1 = 10
Convolution Complexity
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Convolution Complexity
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 9 / 15
y[n] = x[n] h[n]: convolve x[0 : N 1] with h[0 : M 1]
x
Convolution sum:
y[n] =
M1r=0 h[r]x[n r]
y[n] is only non-zero in the range0 n M + N 2
Thus y[n] has onlyM + N 1 non-zero values
Algebraically:
y[0] y[9]
N = 8, M = 3M + N 1 = 10
x[n r] = 0 0 n r N 1
Convolution Complexity
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Convolution Complexity
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 9 / 15
y[n] = x[n] h[n]: convolve x[0 : N 1] with h[0 : M 1]
x
Convolution sum:
y[n] =
M1r=0 h[r]x[n r]
y[n] is only non-zero in the range0 n M + N 2
Thus y[n] has onlyM + N 1 non-zero values
Algebraically:
y[0] y[9]
N = 8, M = 3M + N 1 = 10
x[n r] = 0 0 n r N 1 n + 1 N r n
Convolution Complexity
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Convolution Complexity
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 9 / 15
y[n] = x[n] h[n]: convolve x[0 : N 1] with h[0 : M 1]
x
Convolution sum:
y[n] =
M1r=0 h[r]x[n r]
y[n] is only non-zero in the range0 n M + N 2
Thus y[n] has onlyM + N 1 non-zero values
Algebraically:
y[0] y[9]
N = 8, M = 3M + N 1 = 10
x[n r] = 0 0 n r N 1 n + 1 N r n
Hence: y[n] =min(M1,n))
r=max(0,n+1N) h[r]x[n r]
Convolution Complexity
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Convolution Complexity
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 9 / 15
y[n] = x[n] h[n]: convolve x[0 : N 1] with h[0 : M 1]
x
Convolution sum:
y[n] =
M1r=0 h[r]x[n r]
y[n] is only non-zero in the range0 n M + N 2
Thus y[n] has onlyM + N 1 non-zero values
Algebraically:
y[0] y[9]
N = 8, M = 3M + N 1 = 10
x[n r] = 0 0 n r N 1 n + 1 N r n
Hence: y[n] =min(M1,n))
r=max(0,n+1N) h[r]x[n r]
We must multiply each h[n] by each x[n] and add them to a total
total arithmetic complexity ( or + operations) 2M N
Circular Convolution
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Circular Convolution
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 10 / 15
y[n] = x[n]N h[n]: circ convolve x[0 : N 1] with h[0 : M 1]
x
N
Circular Convolution
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Circular Convolution
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 10 / 15
y[n] = x[n]N h[n]: circ convolve x[0 : N 1] with h[0 : M 1]
x
N
Convolution sum:
yN[n] =M1
r=0 h[r]x[(n r)mod N]
Circular Convolution
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Circular Convolution
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 10 / 15
y[n] = x[n]N h[n]: circ convolve x[0 : N 1] with h[0 : M 1]
x
N
Convolution sum:
yN[n] =M1
r=0 h[r]x[(n r)mod N]
y[0] y[7]
N = 8, M = 3
Circular Convolution
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Circular Convolution
4: Linear Time InvariantSystems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 10 / 15
y[n] = x[n]N h[n]: circ convolve x[0 : N 1] with h[0 : M 1]
x
N
Convolution sum:
yN[n] =M1
r=0 h[r]x[(n r)mod N]
yN[n] has period N
y[0] y[7]
N = 8, M = 3
Circular Convolution
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C cu a Co o ut o
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 10 / 15
y[n] = x[n]N h[n]: circ convolve x[0 : N 1] with h[0 : M 1]
x
N
Convolution sum:
yN[n] =M1
r=0 h[r]x[(n r)mod N]
yN[n] has period N yN[n] has N distinct values
y[0] y[7]
N = 8, M = 3
Circular Convolution
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4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 10 / 15
y[n] = x[n]N h[n]: circ convolve x[0 : N 1] with h[0 : M 1]
x
N
Convolution sum:
yN[n] =M1
r=0 h[r]x[(n r)mod N]
yN[n] has period N yN[n] has N distinct values
y[0] y[7]
N = 8, M = 3
Only the first M 1 values are affected by the circular repetition:
Circular Convolution
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4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 10 / 15
y[n] = x[n]N h[n]: circ convolve x[0 : N 1] with h[0 : M 1]
x
N
Convolution sum:
yN[n] =M1
r=0 h[r]x[(n r)mod N]
yN[n] has period N yN[n] has N distinct values
y[0] y[7]
N = 8, M = 3
Only the first M 1 values are affected by the circular repetition:
yN[n] = y[n] for M 1 n N 1
Circular Convolution
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4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 10 / 15
y[n] = x[n]N h[n]: circ convolve x[0 : N 1] with h[0 : M 1]
x
N
Convolution sum:
yN[n] =M1
r=0 h[r]x[(n r)mod N]
yN[n] has period N yN[n] has N distinct values
y[0] y[7]
N = 8, M = 3
Only the first M 1 values are affected by the circular repetition:
yN[n] = y[n] for M 1 n N 1If we append M 1 zeros (or more) onto x[n], then the circular repetitionhas no effect at all and:
yN+M1 [n] = y[n] for 0 n N + M 2
Frequency-domain convolution
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q y
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 11 / 15
Idea: Use DFT to perform circular convolution - less computation
Frequency-domain convolution
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q y
4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 11 / 15
Idea: Use DFT to perform circular convolution - less computation
(1) Choose L M + N 1 (normally round up to a power of 2)
Frequency-domain convolution
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4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 11 / 15
Idea: Use DFT to perform circular convolution - less computation
(1) Choose L M + N 1 (normally round up to a power of 2)
(2) Zero pad x[n] and h[n] to give sequences of length L: x[n] and h[n]
Frequency-domain convolution
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4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 11 / 15
Idea: Use DFT to perform circular convolution - less computation
(1) Choose L M + N 1 (normally round up to a power of 2)
(2) Zero pad x[n] and h[n] to give sequences of length L: x[n] and h[n]
(3) Use DFT: y[n] = F
1(X[k]H[k]) = x[n]L h[n]
Frequency-domain convolution
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4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 11 / 15
Idea: Use DFT to perform circular convolution - less computation
(1) Choose L M + N 1 (normally round up to a power of 2)
(2) Zero pad x[n] and h[n] to give sequences of length L: x[n] and h[n]
(3) Use DFT: y[n] = F
1(X[k]H[k]) = x[n]L h[n]
(4) y[n] = y[n] for 0 n M + N 2.
Frequency-domain convolution
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4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 11 / 15
Idea: Use DFT to perform circular convolution - less computation
(1) Choose L M + N 1 (normally round up to a power of 2)
(2) Zero pad x[n] and h[n] to give sequences of length L: x[n] and h[n]
(3) Use DFT: y[n] = F
1(X[k]H[k]) = x[n]L h[n]
(4) y[n] = y[n] for 0 n M + N 2.
Arithmetic Complexity:
DFT or IDFT take 4L log2 L operations if L is a power of 2(or 16L log2 L if not).
Frequency-domain convolution
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4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 11 / 15
Idea: Use DFT to perform circular convolution - less computation
(1) Choose L M + N 1 (normally round up to a power of 2)
(2) Zero pad x[n] and h[n] to give sequences of length L: x[n] and h[n]
(3) Use DFT: y[n] = F
1(X[k]H[k]) = x[n]L h[n]
(4) y[n] = y[n] for 0 n M + N 2.
Arithmetic Complexity:
DFT or IDFT take 4L log2 L operations if L is a power of 2(or 16L log2 L if not).Total operations: 12L log2 L 12 (M + N)log2 (M + N)
Frequency-domain convolution
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4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 11 / 15
Idea: Use DFT to perform circular convolution - less computation
(1) Choose L M + N 1 (normally round up to a power of 2)
(2) Zero pad x[n] and h[n] to give sequences of length L: x[n] and h[n]
(3) Use DFT: y[n] = F
1(X[k]H[k]) = x[n]L h[n]
(4) y[n] = y[n] for 0 n M + N 2.
Arithmetic Complexity:
DFT or IDFT take 4L log2 L operations if L is a power of 2(or 16L log2 L if not).Total operations: 12L log2 L 12 (M + N)log2 (M + N)Beneficial if both M and N are > 100 .
Frequency-domain convolution
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4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 11 / 15
Idea: Use DFT to perform circular convolution - less computation
(1) Choose L M + N 1 (normally round up to a power of 2)
(2) Zero pad x[n] and h[n] to give sequences of length L: x[n] and h[n]
(3) Use DFT: y[n] = F
1(X[k]H[k]) = x[n]L h[n]
(4) y[n] = y[n] for 0 n M + N 2.
Arithmetic Complexity:
DFT or IDFT take 4L log2 L operations if L is a power of 2(or 16L log2 L if not).Total operations: 12L log2 L 12 (M + N)log2 (M + N)Beneficial if both M and N are > 100 .
Example: M = 103
, N = 104
:
Frequency-domain convolution
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4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 11 / 15
Idea: Use DFT to perform circular convolution - less computation
(1) Choose L M + N 1 (normally round up to a power of 2)
(2) Zero pad x[n] and h[n] to give sequences of length L: x[n] and h[n]
(3) Use DFT: y[n] = F
1(X[k]H[k]) = x[n]L h[n]
(4) y[n] = y[n] for 0 n M + N 2.
Arithmetic Complexity:
DFT or IDFT take 4L log2 L operations if L is a power of 2(or 16L log2 L if not).Total operations: 12L log2 L 12 (M + N)log2 (M + N)Beneficial if both M and N are > 100 .
Example: M = 103
, N = 104
:Direct: 2M N = 2 107
Frequency-domain convolution
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Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 11 / 15
Idea: Use DFT to perform circular convolution - less computation
(1) Choose L M + N 1 (normally round up to a power of 2)
(2) Zero pad x[n] and h[n] to give sequences of length L: x[n] and h[n]
(3) Use DFT: y[n] = F
1(X[k]H[k]) = x[n]L h[n]
(4) y[n] = y[n] for 0 n M + N 2.
Arithmetic Complexity:
DFT or IDFT take 4L log2 L operations if L is a power of 2(or 16L log2 L if not).Total operations: 12L log2 L 12 (M + N)log2 (M + N)Beneficial if both M and N are > 100 .
Example: M = 103
, N = 104
:Direct: 2M N = 2 107
with DFT: 12 (M + N)log2 (M + N) = 1.8 106
Frequency-domain convolution
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4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 11 / 15
Idea: Use DFT to perform circular convolution - less computation
(1) Choose L M + N 1 (normally round up to a power of 2)
(2) Zero pad x[n] and h[n] to give sequences of length L: x[n] and h[n]
(3) Use DFT: y[n] = F
1(X[k]H[k]) = x[n]L h[n]
(4) y[n] = y[n] for 0 n M + N 2.
Arithmetic Complexity:
DFT or IDFT take 4L log2 L operations if L is a power of 2(or 16L log2 L if not).Total operations: 12L log2 L 12 (M + N)log2 (M + N)Beneficial if both M and N are > 100 .
Example: M = 103
, N = 104
:Direct: 2M N = 2 107
with DFT: 12 (M + N)log2 (M + N) = 1.8 106
But: (a) DFT may be very long if N is large
(b) No output samples until all x[n] has been input.
Overlap Add
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4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 12 / 15
If N is very large:
Overlap Add
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4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 12 / 15
If N is very large:(1) chop x[n] into N
Kchunks of length K
Overlap Add
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4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 12 / 15
If N is very large:(1) chop x[n] into N
Kchunks of length K
(2) convolve each chunk with h[n]
Overlap Add
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4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 12 / 15
If N is very large:(1) chop x[n] into N
Kchunks of length K
(2) convolve each chunk with h[n]
Each output chunk is of length K + M 1 and overlaps the next chunk
Overlap Add
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4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 12 / 15
If N is very large:(1) chop x[n] into N
Kchunks of length K
(2) convolve each chunk with h[n](3) add up the results
Each output chunk is of length K + M 1 and overlaps the next chunk
Overlap Add
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4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 12 / 15
If N is very large:(1) chop x[n] into N
Kchunks of length K
(2) convolve each chunk with h[n](3) add up the results
Each output chunk is of length K + M 1 and overlaps the next chunkOperations: N
K 8 (M + K)log2 (M + K)
Overlap Add
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4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 12 / 15
If N is very large:(1) chop x[n] into N
Kchunks of length K
(2) convolve each chunk with h[n](3) add up the results
Each output chunk is of length K + M 1 and overlaps the next chunkOperations: N
K 8 (M + K)log2 (M + K)
Computational saving if 100 < M K N
Overlap Add
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4: Linear Time Invariant
Systems
LTI Systems
Convolution Properties
BIBO Stability
Frequency Response
Causality
Passive and Lossless
Parallel and Series
Convolution Complexity
Circular Convolution
Frequency-domain
convolution
Overlap Add
Overlap Save
Summary
MATLAB routines
DSP and Digital Filters (2012-2446) LTI Systems: 4 12 / 15
If N is very large:(1) chop x[n] into N
Kchunks of length K
(2) convolve each chunk with h[n](3) add up the results
Each output chunk is of length K + M 1 and overlaps the next chunkOperations: N
K 8 (M + K)log2 (M + K)
Computational saving if 100 < M K N
Example: M = 500, K = 104
, N = 107
Direct: 2M N = 1010
Overlap Add
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4: Linear Time Invariant
Sy