Fourier Series, Convolution, and Filters - University of St....
Transcript of Fourier Series, Convolution, and Filters - University of St....
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Fourier Series, Convolution, and Filters
Patrick J. Van Fleet
Center for Applied MathematicsUniversity of St. Thomas
St. Paul, MN USA
PREP - Wavelet Workshop, 2006
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
OutlineToday’s ScheduleFourier Series
Classical ResultsStudent DifficultiesFinite Length Fourier Series
ConvolutionConvolution DefinedThe Convolution Theorem
FiltersTypes of FiltersLowpass and Highpass Filters
Convolution as a Matrix ProductIn the Classroom
Teaching IdeasComputer UsageStudent Difficulties
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Today’s Schedule
9:00-10:15 Lecture One: Why Wavelets?10:15-10:30 Coffee Break (OSS 235)10:30-11:45 Lecture Two: Digital Images, Measures, and
Huffman Codes12:00-1:00 Lunch (Cafeteria)1:30-2:45 ⇒Lecture Three: Fourier Series, Convolution and
Filters2:45-3:00 Coffee Break (OSS 235)3:00-4:15 Lecture Four: 1D and 2D Haar Transforms5:30-6:30 Dinner (Cafeteria)
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Classical Results
Fourier SeriesClassical Results
I Let’s start by recalling Euler’s formula:
eiω = cos ω + i sin ω
I It can be shown that the set of functions ek (ω) = eikω,k ∈ Z, satisfy ∫ π
−πeikωeijω dω =
{2π j = k
0 j 6= k
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Classical Results
Fourier SeriesClassical Results
I Let’s start by recalling Euler’s formula:
eiω = cos ω + i sin ω
I It can be shown that the set of functions ek (ω) = eikω,k ∈ Z, satisfy ∫ π
−πeikωeijω dω =
{2π j = k
0 j 6= k
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Classical Results
Fourier SeriesClassical Results
I Let’s start by recalling Euler’s formula:
eiω = cos ω + i sin ω
I It can be shown that the set of functions ek (ω) = eikω,k ∈ Z, satisfy ∫ π
−πeikωeijω dω =
{2π j = k
0 j 6= k
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Classical Results
Fourier SeriesClassical Results
I The family {eikω}k∈Z forms a basis for all suitably regular2π-periodic functions.
I A Fourier Series for 2π-periodic function f (ω) is
f (ω) =∑
k
ckeikω
whereck =
12π
∫ π
−πf (ω)e−ikω dω
are called the Fourier coefficients.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Classical Results
Fourier SeriesClassical Results
I The family {eikω}k∈Z forms a basis for all suitably regular2π-periodic functions.
I A Fourier Series for 2π-periodic function f (ω) is
f (ω) =∑
k
ckeikω
whereck =
12π
∫ π
−πf (ω)e−ikω dω
are called the Fourier coefficients.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Classical Results
Fourier SeriesClassical Results
I The family {eikω}k∈Z forms a basis for all suitably regular2π-periodic functions.
I A Fourier Series for 2π-periodic function f (ω) is
f (ω) =∑
k
ckeikω
whereck =
12π
∫ π
−πf (ω)e−ikω dω
are called the Fourier coefficients.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Classical Results
Fourier SeriesClassical Results
If we take f (ω) = ω on the interval [−π, π] and then2π-periodically extend it, we can use integration by parts towrite
f (ω) = i∑k 6=0
(−1)k
keikω = −2
∞∑k=1
(−1)k
ksin(kω)
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Classical Results
Fourier SeriesClassical Results
Here are some partial Fourier series:
Original 0 Terms 3 Terms
10 Terms 20 Terms 50 Terms
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Classical Results
Fourier SeriesClassical Results
I Calculate some Fourier series by hand.I Discover and prove some rules for Fourier series:I Translation Rule: If
f (ω) =∑
k
ckeikω
and g(ω) = f (ω − a), then the Fourier coefficients for g(ω)are e−ikack .
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Classical Results
Fourier SeriesClassical Results
I Calculate some Fourier series by hand.I Discover and prove some rules for Fourier series:I Translation Rule: If
f (ω) =∑
k
ckeikω
and g(ω) = f (ω − a), then the Fourier coefficients for g(ω)are e−ikack .
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Classical Results
Fourier SeriesClassical Results
I Calculate some Fourier series by hand.I Discover and prove some rules for Fourier series:I Translation Rule: If
f (ω) =∑
k
ckeikω
and g(ω) = f (ω − a), then the Fourier coefficients for g(ω)are e−ikack .
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Classical Results
Fourier SeriesClassical Results
I Calculate some Fourier series by hand.I Discover and prove some rules for Fourier series:I Translation Rule: If
f (ω) =∑
k
ckeikω
and g(ω) = f (ω − a), then the Fourier coefficients for g(ω)are e−ikack .
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Student Difficulties
Fourier SeriesStudent Difficulties
I Students, at least at this level, never seem to grasp theideas behind why one would want to compute a Fourierseries.
I You can talk about them in terms of solving differentialequations . . .
I But it’s kind of like Taylor’s series to them: Why take aperfectly good function and make an infinite series out ofit?
I This class gives a wonderful arena for demonstrating theusefulness of Fourier series.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Student Difficulties
Fourier SeriesStudent Difficulties
I Students, at least at this level, never seem to grasp theideas behind why one would want to compute a Fourierseries.
I You can talk about them in terms of solving differentialequations . . .
I But it’s kind of like Taylor’s series to them: Why take aperfectly good function and make an infinite series out ofit?
I This class gives a wonderful arena for demonstrating theusefulness of Fourier series.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Student Difficulties
Fourier SeriesStudent Difficulties
I Students, at least at this level, never seem to grasp theideas behind why one would want to compute a Fourierseries.
I You can talk about them in terms of solving differentialequations . . .
I But it’s kind of like Taylor’s series to them: Why take aperfectly good function and make an infinite series out ofit?
I This class gives a wonderful arena for demonstrating theusefulness of Fourier series.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Student Difficulties
Fourier SeriesStudent Difficulties
I Students, at least at this level, never seem to grasp theideas behind why one would want to compute a Fourierseries.
I You can talk about them in terms of solving differentialequations . . .
I But it’s kind of like Taylor’s series to them: Why take aperfectly good function and make an infinite series out ofit?
I This class gives a wonderful arena for demonstrating theusefulness of Fourier series.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Student Difficulties
Fourier SeriesStudent Difficulties
I Students, at least at this level, never seem to grasp theideas behind why one would want to compute a Fourierseries.
I You can talk about them in terms of solving differentialequations . . .
I But it’s kind of like Taylor’s series to them: Why take aperfectly good function and make an infinite series out ofit?
I This class gives a wonderful arena for demonstrating theusefulness of Fourier series.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Student Difficulties
Fourier SeriesStudent Difficulties
I There are three important uses for Fourier series in thiscourse. Students need:
I to build and manipulate finite length Fourier series and whatthey say about filters. (i.e. what do engineers do with them)
I the ability to extract the coefficients from a Fourier series toobtain a filter.
I to know how to manipulate one Fourier series (say byconjugation, multiplication by a complex exponential,translation) to write down another.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Student Difficulties
Fourier SeriesStudent Difficulties
I There are three important uses for Fourier series in thiscourse. Students need:
I to build and manipulate finite length Fourier series and whatthey say about filters. (i.e. what do engineers do with them)
I the ability to extract the coefficients from a Fourier series toobtain a filter.
I to know how to manipulate one Fourier series (say byconjugation, multiplication by a complex exponential,translation) to write down another.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Student Difficulties
Fourier SeriesStudent Difficulties
I There are three important uses for Fourier series in thiscourse. Students need:
I to build and manipulate finite length Fourier series and whatthey say about filters. (i.e. what do engineers do with them)
I the ability to extract the coefficients from a Fourier series toobtain a filter.
I to know how to manipulate one Fourier series (say byconjugation, multiplication by a complex exponential,translation) to write down another.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Student Difficulties
Fourier SeriesStudent Difficulties
I There are three important uses for Fourier series in thiscourse. Students need:
I to build and manipulate finite length Fourier series and whatthey say about filters. (i.e. what do engineers do with them)
I the ability to extract the coefficients from a Fourier series toobtain a filter.
I to know how to manipulate one Fourier series (say byconjugation, multiplication by a complex exponential,translation) to write down another.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Student Difficulties
Fourier SeriesStudent Difficulties
I There are three important uses for Fourier series in thiscourse. Students need:
I to build and manipulate finite length Fourier series and whatthey say about filters. (i.e. what do engineers do with them)
I the ability to extract the coefficients from a Fourier series toobtain a filter.
I to know how to manipulate one Fourier series (say byconjugation, multiplication by a complex exponential,translation) to write down another.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Finite Length Fourier Series
Fourier SeriesFinite Length Fourier Series
I In a typical math class (like the start of this one!), we give astudent a 2π-periodic function f (ω), they integrate by parts,simplify, and obtain the Fourier coefficients for the Fourierseries of f (ω).
I The engineers do it just the opposite way: They know thatthe coefficients are what’s used to process signals orimages, so they will create a (usually finite) list ofcoefficients (by various means), plug them into a Fourierseries, and analyze the result.
I Let’s look at an example:
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Finite Length Fourier Series
Fourier SeriesFinite Length Fourier Series
I In a typical math class (like the start of this one!), we give astudent a 2π-periodic function f (ω), they integrate by parts,simplify, and obtain the Fourier coefficients for the Fourierseries of f (ω).
I The engineers do it just the opposite way: They know thatthe coefficients are what’s used to process signals orimages, so they will create a (usually finite) list ofcoefficients (by various means), plug them into a Fourierseries, and analyze the result.
I Let’s look at an example:
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Finite Length Fourier Series
Fourier SeriesFinite Length Fourier Series
I In a typical math class (like the start of this one!), we give astudent a 2π-periodic function f (ω), they integrate by parts,simplify, and obtain the Fourier coefficients for the Fourierseries of f (ω).
I The engineers do it just the opposite way: They know thatthe coefficients are what’s used to process signals orimages, so they will create a (usually finite) list ofcoefficients (by various means), plug them into a Fourierseries, and analyze the result.
I Let’s look at an example:
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Finite Length Fourier Series
Fourier SeriesFinite Length Fourier Series
I In a typical math class (like the start of this one!), we give astudent a 2π-periodic function f (ω), they integrate by parts,simplify, and obtain the Fourier coefficients for the Fourierseries of f (ω).
I The engineers do it just the opposite way: They know thatthe coefficients are what’s used to process signals orimages, so they will create a (usually finite) list ofcoefficients (by various means), plug them into a Fourierseries, and analyze the result.
I Let’s look at an example:
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Finite Length Fourier Series
Fourier SeriesFinite Length Fourier Series
I Suppose we have number ck , k ∈ Z, with c0 = c2 = 1/4,c1 = 1/2, and all other ck = 0. Find the Fourier series andplot its modulus.
I We have
C(ω) =14
+12
eiω +14
e2iω
= eiω(14
e−iω +12
+14
eiω)
= eiω(12
+12· eiω + e−iω
2)
= eiω 12(1 + cos ω)
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Finite Length Fourier Series
Fourier SeriesFinite Length Fourier Series
I Suppose we have number ck , k ∈ Z, with c0 = c2 = 1/4,c1 = 1/2, and all other ck = 0. Find the Fourier series andplot its modulus.
I We have
C(ω) =14
+12
eiω +14
e2iω
= eiω(14
e−iω +12
+14
eiω)
= eiω(12
+12· eiω + e−iω
2)
= eiω 12(1 + cos ω)
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Finite Length Fourier Series
Fourier SeriesFinite Length Fourier Series
I Suppose we have number ck , k ∈ Z, with c0 = c2 = 1/4,c1 = 1/2, and all other ck = 0. Find the Fourier series andplot its modulus.
I We have
C(ω) =14
+12
eiω +14
e2iω
= eiω(14
e−iω +12
+14
eiω)
= eiω(12
+12· eiω + e−iω
2)
= eiω 12(1 + cos ω)
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Finite Length Fourier Series
Fourier SeriesFinite Length Fourier Series
|C(ω)| = 12(1 + cos ω) = cos2(ω/2) ≥ 0
|C(ω)|
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Finite Length Fourier Series
Fourier SeriesFinite Length Fourier Series
I The modulus of C(ω) tells the engineer much about thesequence {ck}.
I As we will see in later lectures, these ck ’s are used in amatrix to transform signals or images.
I In the case of this particular sequence, the modulus ismaximized at ω = 0 (lowest oscillation in C(ω)) andminimized at ω = π (highest oscillation in C(ω)). So whenthis sequence is used to process data, it will tend to leavelow oscillations in data largely unchanged and dampenhigh oscillations in data.
I More on this in the filters lecture . . .
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Finite Length Fourier Series
Fourier SeriesFinite Length Fourier Series
I The modulus of C(ω) tells the engineer much about thesequence {ck}.
I As we will see in later lectures, these ck ’s are used in amatrix to transform signals or images.
I In the case of this particular sequence, the modulus ismaximized at ω = 0 (lowest oscillation in C(ω)) andminimized at ω = π (highest oscillation in C(ω)). So whenthis sequence is used to process data, it will tend to leavelow oscillations in data largely unchanged and dampenhigh oscillations in data.
I More on this in the filters lecture . . .
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Finite Length Fourier Series
Fourier SeriesFinite Length Fourier Series
I The modulus of C(ω) tells the engineer much about thesequence {ck}.
I As we will see in later lectures, these ck ’s are used in amatrix to transform signals or images.
I In the case of this particular sequence, the modulus ismaximized at ω = 0 (lowest oscillation in C(ω)) andminimized at ω = π (highest oscillation in C(ω)). So whenthis sequence is used to process data, it will tend to leavelow oscillations in data largely unchanged and dampenhigh oscillations in data.
I More on this in the filters lecture . . .
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Finite Length Fourier Series
Fourier SeriesFinite Length Fourier Series
I The modulus of C(ω) tells the engineer much about thesequence {ck}.
I As we will see in later lectures, these ck ’s are used in amatrix to transform signals or images.
I In the case of this particular sequence, the modulus ismaximized at ω = 0 (lowest oscillation in C(ω)) andminimized at ω = π (highest oscillation in C(ω)). So whenthis sequence is used to process data, it will tend to leavelow oscillations in data largely unchanged and dampenhigh oscillations in data.
I More on this in the filters lecture . . .
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Finite Length Fourier Series
Fourier SeriesFinite Length Fourier Series
I The modulus of C(ω) tells the engineer much about thesequence {ck}.
I As we will see in later lectures, these ck ’s are used in amatrix to transform signals or images.
I In the case of this particular sequence, the modulus ismaximized at ω = 0 (lowest oscillation in C(ω)) andminimized at ω = π (highest oscillation in C(ω)). So whenthis sequence is used to process data, it will tend to leavelow oscillations in data largely unchanged and dampenhigh oscillations in data.
I More on this in the filters lecture . . .
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Finite Length Fourier Series
Fourier SeriesFinite Length Fourier Series
I It is also very important that students learn to “peel” theFourier coefficients off a 2π-periodic function withouthaving to integrate. For example, consider
I What are the Fourier coefficients of
H(ω) = cos2(ω/2)(
1 + sin2(ω/2)(a0 + a1 eiω))
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Finite Length Fourier Series
Fourier SeriesFinite Length Fourier Series
I It is also very important that students learn to “peel” theFourier coefficients off a 2π-periodic function withouthaving to integrate. For example, consider
I What are the Fourier coefficients of
H(ω) = cos2(ω/2)(
1 + sin2(ω/2)(a0 + a1 eiω))
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Finite Length Fourier Series
Fourier SeriesFinite Length Fourier Series
I It is also very important that students learn to “peel” theFourier coefficients off a 2π-periodic function withouthaving to integrate. For example, consider
I What are the Fourier coefficients of
H(ω) = cos2(ω/2)(
1 + sin2(ω/2)(a0 + a1 eiω))
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Finite Length Fourier Series
Fourier SeriesFinite Length Fourier Series
Using the identities
cos(ω) =eiω + e−iω
2and sin(ω) =
eiω − e−iω
2i
and lots of algebra, we have
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Finite Length Fourier Series
Fourier SeriesFinite Length Fourier Series
H(ω) = −a0
16e−2iω +
4 − a1
16e−iω +
8 + 2a0
16
+4 + 2a1
16eiω − a0
16e2iω − a1
16e3iω
so the Fourier coefficients are
(h−2, h−1, h0, h1, h2, h3) = (−a0
16,4 − a1
16,8 + 2a0
16,4 + 2a1
16,
a0
16,−a1
16)
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Finite Length Fourier Series
Fourier SeriesFinite Length Fourier Series
I It is also important to be able to manipulate Fourier series.For example,
I Suppose (h0, h1, h2, h3) are nonzero and let
H(ω) = h0 + h1 eiω + h2 e2iω + h3 e3iω
I What do we have to do to produce the Fourier series
H1(ω) = h0 − h1 eiω + h2 e2iω + h3 e3iω
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Finite Length Fourier Series
Fourier SeriesFinite Length Fourier Series
I It is also important to be able to manipulate Fourier series.For example,
I Suppose (h0, h1, h2, h3) are nonzero and let
H(ω) = h0 + h1 eiω + h2 e2iω + h3 e3iω
I What do we have to do to produce the Fourier series
H1(ω) = h0 − h1 eiω + h2 e2iω + h3 e3iω
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Finite Length Fourier Series
Fourier SeriesFinite Length Fourier Series
I It is also important to be able to manipulate Fourier series.For example,
I Suppose (h0, h1, h2, h3) are nonzero and let
H(ω) = h0 + h1 eiω + h2 e2iω + h3 e3iω
I What do we have to do to produce the Fourier series
H1(ω) = h0 − h1 eiω + h2 e2iω + h3 e3iω
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Finite Length Fourier Series
Fourier SeriesFinite Length Fourier Series
I It is also important to be able to manipulate Fourier series.For example,
I Suppose (h0, h1, h2, h3) are nonzero and let
H(ω) = h0 + h1 eiω + h2 e2iω + h3 e3iω
I What do we have to do to produce the Fourier series
H1(ω) = h0 − h1 eiω + h2 e2iω + h3 e3iω
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Finite Length Fourier Series
Fourier SeriesFinite Length Fourier Series
I OrH2(ω) = h3 + h2 eiω + h1 e2iω + h0 e3iω
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Finite Length Fourier Series
Fourier SeriesFinite Length Fourier Series
I OrH2(ω) = h3 + h2 eiω + h1 e2iω + h0 e3iω
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Finite Length Fourier Series
Fourier SeriesFinite Length Fourier Series
Answers:
H1(ω) = H(ω + π)
H2(ω) = e3iωH(ω)
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution Defined
ConvolutionConvolution Defined
Let h = (. . . , h−1, h0, h1, h2, . . .) and x(. . . , x−1, x0, x1, x2, . . .) betwo bi-infinite sequences. Then the convolution producty = h ∗ x is the bi-infinite sequence whose components aregiven by
yn =∑
k
hkxn−k
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution Defined
ConvolutionConvolution Defined
I In many applications, we think of h as a processor and x isthe input signal.
I Convolution is a standard tool used to process signals andimages.
I The convolution product is commutative.I Modulo a carry algorithm convolution is exactly how we
were taught to multiply numbers in grade school.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution Defined
ConvolutionConvolution Defined
I In many applications, we think of h as a processor and x isthe input signal.
I Convolution is a standard tool used to process signals andimages.
I The convolution product is commutative.I Modulo a carry algorithm convolution is exactly how we
were taught to multiply numbers in grade school.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution Defined
ConvolutionConvolution Defined
I In many applications, we think of h as a processor and x isthe input signal.
I Convolution is a standard tool used to process signals andimages.
I The convolution product is commutative.I Modulo a carry algorithm convolution is exactly how we
were taught to multiply numbers in grade school.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution Defined
ConvolutionConvolution Defined
I In many applications, we think of h as a processor and x isthe input signal.
I Convolution is a standard tool used to process signals andimages.
I The convolution product is commutative.I Modulo a carry algorithm convolution is exactly how we
were taught to multiply numbers in grade school.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution Defined
ConvolutionConvolution Defined
I In many applications, we think of h as a processor and x isthe input signal.
I Convolution is a standard tool used to process signals andimages.
I The convolution product is commutative.I Modulo a carry algorithm convolution is exactly how we
were taught to multiply numbers in grade school.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution Defined
ConvolutionConvolution Defined
Let’s look at an example. Let x be any bi-infinite sequence andsuppose h is the bi-infinite sequence with h0 = h1 = 1
2 and allother hk = 0. Find
y = h ∗ x
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution Defined
ConvolutionConvolution Defined
We have
yn =∑
k
hkxn−k =1∑
k=0
12
xn−k =12(xn + xn−1)
So y is a bi-infinite sequence whose components are averagesof consecutive values of x.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution Defined
ConvolutionConvolution Defined
I Another important convolution product will use g whereg0 = 1
2 , g1 = −12 and all other gk = 0.
I It should be easy to verify that y = g ∗ x where
yn =12(xn − xn−1)
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution Defined
ConvolutionConvolution Defined
I Another important convolution product will use g whereg0 = 1
2 , g1 = −12 and all other gk = 0.
I It should be easy to verify that y = g ∗ x where
yn =12(xn − xn−1)
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution Defined
ConvolutionConvolution Defined
I Another important convolution product will use g whereg0 = 1
2 , g1 = −12 and all other gk = 0.
I It should be easy to verify that y = g ∗ x where
yn =12(xn − xn−1)
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution Defined
ConvolutionConvolution Defined
I Students often have difficulty computing convolutionproducts.
I In this workshop, the processor h will have only finitelymany nonzero elements.
I When this is the case, it is usually better to show them thesliding strip method for convolution:
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution Defined
ConvolutionConvolution Defined
I Students often have difficulty computing convolutionproducts.
I In this workshop, the processor h will have only finitelymany nonzero elements.
I When this is the case, it is usually better to show them thesliding strip method for convolution:
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution Defined
ConvolutionConvolution Defined
I Students often have difficulty computing convolutionproducts.
I In this workshop, the processor h will have only finitelymany nonzero elements.
I When this is the case, it is usually better to show them thesliding strip method for convolution:
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution Defined
ConvolutionConvolution Defined
I Students often have difficulty computing convolutionproducts.
I In this workshop, the processor h will have only finitelymany nonzero elements.
I When this is the case, it is usually better to show them thesliding strip method for convolution:
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution Defined
ConvolutionConvolution Defined
I The component y0 =∑
k hkx−k can be viewed as an innerproduct of h with a reflection of x.
h : · · · h−2 h−1 h0 h1 h2 · · ·· · · x2 x1 x0 x−1 x−2 · · ·
I The component y1 =∑
k hkx1−k can be viewed as an innerproduct of h with the reflection of x shifted one unit right:
h : · · · h−2 h−1 h0 h1 h2 · · ·· · · x3 x2 x1 x0 x−1 · · ·
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution Defined
ConvolutionConvolution Defined
I The component y0 =∑
k hkx−k can be viewed as an innerproduct of h with a reflection of x.
h : · · · h−2 h−1 h0 h1 h2 · · ·· · · x2 x1 x0 x−1 x−2 · · ·
I The component y1 =∑
k hkx1−k can be viewed as an innerproduct of h with the reflection of x shifted one unit right:
h : · · · h−2 h−1 h0 h1 h2 · · ·· · · x3 x2 x1 x0 x−1 · · ·
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution Defined
ConvolutionConvolution Defined
I The component y0 =∑
k hkx−k can be viewed as an innerproduct of h with a reflection of x.
h : · · · h−2 h−1 h0 h1 h2 · · ·· · · x2 x1 x0 x−1 x−2 · · ·
I The component y1 =∑
k hkx1−k can be viewed as an innerproduct of h with the reflection of x shifted one unit right:
h : · · · h−2 h−1 h0 h1 h2 · · ·· · · x3 x2 x1 x0 x−1 · · ·
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
The Convolution Theorem
ConvolutionThe Convolution Theorem
I Convolution is a useful tool for processing signals.I We saw two processors, h and g, that computed averages
and differences of consecutive terms.I While convolution is a useful tool, it is a bit tedious to
analyze. Fortunately there is a result that makes it reallyeasy to analyze in the transform domain:
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
The Convolution Theorem
ConvolutionThe Convolution Theorem
I Convolution is a useful tool for processing signals.I We saw two processors, h and g, that computed averages
and differences of consecutive terms.I While convolution is a useful tool, it is a bit tedious to
analyze. Fortunately there is a result that makes it reallyeasy to analyze in the transform domain:
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
The Convolution Theorem
ConvolutionThe Convolution Theorem
I Convolution is a useful tool for processing signals.I We saw two processors, h and g, that computed averages
and differences of consecutive terms.I While convolution is a useful tool, it is a bit tedious to
analyze. Fortunately there is a result that makes it reallyeasy to analyze in the transform domain:
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
The Convolution Theorem
ConvolutionThe Convolution Theorem
I Convolution is a useful tool for processing signals.I We saw two processors, h and g, that computed averages
and differences of consecutive terms.I While convolution is a useful tool, it is a bit tedious to
analyze. Fortunately there is a result that makes it reallyeasy to analyze in the transform domain:
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
The Convolution Theorem
ConvolutionThe Convolution Theorem
Theorem (The Convolution Theorem)Let h and x be bi-infinite sequences with Fourier series H(ω)and X (ω), respectively. If y = h ∗ x, then
Y (ω) = H(ω)X (ω)
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
The Convolution Theorem
ConvolutionThe Convolution Theorem
I The Convolution Theorem takes convolution to simplemultiplication in the Fourier domain!
I The proof is straightforward and we do it in class.I You can also use the Convolution Theorem to compute
convolutions.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
The Convolution Theorem
ConvolutionThe Convolution Theorem
I The Convolution Theorem takes convolution to simplemultiplication in the Fourier domain!
I The proof is straightforward and we do it in class.I You can also use the Convolution Theorem to compute
convolutions.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
The Convolution Theorem
ConvolutionThe Convolution Theorem
I The Convolution Theorem takes convolution to simplemultiplication in the Fourier domain!
I The proof is straightforward and we do it in class.I You can also use the Convolution Theorem to compute
convolutions.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
The Convolution Theorem
ConvolutionThe Convolution Theorem
I The Convolution Theorem takes convolution to simplemultiplication in the Fourier domain!
I The proof is straightforward and we do it in class.I You can also use the Convolution Theorem to compute
convolutions.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
The Convolution Theorem
ConvolutionThe Convolution Theorem
Suppose h is the bi-infinite sequence with h5 = 1 and all otherhk = 0. Let x be any bi-infinite sequence. Compute
y = h ∗ x
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
The Convolution Theorem
ConvolutionThe Convolution Theorem
The Fourier series for H(ω) is simply H(ω) = e5iω so
Y (ω) = H(ω)X (ω)
=∑
k
xkei(k+5)ω
=∑
k
xk−5eikω
so yn = xn−5.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
The Convolution Theorem
ConvolutionThe Convolution Theorem
Convolve h where h0 = h1 = 1 and all other hk = 0 with itself.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
The Convolution Theorem
ConvolutionThe Convolution Theorem
The Fourier series for H(ω) is H(ω) = 1 + eiω so
Y (ω) = H(ω)H(ω)
= (1 + eiω)(1 + eiω)
= 1 + 2eiω + e2iω
so y0 = y2 = 1, y1 = 2, and all other yn = 0.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Types of Filters
FiltersTypes of Filters
I For our purposes, we will think of a filter as the processor hin the definition of convolution.
I We will identify filters by some characteristics:I A filter h is said to be causal if hk = 0 whenever k < 0.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Types of Filters
FiltersTypes of Filters
I For our purposes, we will think of a filter as the processor hin the definition of convolution.
I We will identify filters by some characteristics:I A filter h is said to be causal if hk = 0 whenever k < 0.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Types of Filters
FiltersTypes of Filters
I For our purposes, we will think of a filter as the processor hin the definition of convolution.
I We will identify filters by some characteristics:I A filter h is said to be causal if hk = 0 whenever k < 0.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Types of Filters
FiltersTypes of Filters
I For our purposes, we will think of a filter as the processor hin the definition of convolution.
I We will identify filters by some characteristics:I A filter h is said to be causal if hk = 0 whenever k < 0.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Types of Filters
FiltersTypes of Filters
I If we convolve x with causal filter h, we obtain
yn =∑
k
hkxn−k
=∞∑
k=0
hkxn−k
= h0xn + h1xn−1 + . . .
I So when h is causal, yn is formed from xn and thepredecessors of xn.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Types of Filters
FiltersTypes of Filters
I If we convolve x with causal filter h, we obtain
yn =∑
k
hkxn−k
=∞∑
k=0
hkxn−k
= h0xn + h1xn−1 + . . .
I So when h is causal, yn is formed from xn and thepredecessors of xn.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Types of Filters
FiltersTypes of Filters
I If we convolve x with causal filter h, we obtain
yn =∑
k
hkxn−k
=∞∑
k=0
hkxn−k
= h0xn + h1xn−1 + . . .
I So when h is causal, yn is formed from xn and thepredecessors of xn.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Types of Filters
FiltersTypes of Filters
I Until we consider biorthogonal wavelet transforms, we willonly work with so-called Finite Impulse Response filters.
I Suppose h is causal and let L > 0, L ∈ Z. If hk = 0 fork > L, h0, hL 6= 0, then we say h is a finite impulseresponse or FIR filter.
I If h is FIR with hk ≥ 0 and∑k
hk = 1, then it is often
convenient to think of h as moving averages - we arecomputing the weighted average of xn, xn−1, . . . , xn−L.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Types of Filters
FiltersTypes of Filters
I Until we consider biorthogonal wavelet transforms, we willonly work with so-called Finite Impulse Response filters.
I Suppose h is causal and let L > 0, L ∈ Z. If hk = 0 fork > L, h0, hL 6= 0, then we say h is a finite impulseresponse or FIR filter.
I If h is FIR with hk ≥ 0 and∑k
hk = 1, then it is often
convenient to think of h as moving averages - we arecomputing the weighted average of xn, xn−1, . . . , xn−L.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Types of Filters
FiltersTypes of Filters
I Until we consider biorthogonal wavelet transforms, we willonly work with so-called Finite Impulse Response filters.
I Suppose h is causal and let L > 0, L ∈ Z. If hk = 0 fork > L, h0, hL 6= 0, then we say h is a finite impulseresponse or FIR filter.
I If h is FIR with hk ≥ 0 and∑k
hk = 1, then it is often
convenient to think of h as moving averages - we arecomputing the weighted average of xn, xn−1, . . . , xn−L.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Types of Filters
FiltersTypes of Filters
I Until we consider biorthogonal wavelet transforms, we willonly work with so-called Finite Impulse Response filters.
I Suppose h is causal and let L > 0, L ∈ Z. If hk = 0 fork > L, h0, hL 6= 0, then we say h is a finite impulseresponse or FIR filter.
I If h is FIR with hk ≥ 0 and∑k
hk = 1, then it is often
convenient to think of h as moving averages - we arecomputing the weighted average of xn, xn−1, . . . , xn−L.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
I Recall the averaging filter h = (h0, h1) = (12 , 1
2).I If we convolve h with a bi-infinite sequence u consisting
entirely of 1’s, we obtain
yn =∑
k
hkun−k =12(un + un−1) = 1 = un
I If we convolve h with v where vk = (−1)k , then we obtain
yn =∑
k
hkvn−k =12(vn + vn−1) = 0
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
I Recall the averaging filter h = (h0, h1) = (12 , 1
2).I If we convolve h with a bi-infinite sequence u consisting
entirely of 1’s, we obtain
yn =∑
k
hkun−k =12(un + un−1) = 1 = un
I If we convolve h with v where vk = (−1)k , then we obtain
yn =∑
k
hkvn−k =12(vn + vn−1) = 0
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
I Recall the averaging filter h = (h0, h1) = (12 , 1
2).I If we convolve h with a bi-infinite sequence u consisting
entirely of 1’s, we obtain
yn =∑
k
hkun−k =12(un + un−1) = 1 = un
I If we convolve h with v where vk = (−1)k , then we obtain
yn =∑
k
hkvn−k =12(vn + vn−1) = 0
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
I Recall the averaging filter h = (h0, h1) = (12 , 1
2).I If we convolve h with a bi-infinite sequence u consisting
entirely of 1’s, we obtain
yn =∑
k
hkun−k =12(un + un−1) = 1 = un
I If we convolve h with v where vk = (−1)k , then we obtain
yn =∑
k
hkvn−k =12(vn + vn−1) = 0
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
I u and v are important testing sequences for filters. Thesequence u contains no oscillations whatsoever - it isconstantly 1.
I On the other hand, elements of v change sign at eachelement - it is highly oscillatory.
I Note that h applied to u returned u while h applied to vproduced the 0 sequence.
I Filters that generally reproduce low oscillatory signals andannihilate (or dampen) high oscillatory signals are knownas lowpass filters.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
I u and v are important testing sequences for filters. Thesequence u contains no oscillations whatsoever - it isconstantly 1.
I On the other hand, elements of v change sign at eachelement - it is highly oscillatory.
I Note that h applied to u returned u while h applied to vproduced the 0 sequence.
I Filters that generally reproduce low oscillatory signals andannihilate (or dampen) high oscillatory signals are knownas lowpass filters.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
I u and v are important testing sequences for filters. Thesequence u contains no oscillations whatsoever - it isconstantly 1.
I On the other hand, elements of v change sign at eachelement - it is highly oscillatory.
I Note that h applied to u returned u while h applied to vproduced the 0 sequence.
I Filters that generally reproduce low oscillatory signals andannihilate (or dampen) high oscillatory signals are knownas lowpass filters.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
I u and v are important testing sequences for filters. Thesequence u contains no oscillations whatsoever - it isconstantly 1.
I On the other hand, elements of v change sign at eachelement - it is highly oscillatory.
I Note that h applied to u returned u while h applied to vproduced the 0 sequence.
I Filters that generally reproduce low oscillatory signals andannihilate (or dampen) high oscillatory signals are knownas lowpass filters.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
I u and v are important testing sequences for filters. Thesequence u contains no oscillations whatsoever - it isconstantly 1.
I On the other hand, elements of v change sign at eachelement - it is highly oscillatory.
I Note that h applied to u returned u while h applied to vproduced the 0 sequence.
I Filters that generally reproduce low oscillatory signals andannihilate (or dampen) high oscillatory signals are knownas lowpass filters.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
The modulus |H(ω)| of a lowpass filter typically looks like:
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
In general, we will ask that if h is a lowpass filter, then
H(0) = 1 and H(π) = 0
This is equivalent to
L∑k=0
hk = 1 andL∑
k=0
(−1)khk = 0
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
Note that the ideal lowpass filter would look like
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
But it is easy to show that this 2π-periodic functions has aFourier series consisting of infinitely many nonzero values so itis not desirable for applications.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
I The filter g = (g0, g1) = (12 ,−1
2), when convolved with uproduces
yn =∑
k
gkun−k =12(un − un−1) = 0
I If we convolve g with v we have
yn =∑
k
gkvn−k =12(vn − vn−1) = (−1)n = vn
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
I The filter g = (g0, g1) = (12 ,−1
2), when convolved with uproduces
yn =∑
k
gkun−k =12(un − un−1) = 0
I If we convolve g with v we have
yn =∑
k
gkvn−k =12(vn − vn−1) = (−1)n = vn
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
I The filter g = (g0, g1) = (12 ,−1
2), when convolved with uproduces
yn =∑
k
gkun−k =12(un − un−1) = 0
I If we convolve g with v we have
yn =∑
k
gkvn−k =12(vn − vn−1) = (−1)n = vn
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
I Thus g annihilates the non-oscillatory signal u andpreserves the highly oscillatory signal v.
I Such a filter is called a highpass filter.I The Fourier series for g is
G(ω) =12− 1
2eiω
and note that G(0) = 0 and G(π) = 1.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
I Thus g annihilates the non-oscillatory signal u andpreserves the highly oscillatory signal v.
I Such a filter is called a highpass filter.I The Fourier series for g is
G(ω) =12− 1
2eiω
and note that G(0) = 0 and G(π) = 1.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
I Thus g annihilates the non-oscillatory signal u andpreserves the highly oscillatory signal v.
I Such a filter is called a highpass filter.I The Fourier series for g is
G(ω) =12− 1
2eiω
and note that G(0) = 0 and G(π) = 1.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
I Thus g annihilates the non-oscillatory signal u andpreserves the highly oscillatory signal v.
I Such a filter is called a highpass filter.I The Fourier series for g is
G(ω) =12− 1
2eiω
and note that G(0) = 0 and G(π) = 1.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
The graph of |G(ω)| is
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
I It is easy to build a highpass filter from a lowpass filter.I If h is a lowpass filter with H(0) = 1 and H(π) = 0, thenI Take G(ω) = H(ω + π). Then G(0) = H(π) = 0 and
G(π) = H(2π) = H(0) = 1.I Note that
G(ω) = H(ω + π) =∑
k
hkeik(ω+π) =∑
k
hk (−1)keikω
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
I It is easy to build a highpass filter from a lowpass filter.I If h is a lowpass filter with H(0) = 1 and H(π) = 0, thenI Take G(ω) = H(ω + π). Then G(0) = H(π) = 0 and
G(π) = H(2π) = H(0) = 1.I Note that
G(ω) = H(ω + π) =∑
k
hkeik(ω+π) =∑
k
hk (−1)keikω
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
I It is easy to build a highpass filter from a lowpass filter.I If h is a lowpass filter with H(0) = 1 and H(π) = 0, thenI Take G(ω) = H(ω + π). Then G(0) = H(π) = 0 and
G(π) = H(2π) = H(0) = 1.I Note that
G(ω) = H(ω + π) =∑
k
hkeik(ω+π) =∑
k
hk (−1)keikω
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
I It is easy to build a highpass filter from a lowpass filter.I If h is a lowpass filter with H(0) = 1 and H(π) = 0, thenI Take G(ω) = H(ω + π). Then G(0) = H(π) = 0 and
G(π) = H(2π) = H(0) = 1.I Note that
G(ω) = H(ω + π) =∑
k
hkeik(ω+π) =∑
k
hk (−1)keikω
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
I It is easy to build a highpass filter from a lowpass filter.I If h is a lowpass filter with H(0) = 1 and H(π) = 0, thenI Take G(ω) = H(ω + π). Then G(0) = H(π) = 0 and
G(π) = H(2π) = H(0) = 1.I Note that
G(ω) = H(ω + π) =∑
k
hkeik(ω+π) =∑
k
hk (−1)keikω
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
I We will build our wavelet transforms from lowpass andhighpass filters.
I Lowpass filters tend to produce a good approximation ofthe original signal in areas where the values of the signalsare homogeneous.
I Highpass filters tend to show us where significant changein the signal is taking place.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
I We will build our wavelet transforms from lowpass andhighpass filters.
I Lowpass filters tend to produce a good approximation ofthe original signal in areas where the values of the signalsare homogeneous.
I Highpass filters tend to show us where significant changein the signal is taking place.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
I We will build our wavelet transforms from lowpass andhighpass filters.
I Lowpass filters tend to produce a good approximation ofthe original signal in areas where the values of the signalsare homogeneous.
I Highpass filters tend to show us where significant changein the signal is taking place.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Lowpass and Highpass Filters
FiltersLowpass and Highpass Filters
I We will build our wavelet transforms from lowpass andhighpass filters.
I Lowpass filters tend to produce a good approximation ofthe original signal in areas where the values of the signalsare homogeneous.
I Highpass filters tend to show us where significant changein the signal is taking place.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution as a Matrix ProductThe Convolution Matrix
I Since we will build our wavelet transforms usingconvolutions with lowpass and highpass filters, it is naturalto ask what convolution looks like as a matrix.
I Let’s consider a FIR filter (h0, . . . , hL).I If we think of y = Hx, we have
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution as a Matrix ProductThe Convolution Matrix
I Since we will build our wavelet transforms usingconvolutions with lowpass and highpass filters, it is naturalto ask what convolution looks like as a matrix.
I Let’s consider a FIR filter (h0, . . . , hL).I If we think of y = Hx, we have
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution as a Matrix ProductThe Convolution Matrix
I Since we will build our wavelet transforms usingconvolutions with lowpass and highpass filters, it is naturalto ask what convolution looks like as a matrix.
I Let’s consider a FIR filter (h0, . . . , hL).I If we think of y = Hx, we have
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution as a Matrix ProductThe Convolution Matrix
I Since we will build our wavelet transforms usingconvolutions with lowpass and highpass filters, it is naturalto ask what convolution looks like as a matrix.
I Let’s consider a FIR filter (h0, . . . , hL).I If we think of y = Hx, we have
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution as a Matrix ProductThe Convolution Matrix
...y−2y−1y0y1...
yn...
= H
...x−2x−1x0x1...
xn...
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution as a Matrix ProductThe Convolution Matrix
H =
. . .. . .
. . .. . .
. . .hL . . . h2 h1 h0 0 0 0 0 0 0 0 00 hL . . . h2 h1 h0 0 0 0 0 0 0 0
. . . 0 0 hL . . . h2 h1 h0 0 0 0 0 0 0 . . .0 0 0 hL . . . h2 h1 h0 0 0 0 0 00 0 0 0 hL . . . h2 h1 h0 0 0 0 0
. . .. . .
. . .. . .
. . .. . .
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution as a Matrix ProductThe Convolution Matrix
So in the case where h is an FIR filter, the matrix H is a lowertriangular matrix with h0 down the main diagonal, h1 on the firstsubdiagonal and so on.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution as a Matrix ProductDe-Convolution
I It is natural to ask if we can invert or de-convolve theprocess.
I We can appeal to the convolution theorem here:
Y (ω) = H(ω)X (ω) ⇒ X (ω) =Y (ω)
H(ω)
I As long as H(ω) 6= 0, then theoretically we could write 1H(ω)
as a Fourier series M(ω) and the Fourier coefficients mkconvolved with yk would produce xk .
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution as a Matrix ProductDe-Convolution
I It is natural to ask if we can invert or de-convolve theprocess.
I We can appeal to the convolution theorem here:
Y (ω) = H(ω)X (ω) ⇒ X (ω) =Y (ω)
H(ω)
I As long as H(ω) 6= 0, then theoretically we could write 1H(ω)
as a Fourier series M(ω) and the Fourier coefficients mkconvolved with yk would produce xk .
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution as a Matrix ProductDe-Convolution
I It is natural to ask if we can invert or de-convolve theprocess.
I We can appeal to the convolution theorem here:
Y (ω) = H(ω)X (ω) ⇒ X (ω) =Y (ω)
H(ω)
I As long as H(ω) 6= 0, then theoretically we could write 1H(ω)
as a Fourier series M(ω) and the Fourier coefficients mkconvolved with yk would produce xk .
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution as a Matrix ProductDe-Convolution
I It is natural to ask if we can invert or de-convolve theprocess.
I We can appeal to the convolution theorem here:
Y (ω) = H(ω)X (ω) ⇒ X (ω) =Y (ω)
H(ω)
I As long as H(ω) 6= 0, then theoretically we could write 1H(ω)
as a Fourier series M(ω) and the Fourier coefficients mkconvolved with yk would produce xk .
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution as a Matrix ProductDe-Convolution
I There’s only one small hitch here . . .
I We want to convolve with lowpass filter h and highpassfilter g, but
I h lowpass ⇒ H(π) = 0I g highpass ⇒ G(0) = 0
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution as a Matrix ProductDe-Convolution
I There’s only one small hitch here . . .
I We want to convolve with lowpass filter h and highpassfilter g, but
I h lowpass ⇒ H(π) = 0I g highpass ⇒ G(0) = 0
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution as a Matrix ProductDe-Convolution
I There’s only one small hitch here . . .
I We want to convolve with lowpass filter h and highpassfilter g, but
I h lowpass ⇒ H(π) = 0I g highpass ⇒ G(0) = 0
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution as a Matrix ProductDe-Convolution
I There’s only one small hitch here . . .
I We want to convolve with lowpass filter h and highpassfilter g, but
I h lowpass ⇒ H(π) = 0I g highpass ⇒ G(0) = 0
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution as a Matrix ProductDe-Convolution
I There’s only one small hitch here . . .
I We want to convolve with lowpass filter h and highpassfilter g, but
I h lowpass ⇒ H(π) = 0I g highpass ⇒ G(0) = 0
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution as a Matrix ProductDe-Convolution
I So we can’t de-convolve when using lowpass or highpassfilters!
I This should make sense - if you are given a list of averagesof consecutive numbers, you have no way of knowing whatthe original numbers are.
I We’ve hit a roadblock - that means . . .
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution as a Matrix ProductDe-Convolution
I So we can’t de-convolve when using lowpass or highpassfilters!
I This should make sense - if you are given a list of averagesof consecutive numbers, you have no way of knowing whatthe original numbers are.
I We’ve hit a roadblock - that means . . .
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution as a Matrix ProductDe-Convolution
I So we can’t de-convolve when using lowpass or highpassfilters!
I This should make sense - if you are given a list of averagesof consecutive numbers, you have no way of knowing whatthe original numbers are.
I We’ve hit a roadblock - that means . . .
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution as a Matrix ProductDe-Convolution
I So we can’t de-convolve when using lowpass or highpassfilters!
I This should make sense - if you are given a list of averagesof consecutive numbers, you have no way of knowing whatthe original numbers are.
I We’ve hit a roadblock - that means . . .
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution as a Matrix ProductDe-Convolution
I Workshop Over! We’ll just goof off for the next 3 days.I Just kidding . . . let’s look at some student issues before
breaking.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution as a Matrix ProductDe-Convolution
I Workshop Over! We’ll just goof off for the next 3 days.I Just kidding . . . let’s look at some student issues before
breaking.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Convolution as a Matrix ProductDe-Convolution
I Workshop Over! We’ll just goof off for the next 3 days.I Just kidding . . . let’s look at some student issues before
breaking.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Teaching Ideas
In The ClassroomTeaching Ideas
I We work lots and lots of basic convolution products.I I make sure to show students that multiplication is
convolution - they really like this - see Exercise 3 on page158.
I The conditions H(0) = 1 ⇒∑
hk = 1 andH(π) = 0 ⇒
∑(−1)khk = 0 come up repeatedly during the
remainder of the course.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Teaching Ideas
In The ClassroomTeaching Ideas
I We work lots and lots of basic convolution products.I I make sure to show students that multiplication is
convolution - they really like this - see Exercise 3 on page158.
I The conditions H(0) = 1 ⇒∑
hk = 1 andH(π) = 0 ⇒
∑(−1)khk = 0 come up repeatedly during the
remainder of the course.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Teaching Ideas
In The ClassroomTeaching Ideas
I We work lots and lots of basic convolution products.I I make sure to show students that multiplication is
convolution - they really like this - see Exercise 3 on page158.
I The conditions H(0) = 1 ⇒∑
hk = 1 andH(π) = 0 ⇒
∑(−1)khk = 0 come up repeatedly during the
remainder of the course.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Teaching Ideas
In The ClassroomTeaching Ideas
I We work lots and lots of basic convolution products.I I make sure to show students that multiplication is
convolution - they really like this - see Exercise 3 on page158.
I The conditions H(0) = 1 ⇒∑
hk = 1 andH(π) = 0 ⇒
∑(−1)khk = 0 come up repeatedly during the
remainder of the course.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Teaching Ideas
In The ClassroomTeaching Ideas
I I usually have them try to build their own lowpass/highpassfilters. You can add some conditions too - like a lowpassfilter with H ′(π) = 0.
I It is very important that they make the connect - thenumbers in the convolution matrix are the filter elementsreversed and they are also the Fourier coefficients. There’sa lot going on here.
I For future work, it is important to have them work throughproblems 6-11 in Section 5.1.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Teaching Ideas
In The ClassroomTeaching Ideas
I I usually have them try to build their own lowpass/highpassfilters. You can add some conditions too - like a lowpassfilter with H ′(π) = 0.
I It is very important that they make the connect - thenumbers in the convolution matrix are the filter elementsreversed and they are also the Fourier coefficients. There’sa lot going on here.
I For future work, it is important to have them work throughproblems 6-11 in Section 5.1.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Teaching Ideas
In The ClassroomTeaching Ideas
I I usually have them try to build their own lowpass/highpassfilters. You can add some conditions too - like a lowpassfilter with H ′(π) = 0.
I It is very important that they make the connect - thenumbers in the convolution matrix are the filter elementsreversed and they are also the Fourier coefficients. There’sa lot going on here.
I For future work, it is important to have them work throughproblems 6-11 in Section 5.1.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Teaching Ideas
In The ClassroomTeaching Ideas
I I usually have them try to build their own lowpass/highpassfilters. You can add some conditions too - like a lowpassfilter with H ′(π) = 0.
I It is very important that they make the connect - thenumbers in the convolution matrix are the filter elementsreversed and they are also the Fourier coefficients. There’sa lot going on here.
I For future work, it is important to have them work throughproblems 6-11 in Section 5.1.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Computer Usage
In The ClassroomComputer Usage
I I am in the process of developing an animation routine forconvolving vectors, but other than that I don’t use thecomputer for much more than graphing moduli of Fourierseries of filters and manipulating Fourier series and seeingthe results visually.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Student Difficulties
In The ClassroomStudent Difficulties
I Getting the convolutions correct when both h and x are offinite length takes a lot of practice.
I The properties on H(ω), G(ω) to make the filters lowpass,highpass, respectively, never seems to seek in. This isused throughout the course and it gets frustrating having torepeat them.
I Students seem generally uncomfortable manipulating finitelength Fourier series. They are fine with looking at a graphof the modulus and determining if a filter islowpass/highpass, but they are hesitant to manipulate aFourier series to produce say a highpass filter from alowpass filter.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Student Difficulties
In The ClassroomStudent Difficulties
I Getting the convolutions correct when both h and x are offinite length takes a lot of practice.
I The properties on H(ω), G(ω) to make the filters lowpass,highpass, respectively, never seems to seek in. This isused throughout the course and it gets frustrating having torepeat them.
I Students seem generally uncomfortable manipulating finitelength Fourier series. They are fine with looking at a graphof the modulus and determining if a filter islowpass/highpass, but they are hesitant to manipulate aFourier series to produce say a highpass filter from alowpass filter.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Student Difficulties
In The ClassroomStudent Difficulties
I Getting the convolutions correct when both h and x are offinite length takes a lot of practice.
I The properties on H(ω), G(ω) to make the filters lowpass,highpass, respectively, never seems to seek in. This isused throughout the course and it gets frustrating having torepeat them.
I Students seem generally uncomfortable manipulating finitelength Fourier series. They are fine with looking at a graphof the modulus and determining if a filter islowpass/highpass, but they are hesitant to manipulate aFourier series to produce say a highpass filter from alowpass filter.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Student Difficulties
In The ClassroomStudent Difficulties
I Getting the convolutions correct when both h and x are offinite length takes a lot of practice.
I The properties on H(ω), G(ω) to make the filters lowpass,highpass, respectively, never seems to seek in. This isused throughout the course and it gets frustrating having torepeat them.
I Students seem generally uncomfortable manipulating finitelength Fourier series. They are fine with looking at a graphof the modulus and determining if a filter islowpass/highpass, but they are hesitant to manipulate aFourier series to produce say a highpass filter from alowpass filter.
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters
Today’s Schedule Fourier Series Convolution Filters Convolution as a Matrix Product In the Classroom
Today’s Schedule
9:00-10:15 Lecture One: Why Wavelets?10:15-10:30 Coffee Break (OSS 235)10:30-11:45 Lecture Two: Digital Images, Measures, and
Huffman Codes12:00-1:00 Lunch (Cafeteria)1:30-2:45 Lecture Three: Fourier Series, Convolution and
Filters2:45-3:00 ⇒Coffee Break (OSS 235)3:00-4:15 Lecture Four: 1D and 2D Haar Transforms5:30-6:30 Dinner (Cafeteria)
Wednesday, 7 June, 2006 Lecture 3
Fourier Series, Convolution, and Filters