8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
8: Correlation
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 1 / 11
Cross-Correlation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
The cross-correlation between two signals u(t) and v(t) is
w(t) = u(t) ⊗ v(t) ,
E1.10 Fourier Series and Transforms (2015-5585)
R∞
∗
u
(τ )v(τ + t)dτ
−∞
Fourier Transform - Correlation: 8 – 2 / 11
Cross-Correlation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
The cross-correlation between two signals u(t) and v(t) is
w(t) = u(t) ⊗ v(t) ,
E1.10 Fourier Series and Transforms (2015-5585)
=
R∞
∗
u
(τ )v(τ + t)dτ
−∞
R∞
−∞
u∗ (τ − t)v(τ )dτ
[sub: τ → τ − t]
Fourier Transform - Correlation: 8 – 2 / 11
Cross-Correlation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
The cross-correlation between two signals u(t) and v(t) is
w(t) = u(t) ⊗ v(t) ,
=
R∞
∗
u
(τ )v(τ + t)dτ
−∞
R∞
−∞
u∗ (τ − t)v(τ )dτ
[sub: τ → τ − t]
The complex conjugate, u∗ (τ ) makes no difference if u(t) is real-valued
but makes the definition work even if u(t) is complex-valued.
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 2 / 11
Cross-Correlation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
The cross-correlation between two signals u(t) and v(t) is
w(t) = u(t) ⊗ v(t) ,
=
R∞
∗
u
(τ )v(τ + t)dτ
−∞
R∞
−∞
u∗ (τ − t)v(τ )dτ
[sub: τ → τ − t]
The complex conjugate, u∗ (τ ) makes no difference if u(t) is real-valued
but makes the definition work even if u(t) is complex-valued.
Correlation versus Convolution:
R∞
u(t) ⊗ v(t) = −∞ u∗ (τ )v(τ + t)dτ
R∞
u(t) ∗ v(t) = −∞ u(τ )v(t − τ )dτ
E1.10 Fourier Series and Transforms (2015-5585)
[correlation]
[convolution]
Fourier Transform - Correlation: 8 – 2 / 11
Cross-Correlation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
The cross-correlation between two signals u(t) and v(t) is
w(t) = u(t) ⊗ v(t) ,
=
R∞
∗
u
(τ )v(τ + t)dτ
−∞
R∞
−∞
u∗ (τ − t)v(τ )dτ
[sub: τ → τ − t]
The complex conjugate, u∗ (τ ) makes no difference if u(t) is real-valued
but makes the definition work even if u(t) is complex-valued.
Correlation versus Convolution:
R∞
u(t) ⊗ v(t) = −∞ u∗ (τ )v(τ + t)dτ
[correlation]
R∞
u(t) ∗ v(t) = −∞ u(τ )v(t − τ )dτ
[convolution]
Unlike convolution, the integration variable, τ , has the same sign in the
arguments of u(· · · ) and v(· · · ) so the arguments have a constant
difference instead of a constant sum (i.e. v(t) is not time-flipped).
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 2 / 11
Cross-Correlation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
The cross-correlation between two signals u(t) and v(t) is
w(t) = u(t) ⊗ v(t) ,
=
R∞
∗
u
(τ )v(τ + t)dτ
−∞
R∞
−∞
u∗ (τ − t)v(τ )dτ
[sub: τ → τ − t]
The complex conjugate, u∗ (τ ) makes no difference if u(t) is real-valued
but makes the definition work even if u(t) is complex-valued.
Correlation versus Convolution:
R∞
u(t) ⊗ v(t) = −∞ u∗ (τ )v(τ + t)dτ
[correlation]
R∞
u(t) ∗ v(t) = −∞ u(τ )v(t − τ )dτ
[convolution]
Unlike convolution, the integration variable, τ , has the same sign in the
arguments of u(· · · ) and v(· · · ) so the arguments have a constant
difference instead of a constant sum (i.e. v(t) is not time-flipped).
Notes: (a) The argument of w(t) is called the “lag” (= delay of u versus v ).
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 2 / 11
Cross-Correlation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
The cross-correlation between two signals u(t) and v(t) is
w(t) = u(t) ⊗ v(t) ,
=
R∞
∗
u
(τ )v(τ + t)dτ
−∞
R∞
−∞
u∗ (τ − t)v(τ )dτ
[sub: τ → τ − t]
The complex conjugate, u∗ (τ ) makes no difference if u(t) is real-valued
but makes the definition work even if u(t) is complex-valued.
Correlation versus Convolution:
R∞
u(t) ⊗ v(t) = −∞ u∗ (τ )v(τ + t)dτ
[correlation]
R∞
u(t) ∗ v(t) = −∞ u(τ )v(t − τ )dτ
[convolution]
Unlike convolution, the integration variable, τ , has the same sign in the
arguments of u(· · · ) and v(· · · ) so the arguments have a constant
difference instead of a constant sum (i.e. v(t) is not time-flipped).
Notes: (a) The argument of w(t) is called the “lag” (= delay of u versus v ).
(b) Some people write u(t) ⋆ v(t) instead of u(t) ⊗ v(t).
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 2 / 11
Cross-Correlation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
The cross-correlation between two signals u(t) and v(t) is
w(t) = u(t) ⊗ v(t) ,
=
R∞
∗
u
(τ )v(τ + t)dτ
−∞
R∞
−∞
u∗ (τ − t)v(τ )dτ
[sub: τ → τ − t]
The complex conjugate, u∗ (τ ) makes no difference if u(t) is real-valued
but makes the definition work even if u(t) is complex-valued.
Correlation versus Convolution:
R∞
u(t) ⊗ v(t) = −∞ u∗ (τ )v(τ + t)dτ
[correlation]
R∞
u(t) ∗ v(t) = −∞ u(τ )v(t − τ )dτ
[convolution]
Unlike convolution, the integration variable, τ , has the same sign in the
arguments of u(· · · ) and v(· · · ) so the arguments have a constant
difference instead of a constant sum (i.e. v(t) is not time-flipped).
Notes: (a) The argument of w(t) is called the “lag” (= delay of u versus v ).
(b) Some people write u(t) ⋆ v(t) instead of u(t) ⊗ v(t).
(c) Some swap u and v and/or negate t in the integral.
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 2 / 11
Cross-Correlation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
The cross-correlation between two signals u(t) and v(t) is
w(t) = u(t) ⊗ v(t) ,
=
R∞
∗
u
(τ )v(τ + t)dτ
−∞
R∞
−∞
u∗ (τ − t)v(τ )dτ
[sub: τ → τ − t]
The complex conjugate, u∗ (τ ) makes no difference if u(t) is real-valued
but makes the definition work even if u(t) is complex-valued.
Correlation versus Convolution:
R∞
u(t) ⊗ v(t) = −∞ u∗ (τ )v(τ + t)dτ
[correlation]
R∞
u(t) ∗ v(t) = −∞ u(τ )v(t − τ )dτ
[convolution]
Unlike convolution, the integration variable, τ , has the same sign in the
arguments of u(· · · ) and v(· · · ) so the arguments have a constant
difference instead of a constant sum (i.e. v(t) is not time-flipped).
Notes: (a) The argument of w(t) is called the “lag” (= delay of u versus v ).
(b) Some people write u(t) ⋆ v(t) instead of u(t) ⊗ v(t).
(c) Some swap u and v and/or negate t in the integral.
It is all rather inconsistent /.
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 2 / 11
Signal Matching
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
Cross correlation is used to find where two
signals match
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 3 / 11
Signal Matching
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
Cross correlation is used to find where two
signals match: u(t) is the test waveform.
E1.10 Fourier Series and Transforms (2015-5585)
1
u(t)
8: Correlation
0.5
0
0
200
400
600
Fourier Transform - Correlation: 8 – 3 / 11
Signal Matching
Cross correlation is used to find where two
signals match: u(t) is the test waveform.
1
u(t)
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
0.5
0
0
Example 1:
v(t) contains u(t) with an unknown delay
and added noise.
E1.10 Fourier Series and Transforms (2015-5585)
200
400
600
1
v(t)
8: Correlation
0.5
0
0
200
400
600
800
Fourier Transform - Correlation: 8 – 3 / 11
Signal Matching
Cross correlation is used to find where two
signals match: u(t) is the test waveform.
1
u(t)
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
0.5
0
0
Example 1:
v(t) contains u(t) with an unknown delay
and added noise.
200
400
600
1
v(t)
8: Correlation
0.5
0
0
200
400
600
800
w(t) = u(t) ⊗ v(t)
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 3 / 11
Signal Matching
Cross correlation is used to find where two
signals match: u(t) is the test waveform.
1
u(t)
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
0.5
0
0
Example 1:
v(t) contains u(t) with an unknown delay
and added noise.
200
400
600
1
v(t)
8: Correlation
0.5
0
0
200
400
600
800
w(t) = u(t)
R ∗ ⊗ v(t)
= u (τ − t)v(τ )dt
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 3 / 11
Signal Matching
1
u(t)
Cross correlation is used to find where two
signals match: u(t) is the test waveform.
0.5
0
0
Example 1:
v(t) contains u(t) with an unknown delay
and added noise.
w(t) = u(t)
R ∗ ⊗ v(t)
= u (τ − t)v(τ )dt
E1.10 Fourier Series and Transforms (2015-5585)
200
400
600
1
v(t)
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
0.5
0
0
200
400
600
800
0
200
400
600
800
20
w(t)
8: Correlation
10
0
Fourier Transform - Correlation: 8 – 3 / 11
Signal Matching
1
u(t)
Cross correlation is used to find where two
signals match: u(t) is the test waveform.
0.5
0
0
Example 1:
v(t) contains u(t) with an unknown delay
and added noise.
w(t) = u(t)
R ∗ ⊗ v(t)
= u (τ − t)v(τ )dt gives a peak
at the time lag where u(τ − t) best
matches v(τ )
E1.10 Fourier Series and Transforms (2015-5585)
200
400
600
1
v(t)
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
0.5
0
0
200
400
600
800
0
200
400
600
800
20
w(t)
8: Correlation
10
0
Fourier Transform - Correlation: 8 – 3 / 11
Signal Matching
1
u(t)
Cross correlation is used to find where two
signals match: u(t) is the test waveform.
0.5
0
0
Example 1:
v(t) contains u(t) with an unknown delay
and added noise.
w(t) = u(t)
R ∗ ⊗ v(t)
= u (τ − t)v(τ )dt gives a peak
at the time lag where u(τ − t) best
matches v(τ ); in this case at t = 450
E1.10 Fourier Series and Transforms (2015-5585)
200
400
600
1
v(t)
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
0.5
0
0
200
400
600
800
0
200
400
600
800
20
w(t)
8: Correlation
10
0
Fourier Transform - Correlation: 8 – 3 / 11
Signal Matching
1
u(t)
Cross correlation is used to find where two
signals match: u(t) is the test waveform.
0.5
0
0
w(t) = u(t)
R ∗ ⊗ v(t)
= u (τ − t)v(τ )dt gives a peak
at the time lag where u(τ − t) best
matches v(τ ); in this case at t = 450
Example 2:
y(t) is the same as v(t) with more noise
E1.10 Fourier Series and Transforms (2015-5585)
200
400
600
1
v(t)
Example 1:
v(t) contains u(t) with an unknown delay
and added noise.
0.5
0
0
200
400
600
800
0
200
400
600
800
0
200
400
600
800
20
w(t)
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
10
0
y(t)
8: Correlation
2
0
-2
Fourier Transform - Correlation: 8 – 3 / 11
Signal Matching
1
u(t)
Cross correlation is used to find where two
signals match: u(t) is the test waveform.
0.5
0
0
w(t) = u(t)
R ∗ ⊗ v(t)
= u (τ − t)v(τ )dt gives a peak
at the time lag where u(τ − t) best
matches v(τ ); in this case at t = 450
Example 2:
y(t) is the same as v(t) with more noise
z(t) = u(t) ⊗ y(t) can still detect the
correct time delay (hard for humans)
E1.10 Fourier Series and Transforms (2015-5585)
200
400
600
v(t)
1
0.5
0
0
200
400
600
800
0
200
400
600
800
0
200
400
600
800
0
200
400
600
800
20
w(t)
Example 1:
v(t) contains u(t) with an unknown delay
and added noise.
10
0
y(t)
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
2
0
-2
20
z(t)
8: Correlation
10
0
-10
Fourier Transform - Correlation: 8 – 3 / 11
Signal Matching
1
u(t)
Cross correlation is used to find where two
signals match: u(t) is the test waveform.
0.5
0
0
w(t) = u(t)
R ∗ ⊗ v(t)
= u (τ − t)v(τ )dt gives a peak
at the time lag where u(τ − t) best
matches v(τ ); in this case at t = 450
Example 2:
y(t) is the same as v(t) with more noise
z(t) = u(t) ⊗ y(t) can still detect the
correct time delay (hard for humans)
200
400
600
v(t)
1
0.5
0
0
200
400
600
800
0
200
400
600
800
0
200
400
600
800
0
200
400
600
800
0
200
400
600
800
20
w(t)
Example 1:
v(t) contains u(t) with an unknown delay
and added noise.
10
0
y(t)
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
2
0
-2
20
10
z(t)
8: Correlation
0
-10
Example 3:
p(t) contains −u(t)
E1.10 Fourier Series and Transforms (2015-5585)
p(t)
1
0
-1
Fourier Transform - Correlation: 8 – 3 / 11
Signal Matching
1
u(t)
Cross correlation is used to find where two
signals match: u(t) is the test waveform.
0.5
0
0
w(t) = u(t)
R ∗ ⊗ v(t)
= u (τ − t)v(τ )dt gives a peak
at the time lag where u(τ − t) best
matches v(τ ); in this case at t = 450
Example 2:
y(t) is the same as v(t) with more noise
z(t) = u(t) ⊗ y(t) can still detect the
correct time delay (hard for humans)
200
400
600
v(t)
1
0.5
0
0
200
400
600
800
0
200
400
600
800
0
200
400
600
800
0
200
400
600
800
0
200
400
600
800
20
w(t)
Example 1:
v(t) contains u(t) with an unknown delay
and added noise.
10
0
y(t)
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
2
0
-2
20
10
z(t)
8: Correlation
0
-10
Example 3:
p(t) contains −u(t) so that
q(t) = u(t) ⊗ p(t) has a negative peak
E1.10 Fourier Series and Transforms (2015-5585)
p(t)
1
0
-1
Fourier Transform - Correlation: 8 – 3 / 11
Signal Matching
1
u(t)
Cross correlation is used to find where two
signals match: u(t) is the test waveform.
0.5
0
0
w(t) = u(t)
R ∗ ⊗ v(t)
= u (τ − t)v(τ )dt gives a peak
at the time lag where u(τ − t) best
matches v(τ ); in this case at t = 450
Example 2:
y(t) is the same as v(t) with more noise
z(t) = u(t) ⊗ y(t) can still detect the
correct time delay (hard for humans)
200
400
600
v(t)
1
0.5
0
0
200
400
600
800
0
200
400
600
800
0
200
400
600
800
0
200
400
600
800
0
200
400
600
800
0
200
400
600
800
20
w(t)
Example 1:
v(t) contains u(t) with an unknown delay
and added noise.
10
0
y(t)
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
2
0
-2
20
10
z(t)
8: Correlation
0
-10
E1.10 Fourier Series and Transforms (2015-5585)
0
-1
0
q(t)
Example 3:
p(t) contains −u(t) so that
q(t) = u(t) ⊗ p(t) has a negative peak
p(t)
1
-10
-20
Fourier Transform - Correlation: 8 – 3 / 11
Cross-correlation as Convolution
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 4 / 11
Cross-correlation as Convolution
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define x(t) = u∗ (−t) then
x(t) ∗ v(t) ,
E1.10 Fourier Series and Transforms (2015-5585)
R∞
−∞
x(t − τ )v(τ )dτ
Fourier Transform - Correlation: 8 – 4 / 11
Cross-correlation as Convolution
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define x(t) = u∗ (−t) then
x(t) ∗ v(t) ,
E1.10 Fourier Series and Transforms (2015-5585)
R∞
x(t − τ )v(τ )dτ =
−∞
R∞
∗
u
(τ − t)v(τ )dτ
−∞
Fourier Transform - Correlation: 8 – 4 / 11
Cross-correlation as Convolution
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define x(t) = u∗ (−t) then
R∞
x(t) ∗ v(t) , −∞ x(t − τ )v(τ )dτ =
= u(t) ⊗ v(t)
E1.10 Fourier Series and Transforms (2015-5585)
R∞
∗
u
(τ − t)v(τ )dτ
−∞
Fourier Transform - Correlation: 8 – 4 / 11
Cross-correlation as Convolution
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define x(t) = u∗ (−t) then
R∞
x(t) ∗ v(t) , −∞ x(t − τ )v(τ )dτ =
= u(t) ⊗ v(t)
R∞
∗
u
(τ − t)v(τ )dτ
−∞
Fourier Transform of x(t):
X(f ) =
R∞
E1.10 Fourier Series and Transforms (2015-5585)
−i2πf t
x(t)e
dt
−∞
Fourier Transform - Correlation: 8 – 4 / 11
Cross-correlation as Convolution
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define x(t) = u∗ (−t) then
R∞
x(t) ∗ v(t) , −∞ x(t − τ )v(τ )dτ =
= u(t) ⊗ v(t)
Fourier Transform of x(t):
X(f ) =
R∞
E1.10 Fourier Series and Transforms (2015-5585)
−i2πf t
x(t)e
−∞
dt =
R∞
∗
u
(τ − t)v(τ )dτ
−∞
R∞
∗
−i2πf t
u
(−t)e
dt
−∞
Fourier Transform - Correlation: 8 – 4 / 11
Cross-correlation as Convolution
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define x(t) = u∗ (−t) then
R∞
x(t) ∗ v(t) , −∞ x(t − τ )v(τ )dτ =
= u(t) ⊗ v(t)
Fourier Transform of x(t):
X(f ) =
=
R∞
−i2πf t
x(t)e
−∞
R∞
dt =
R∞
∗
u
(τ − t)v(τ )dτ
−∞
R∞
∗
−i2πf t
u
(−t)e
dt
−∞
∗
i2πf t
u
(t)e
dt
−∞
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 4 / 11
Cross-correlation as Convolution
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define x(t) = u∗ (−t) then
R∞
x(t) ∗ v(t) , −∞ x(t − τ )v(τ )dτ =
= u(t) ⊗ v(t)
Fourier Transform of x(t):
R∞
R∞
∗
u
(τ − t)v(τ )dτ
−∞
R∞
dt = −∞ u∗ (−t)e−i2πf t dt
∗
R
R∞ ∗
∞
−i2πf t
i2πf t
dt
dt = −∞ u(t)e
= −∞ u (t)e
X(f ) =
E1.10 Fourier Series and Transforms (2015-5585)
−i2πf t
x(t)e
−∞
Fourier Transform - Correlation: 8 – 4 / 11
Cross-correlation as Convolution
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define x(t) = u∗ (−t) then
R∞
x(t) ∗ v(t) , −∞ x(t − τ )v(τ )dτ =
= u(t) ⊗ v(t)
Fourier Transform of x(t):
R∞
R∞
∗
u
(τ − t)v(τ )dτ
−∞
R∞
dt = −∞ u∗ (−t)e−i2πf t dt
∗
R
R∞ ∗
∞
−i2πf t
i2πf t
dt
dt = −∞ u(t)e
= −∞ u (t)e
X(f ) =
−i2πf t
x(t)e
−∞
= U ∗ (f )
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 4 / 11
Cross-correlation as Convolution
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define x(t) = u∗ (−t) then
R∞
x(t) ∗ v(t) , −∞ x(t − τ )v(τ )dτ =
= u(t) ⊗ v(t)
Fourier Transform of x(t):
R∞
R∞
∗
u
(τ − t)v(τ )dτ
−∞
R∞
dt = −∞ u∗ (−t)e−i2πf t dt
∗
R
R∞ ∗
∞
−i2πf t
i2πf t
dt
dt = −∞ u(t)e
= −∞ u (t)e
X(f ) =
−i2πf t
x(t)e
−∞
= U ∗ (f )
So w(t) = x(t) ∗ v(t) ⇒ W (f ) = X(f )V (f )
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 4 / 11
Cross-correlation as Convolution
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define x(t) = u∗ (−t) then
R∞
x(t) ∗ v(t) , −∞ x(t − τ )v(τ )dτ =
= u(t) ⊗ v(t)
Fourier Transform of x(t):
R∞
R∞
∗
u
(τ − t)v(τ )dτ
−∞
R∞
dt = −∞ u∗ (−t)e−i2πf t dt
∗
R
R∞ ∗
∞
−i2πf t
i2πf t
dt
dt = −∞ u(t)e
= −∞ u (t)e
X(f ) =
−i2πf t
x(t)e
−∞
= U ∗ (f )
So w(t) = x(t) ∗ v(t) ⇒ W (f ) = X(f )V (f ) = U ∗ (f )V (f )
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 4 / 11
Cross-correlation as Convolution
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define x(t) = u∗ (−t) then
R∞
x(t) ∗ v(t) , −∞ x(t − τ )v(τ )dτ =
= u(t) ⊗ v(t)
Fourier Transform of x(t):
R∞
R∞
∗
u
(τ − t)v(τ )dτ
−∞
R∞
dt = −∞ u∗ (−t)e−i2πf t dt
∗
R
R∞ ∗
∞
−i2πf t
i2πf t
dt
dt = −∞ u(t)e
= −∞ u (t)e
X(f ) =
−i2πf t
x(t)e
−∞
= U ∗ (f )
So w(t) = x(t) ∗ v(t) ⇒ W (f ) = X(f )V (f ) = U ∗ (f )V (f )
Hence the Cross-correlation theorem:
w(t) = u(t) ⊗ v(t)
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 4 / 11
Cross-correlation as Convolution
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define x(t) = u∗ (−t) then
R∞
x(t) ∗ v(t) , −∞ x(t − τ )v(τ )dτ =
= u(t) ⊗ v(t)
Fourier Transform of x(t):
R∞
R∞
∗
u
(τ − t)v(τ )dτ
−∞
R∞
dt = −∞ u∗ (−t)e−i2πf t dt
∗
R
R∞ ∗
∞
−i2πf t
i2πf t
dt
dt = −∞ u(t)e
= −∞ u (t)e
X(f ) =
−i2πf t
x(t)e
−∞
= U ∗ (f )
So w(t) = x(t) ∗ v(t) ⇒ W (f ) = X(f )V (f ) = U ∗ (f )V (f )
Hence the Cross-correlation theorem:
w(t) = u(t) ⊗ v(t)
= u∗ (−t) ∗ v(t)
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 4 / 11
Cross-correlation as Convolution
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define x(t) = u∗ (−t) then
R∞
x(t) ∗ v(t) , −∞ x(t − τ )v(τ )dτ =
= u(t) ⊗ v(t)
Fourier Transform of x(t):
R∞
R∞
∗
u
(τ − t)v(τ )dτ
−∞
R∞
dt = −∞ u∗ (−t)e−i2πf t dt
∗
R
R∞ ∗
∞
−i2πf t
i2πf t
dt
dt = −∞ u(t)e
= −∞ u (t)e
X(f ) =
−i2πf t
x(t)e
−∞
= U ∗ (f )
So w(t) = x(t) ∗ v(t) ⇒ W (f ) = X(f )V (f ) = U ∗ (f )V (f )
Hence the Cross-correlation theorem:
w(t) = u(t) ⊗ v(t)
= u∗ (−t) ∗ v(t)
E1.10 Fourier Series and Transforms (2015-5585)
⇔
W (f ) = U ∗ (f )V (f )
Fourier Transform - Correlation: 8 – 4 / 11
Cross-correlation as Convolution
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define x(t) = u∗ (−t) then
R∞
x(t) ∗ v(t) , −∞ x(t − τ )v(τ )dτ =
= u(t) ⊗ v(t)
Fourier Transform of x(t):
R∞
R∞
∗
u
(τ − t)v(τ )dτ
−∞
R∞
dt = −∞ u∗ (−t)e−i2πf t dt
∗
R
R∞ ∗
∞
−i2πf t
i2πf t
dt
dt = −∞ u(t)e
= −∞ u (t)e
X(f ) =
−i2πf t
x(t)e
−∞
= U ∗ (f )
So w(t) = x(t) ∗ v(t) ⇒ W (f ) = X(f )V (f ) = U ∗ (f )V (f )
Hence the Cross-correlation theorem:
w(t) = u(t) ⊗ v(t)
= u∗ (−t) ∗ v(t)
⇔
W (f ) = U ∗ (f )V (f )
Note that, unlike convolution, correlation is not associative or commutative:
v(t) ⊗ u(t) = v ∗ (−t) ∗ u(t)
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 4 / 11
Cross-correlation as Convolution
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define x(t) = u∗ (−t) then
R∞
x(t) ∗ v(t) , −∞ x(t − τ )v(τ )dτ =
= u(t) ⊗ v(t)
Fourier Transform of x(t):
R∞
R∞
∗
u
(τ − t)v(τ )dτ
−∞
R∞
dt = −∞ u∗ (−t)e−i2πf t dt
∗
R
R∞ ∗
∞
−i2πf t
i2πf t
dt
dt = −∞ u(t)e
= −∞ u (t)e
X(f ) =
−i2πf t
x(t)e
−∞
= U ∗ (f )
So w(t) = x(t) ∗ v(t) ⇒ W (f ) = X(f )V (f ) = U ∗ (f )V (f )
Hence the Cross-correlation theorem:
w(t) = u(t) ⊗ v(t)
= u∗ (−t) ∗ v(t)
⇔
W (f ) = U ∗ (f )V (f )
Note that, unlike convolution, correlation is not associative or commutative:
v(t) ⊗ u(t) = v ∗ (−t) ∗ u(t) = u(t) ∗ v ∗ (−t)
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 4 / 11
Cross-correlation as Convolution
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define x(t) = u∗ (−t) then
R∞
x(t) ∗ v(t) , −∞ x(t − τ )v(τ )dτ =
= u(t) ⊗ v(t)
Fourier Transform of x(t):
R∞
R∞
∗
u
(τ − t)v(τ )dτ
−∞
R∞
dt = −∞ u∗ (−t)e−i2πf t dt
∗
R
R∞ ∗
∞
−i2πf t
i2πf t
dt
dt = −∞ u(t)e
= −∞ u (t)e
X(f ) =
−i2πf t
x(t)e
−∞
= U ∗ (f )
So w(t) = x(t) ∗ v(t) ⇒ W (f ) = X(f )V (f ) = U ∗ (f )V (f )
Hence the Cross-correlation theorem:
w(t) = u(t) ⊗ v(t)
= u∗ (−t) ∗ v(t)
⇔
W (f ) = U ∗ (f )V (f )
Note that, unlike convolution, correlation is not associative or commutative:
v(t) ⊗ u(t) = v ∗ (−t) ∗ u(t) = u(t) ∗ v ∗ (−t) = w∗ (−t)
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 4 / 11
Normalized Cross-correlation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 5 / 11
Normalized Cross-correlation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define y(t) = u(t − t0 ) for some fixed t0 , then Ey = Eu :
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 5 / 11
Normalized Cross-correlation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define y(t) = u(t − t0 ) for some fixed t0 , then Ey = Eu :
Ey =
R∞
2
|y(t)| dt
−∞
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 5 / 11
Normalized Cross-correlation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define y(t) = u(t − t0 ) for some fixed t0 , then Ey = Eu :
Ey =
R∞
2
|y(t)| dt =
−∞
E1.10 Fourier Series and Transforms (2015-5585)
R∞
2
|u(t − t0 )| dt
−∞
Fourier Transform - Correlation: 8 – 5 / 11
Normalized Cross-correlation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define y(t) = u(t − t0 ) for some fixed t0 , then Ey = Eu :
Ey =
R∞
2
2
|u(t − t0 )| dt
−∞
R∞
2
= −∞ |u(τ )| dτ = Eu
|y(t)| dt =
−∞
E1.10 Fourier Series and Transforms (2015-5585)
R∞
[ t → τ + t0 ]
Fourier Transform - Correlation: 8 – 5 / 11
Normalized Cross-correlation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define y(t) = u(t − t0 ) for some fixed t0 , then Ey = Eu :
R∞
2
R∞
2
|u(t − t0 )| dt
−∞
R∞
2
= −∞ |u(τ )| dτ = Eu
[ t → τ + t0 ]
R
2
∞ ∗
Cauchy-Schwarz inequality: −∞ y (τ )v(τ )dτ ≤ Ey Ev
Ey =
|y(t)| dt =
−∞
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 5 / 11
Normalized Cross-correlation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define y(t) = u(t − t0 ) for some fixed t0 , then Ey = Eu :
R∞
2
R∞
2
|u(t − t0 )| dt
−∞
R∞
2
= −∞ |u(τ )| dτ = Eu
[ t → τ + t0 ]
R
2
∞ ∗
Cauchy-Schwarz inequality: −∞ y (τ )v(τ )dτ ≤ Ey Ev
R
2
∞ ∗
2
⇒ |w(t0 )| = −∞ u (τ − t0 )v(τ )dτ Ey =
|y(t)| dt =
−∞
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 5 / 11
Normalized Cross-correlation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define y(t) = u(t − t0 ) for some fixed t0 , then Ey = Eu :
R∞
2
R∞
2
|u(t − t0 )| dt
−∞
R∞
2
= −∞ |u(τ )| dτ = Eu
[ t → τ + t0 ]
R
2
∞ ∗
Cauchy-Schwarz inequality: −∞ y (τ )v(τ )dτ ≤ Ey Ev
R
2
∞ ∗
2
⇒ |w(t0 )| = −∞ u (τ − t0 )v(τ )dτ ≤ Ey Ev
Ey =
|y(t)| dt =
−∞
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 5 / 11
Normalized Cross-correlation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define y(t) = u(t − t0 ) for some fixed t0 , then Ey = Eu :
R∞
2
R∞
2
|u(t − t0 )| dt
−∞
R∞
2
= −∞ |u(τ )| dτ = Eu
[ t → τ + t0 ]
R
2
∞ ∗
Cauchy-Schwarz inequality: −∞ y (τ )v(τ )dτ ≤ Ey Ev
R
2
∞ ∗
2
⇒ |w(t0 )| = −∞ u (τ − t0 )v(τ )dτ ≤ Ey Ev = Eu Ev
Ey =
|y(t)| dt =
−∞
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 5 / 11
Normalized Cross-correlation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define y(t) = u(t − t0 ) for some fixed t0 , then Ey = Eu :
R∞
2
R∞
2
|u(t − t0 )| dt
−∞
R∞
2
= −∞ |u(τ )| dτ = Eu
[ t → τ + t0 ]
R
2
∞ ∗
Cauchy-Schwarz inequality: −∞ y (τ )v(τ )dτ ≤ Ey Ev
R
2
∞ ∗
2
⇒ |w(t0 )| = −∞ u (τ − t0 )v(τ )dτ ≤ Ey Ev = Eu Ev
√
but t0 was arbitrary, so we must have |w(t)| ≤ Eu Ev for all t
Ey =
|y(t)| dt =
−∞
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 5 / 11
Normalized Cross-correlation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define y(t) = u(t − t0 ) for some fixed t0 , then Ey = Eu :
R∞
2
R∞
2
|u(t − t0 )| dt
−∞
R∞
2
= −∞ |u(τ )| dτ = Eu
[ t → τ + t0 ]
R
2
∞ ∗
Cauchy-Schwarz inequality: −∞ y (τ )v(τ )dτ ≤ Ey Ev
R
2
∞ ∗
2
⇒ |w(t0 )| = −∞ u (τ − t0 )v(τ )dτ ≤ Ey Ev = Eu Ev
√
but t0 was arbitrary, so we must have |w(t)| ≤ Eu Ev for all t
Ey =
|y(t)| dt =
−∞
We can define the normalized cross-correlation
z(t) =
E1.10 Fourier Series and Transforms (2015-5585)
u(t)⊗v(t)
√
Eu Ev
Fourier Transform - Correlation: 8 – 5 / 11
Normalized Cross-correlation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define y(t) = u(t − t0 ) for some fixed t0 , then Ey = Eu :
R∞
2
R∞
2
|u(t − t0 )| dt
−∞
R∞
2
= −∞ |u(τ )| dτ = Eu
[ t → τ + t0 ]
R
2
∞ ∗
Cauchy-Schwarz inequality: −∞ y (τ )v(τ )dτ ≤ Ey Ev
R
2
∞ ∗
2
⇒ |w(t0 )| = −∞ u (τ − t0 )v(τ )dτ ≤ Ey Ev = Eu Ev
√
but t0 was arbitrary, so we must have |w(t)| ≤ Eu Ev for all t
Ey =
|y(t)| dt =
−∞
We can define the normalized cross-correlation
z(t) =
u(t)⊗v(t)
√
Eu Ev
with properties: (1) |z(t)| ≤ 1 for all t
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 5 / 11
Normalized Cross-correlation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
If we define y(t) = u(t − t0 ) for some fixed t0 , then Ey = Eu :
R∞
2
R∞
2
|u(t − t0 )| dt
−∞
R∞
2
= −∞ |u(τ )| dτ = Eu
[ t → τ + t0 ]
R
2
∞ ∗
Cauchy-Schwarz inequality: −∞ y (τ )v(τ )dτ ≤ Ey Ev
R
2
∞ ∗
2
⇒ |w(t0 )| = −∞ u (τ − t0 )v(τ )dτ ≤ Ey Ev = Eu Ev
√
but t0 was arbitrary, so we must have |w(t)| ≤ Eu Ev for all t
Ey =
|y(t)| dt =
−∞
We can define the normalized cross-correlation
z(t) =
u(t)⊗v(t)
√
Eu Ev
with properties: (1) |z(t)| ≤ 1 for all t
(2) |z(t0 )| = 1 ⇔ v(τ ) = αu(τ − t0 ) with α constant
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 5 / 11
Autocorrelation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
The correlation of a signal with itself is its autocorrelation:
w(t) = u(t) ⊗ u(t)
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 6 / 11
Autocorrelation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
The correlation of a signal with itself is its autocorrelation:
w(t) = u(t) ⊗ u(t) =
E1.10 Fourier Series and Transforms (2015-5585)
R∞
∗
u
(τ − t)u(τ )dτ
−∞
Fourier Transform - Correlation: 8 – 6 / 11
Autocorrelation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
The correlation of a signal with itself is its autocorrelation:
w(t) = u(t) ⊗ u(t) =
R∞
∗
u
(τ − t)u(τ )dτ
−∞
The autocorrelation at zero lag:
w(0) =
E1.10 Fourier Series and Transforms (2015-5585)
R∞
∗
u
(τ − 0)u(τ )dτ
−∞
Fourier Transform - Correlation: 8 – 6 / 11
Autocorrelation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
The correlation of a signal with itself is its autocorrelation:
w(t) = u(t) ⊗ u(t) =
R∞
∗
u
(τ − t)u(τ )dτ
−∞
The autocorrelation at zero lag:
∗
u
(τ − 0)u(τ )dτ
−∞
R∞ ∗
= −∞ u (τ )u(τ )dτ
w(0) =
E1.10 Fourier Series and Transforms (2015-5585)
R∞
Fourier Transform - Correlation: 8 – 6 / 11
Autocorrelation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
The correlation of a signal with itself is its autocorrelation:
w(t) = u(t) ⊗ u(t) =
R∞
∗
u
(τ − t)u(τ )dτ
−∞
The autocorrelation at zero lag:
∗
u
(τ − 0)u(τ )dτ
−∞
R∞ ∗
= −∞ u (τ )u(τ )dτ
R∞
2
= −∞ |u(τ )| dτ = Eu
w(0) =
E1.10 Fourier Series and Transforms (2015-5585)
R∞
Fourier Transform - Correlation: 8 – 6 / 11
Autocorrelation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
The correlation of a signal with itself is its autocorrelation:
w(t) = u(t) ⊗ u(t) =
R∞
∗
u
(τ − t)u(τ )dτ
−∞
The autocorrelation at zero lag:
R∞
∗
u
(τ − 0)u(τ )dτ
−∞
R∞ ∗
= −∞ u (τ )u(τ )dτ
R∞
2
= −∞ |u(τ )| dτ = Eu
The autocorrelation at zero lag, w(0), is the energy of the signal.
w(0) =
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 6 / 11
Autocorrelation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
The correlation of a signal with itself is its autocorrelation:
w(t) = u(t) ⊗ u(t) =
R∞
∗
u
(τ − t)u(τ )dτ
−∞
The autocorrelation at zero lag:
R∞
∗
u
(τ − 0)u(τ )dτ
−∞
R∞ ∗
= −∞ u (τ )u(τ )dτ
R∞
2
= −∞ |u(τ )| dτ = Eu
The autocorrelation at zero lag, w(0), is the energy of the signal.
w(0) =
The normalized autocorrelation:
E1.10 Fourier Series and Transforms (2015-5585)
z(t) =
u(t)⊗u(t)
Eu
Fourier Transform - Correlation: 8 – 6 / 11
Autocorrelation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
The correlation of a signal with itself is its autocorrelation:
w(t) = u(t) ⊗ u(t) =
R∞
∗
u
(τ − t)u(τ )dτ
−∞
The autocorrelation at zero lag:
R∞
∗
u
(τ − 0)u(τ )dτ
−∞
R∞ ∗
= −∞ u (τ )u(τ )dτ
R∞
2
= −∞ |u(τ )| dτ = Eu
The autocorrelation at zero lag, w(0), is the energy of the signal.
w(0) =
z(t) = u(t)⊗u(t)
Eu
satisfies z(0) = 1 and |z(t)| ≤ 1 for any t.
The normalized autocorrelation:
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 6 / 11
Autocorrelation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
The correlation of a signal with itself is its autocorrelation:
w(t) = u(t) ⊗ u(t) =
R∞
∗
u
(τ − t)u(τ )dτ
−∞
The autocorrelation at zero lag:
R∞
∗
u
(τ − 0)u(τ )dτ
−∞
R∞ ∗
= −∞ u (τ )u(τ )dτ
R∞
2
= −∞ |u(τ )| dτ = Eu
The autocorrelation at zero lag, w(0), is the energy of the signal.
w(0) =
z(t) = u(t)⊗u(t)
Eu
satisfies z(0) = 1 and |z(t)| ≤ 1 for any t.
The normalized autocorrelation:
Wiener-Khinchin Theorem: [Cross-correlation theorem when v(t) = u(t)]
w(t) = u(t) ⊗ u(t)
E1.10 Fourier Series and Transforms (2015-5585)
⇔
W (f ) = U ∗ (f )U (f )
Fourier Transform - Correlation: 8 – 6 / 11
Autocorrelation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
The correlation of a signal with itself is its autocorrelation:
w(t) = u(t) ⊗ u(t) =
R∞
∗
u
(τ − t)u(τ )dτ
−∞
The autocorrelation at zero lag:
R∞
∗
u
(τ − 0)u(τ )dτ
−∞
R∞ ∗
= −∞ u (τ )u(τ )dτ
R∞
2
= −∞ |u(τ )| dτ = Eu
The autocorrelation at zero lag, w(0), is the energy of the signal.
w(0) =
z(t) = u(t)⊗u(t)
Eu
satisfies z(0) = 1 and |z(t)| ≤ 1 for any t.
The normalized autocorrelation:
Wiener-Khinchin Theorem: [Cross-correlation theorem when v(t) = u(t)]
w(t) = u(t) ⊗ u(t)
E1.10 Fourier Series and Transforms (2015-5585)
⇔
W (f ) = U ∗ (f )U (f ) = |U (f )|
2
Fourier Transform - Correlation: 8 – 6 / 11
Autocorrelation
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
The correlation of a signal with itself is its autocorrelation:
w(t) = u(t) ⊗ u(t) =
R∞
∗
u
(τ − t)u(τ )dτ
−∞
The autocorrelation at zero lag:
R∞
∗
u
(τ − 0)u(τ )dτ
−∞
R∞ ∗
= −∞ u (τ )u(τ )dτ
R∞
2
= −∞ |u(τ )| dτ = Eu
The autocorrelation at zero lag, w(0), is the energy of the signal.
w(0) =
z(t) = u(t)⊗u(t)
Eu
satisfies z(0) = 1 and |z(t)| ≤ 1 for any t.
The normalized autocorrelation:
Wiener-Khinchin Theorem: [Cross-correlation theorem when v(t) = u(t)]
w(t) = u(t) ⊗ u(t)
⇔
W (f ) = U ∗ (f )U (f ) = |U (f )|
2
The Fourier transform of the autocorrelation is the energy spectrum.
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 6 / 11
Autocorrelation example
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
Cross-correlation is used to find when two different signals are similar.
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 7 / 11
Autocorrelation example
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
Cross-correlation is used to find when two different signals are similar.
Autocorrelation is used to find when a signal is similar to itself delayed.
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 7 / 11
Autocorrelation example
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
Cross-correlation is used to find when two different signals are similar.
Autocorrelation is used to find when a signal is similar to itself delayed.
First graph shows s(t) a segment of the microphone signal from the initial
vowel of “early” spoken by me.
Speech s(t)
1
0.5
0
-0.5
-1
10
20
30
Time (ms)
E1.10 Fourier Series and Transforms (2015-5585)
40
50
Fourier Transform - Correlation: 8 – 7 / 11
Autocorrelation example
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
Cross-correlation is used to find when two different signals are similar.
Autocorrelation is used to find when a signal is similar to itself delayed.
First graph shows s(t) a segment of the microphone signal from the initial
vowel of “early” spoken by me. The waveform is “quasi-periodic” =
“almost periodic but not quite”.
Speech s(t)
1
0.5
0
-0.5
-1
10
20
30
Time (ms)
E1.10 Fourier Series and Transforms (2015-5585)
40
50
Fourier Transform - Correlation: 8 – 7 / 11
Autocorrelation example
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
Cross-correlation is used to find when two different signals are similar.
Autocorrelation is used to find when a signal is similar to itself delayed.
First graph shows s(t) a segment of the microphone signal from the initial
vowel of “early” spoken by me. The waveform is “quasi-periodic” =
“almost periodic but not quite”.
Second graph shows normalized autocorrelation, z(t) =
Norm Autocorr, z(t)
Speech s(t)
1
0.5
0
-0.5
-1
10
20
30
Time (ms)
E1.10 Fourier Series and Transforms (2015-5585)
40
50
s(t)⊗s(t)
.
Es
1
0.82
Lag = 6.2 ms (161 Hz)
0.5
0
0
5
10
Lag: t (ms)
15
20
Fourier Transform - Correlation: 8 – 7 / 11
Autocorrelation example
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
Cross-correlation is used to find when two different signals are similar.
Autocorrelation is used to find when a signal is similar to itself delayed.
First graph shows s(t) a segment of the microphone signal from the initial
vowel of “early” spoken by me. The waveform is “quasi-periodic” =
“almost periodic but not quite”.
Second graph shows normalized autocorrelation, z(t) =
s(t)⊗s(t)
.
Es
z(0) = 1 for t = 0 since a signal always matches itself exactly.
Norm Autocorr, z(t)
Speech s(t)
1
0.5
0
-0.5
-1
10
20
30
Time (ms)
E1.10 Fourier Series and Transforms (2015-5585)
40
50
1
0.82
Lag = 6.2 ms (161 Hz)
0.5
0
0
5
10
Lag: t (ms)
15
20
Fourier Transform - Correlation: 8 – 7 / 11
Autocorrelation example
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
Cross-correlation is used to find when two different signals are similar.
Autocorrelation is used to find when a signal is similar to itself delayed.
First graph shows s(t) a segment of the microphone signal from the initial
vowel of “early” spoken by me. The waveform is “quasi-periodic” =
“almost periodic but not quite”.
Second graph shows normalized autocorrelation, z(t) =
s(t)⊗s(t)
.
Es
z(0) = 1 for t = 0 since a signal always matches itself exactly.
z(t) = 0.82 for t = 6.2 ms = one period lag (not an exact match).
Norm Autocorr, z(t)
Speech s(t)
1
0.5
0
-0.5
-1
10
20
30
Time (ms)
E1.10 Fourier Series and Transforms (2015-5585)
40
50
1
0.82
Lag = 6.2 ms (161 Hz)
0.5
0
0
5
10
Lag: t (ms)
15
20
Fourier Transform - Correlation: 8 – 7 / 11
Autocorrelation example
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
Cross-correlation is used to find when two different signals are similar.
Autocorrelation is used to find when a signal is similar to itself delayed.
First graph shows s(t) a segment of the microphone signal from the initial
vowel of “early” spoken by me. The waveform is “quasi-periodic” =
“almost periodic but not quite”.
Second graph shows normalized autocorrelation, z(t) =
s(t)⊗s(t)
.
Es
z(0) = 1 for t = 0 since a signal always matches itself exactly.
z(t) = 0.82 for t = 6.2 ms = one period lag (not an exact match).
z(t) = 0.53 for t = 12.4 ms = two periods lag (even worse match).
Norm Autocorr, z(t)
Speech s(t)
1
0.5
0
-0.5
-1
10
20
30
Time (ms)
E1.10 Fourier Series and Transforms (2015-5585)
40
50
1
0.82
Lag = 6.2 ms (161 Hz)
0.5
0
0
5
10
Lag: t (ms)
15
20
Fourier Transform - Correlation: 8 – 7 / 11
Fourier Transform Variants
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
There are three different versions of the Fourier Transform in current use.
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 8 / 11
Fourier Transform Variants
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
There are three different versions of the Fourier Transform in current use.
(1) Frequency version (we have used this in lectures)
U (f ) =
R∞
−i2πf t
u(t)e
−∞
E1.10 Fourier Series and Transforms (2015-5585)
dt
u(t) =
R∞
i2πf t
U
(f
)e
df
−∞
Fourier Transform - Correlation: 8 – 8 / 11
Fourier Transform Variants
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
There are three different versions of the Fourier Transform in current use.
(1) Frequency version (we have used this in lectures)
R∞
R∞
dt
u(t) = −∞ U (f )ei2πf t df
U (f ) = −∞ u(t)e
• Used in the communications/broadcasting industry and textbooks.
• The formulae do not need scale factors of 2π anywhere.
,,,
E1.10 Fourier Series and Transforms (2015-5585)
−i2πf t
Fourier Transform - Correlation: 8 – 8 / 11
Fourier Transform Variants
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
There are three different versions of the Fourier Transform in current use.
(1) Frequency version (we have used this in lectures)
R∞
R∞
dt
u(t) = −∞ U (f )ei2πf t df
U (f ) = −∞ u(t)e
• Used in the communications/broadcasting industry and textbooks.
• The formulae do not need scale factors of 2π anywhere.
,,,
−i2πf t
(2) Angular frequency version
e (ω) =
U
R∞
−∞
E1.10 Fourier Series and Transforms (2015-5585)
−iωt
u(t)e
dt
u(t) =
1
2π
R∞
−∞
e (ω)eiωt dω
U
Fourier Transform - Correlation: 8 – 8 / 11
Fourier Transform Variants
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
There are three different versions of the Fourier Transform in current use.
(1) Frequency version (we have used this in lectures)
R∞
R∞
dt
u(t) = −∞ U (f )ei2πf t df
U (f ) = −∞ u(t)e
• Used in the communications/broadcasting industry and textbooks.
• The formulae do not need scale factors of 2π anywhere.
,,,
−i2πf t
(2) Angular frequency version
e (ω) =
U
R∞
R∞
e (ω)eiωt dω
u(t) =
U
−∞
e (ω) = U (f ) = U ( ω )
Continuous spectra are unchanged: U
−∞
E1.10 Fourier Series and Transforms (2015-5585)
−iωt
u(t)e
dt
1
2π
2π
Fourier Transform - Correlation: 8 – 8 / 11
Fourier Transform Variants
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
There are three different versions of the Fourier Transform in current use.
(1) Frequency version (we have used this in lectures)
R∞
R∞
dt
u(t) = −∞ U (f )ei2πf t df
U (f ) = −∞ u(t)e
• Used in the communications/broadcasting industry and textbooks.
• The formulae do not need scale factors of 2π anywhere.
,,,
−i2πf t
(2) Angular frequency version
e (ω) =
U
R∞
R∞
e (ω)eiωt dω
u(t) =
U
−∞
e (ω) = U (f ) = U ( ω )
Continuous spectra are unchanged: U
2π
However δ -function spectral components are multiplied by 2π so that
e (ω) = 2π × δ(ω − 2πf0 )
U (f ) = δ(f − f0 ) ⇒ U
−∞
E1.10 Fourier Series and Transforms (2015-5585)
−iωt
u(t)e
dt
1
2π
Fourier Transform - Correlation: 8 – 8 / 11
Fourier Transform Variants
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
There are three different versions of the Fourier Transform in current use.
(1) Frequency version (we have used this in lectures)
R∞
R∞
dt
u(t) = −∞ U (f )ei2πf t df
U (f ) = −∞ u(t)e
• Used in the communications/broadcasting industry and textbooks.
• The formulae do not need scale factors of 2π anywhere.
,,,
−i2πf t
(2) Angular frequency version
e (ω) =
U
R∞
R∞
e (ω)eiωt dω
u(t) =
U
−∞
e (ω) = U (f ) = U ( ω )
Continuous spectra are unchanged: U
2π
However δ -function spectral components are multiplied by 2π so that
e (ω) = 2π × δ(ω − 2πf0 )
U (f ) = δ(f − f0 ) ⇒ U
• Used in most signal processing and control theory textbooks.
−∞
E1.10 Fourier Series and Transforms (2015-5585)
−iωt
u(t)e
dt
1
2π
Fourier Transform - Correlation: 8 – 8 / 11
Fourier Transform Variants
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
There are three different versions of the Fourier Transform in current use.
(1) Frequency version (we have used this in lectures)
R∞
R∞
dt
u(t) = −∞ U (f )ei2πf t df
U (f ) = −∞ u(t)e
• Used in the communications/broadcasting industry and textbooks.
• The formulae do not need scale factors of 2π anywhere.
,,,
−i2πf t
(2) Angular frequency version
e (ω) =
U
R∞
R∞
e (ω)eiωt dω
u(t) =
U
−∞
e (ω) = U (f ) = U ( ω )
Continuous spectra are unchanged: U
2π
However δ -function spectral components are multiplied by 2π so that
e (ω) = 2π × δ(ω − 2πf0 )
U (f ) = δ(f − f0 ) ⇒ U
• Used in most signal processing and control theory textbooks.
−∞
−iωt
u(t)e
1
2π
dt
(3) Angular frequency + symmetrical scale factor
b (ω) =
U
√1
2π
E1.10 Fourier Series and Transforms (2015-5585)
R∞
−iωt
u(t)e
−∞
dt
u(t) =
√1
2π
R∞
b (ω)eiωt dω
U
−∞
Fourier Transform - Correlation: 8 – 8 / 11
Fourier Transform Variants
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
There are three different versions of the Fourier Transform in current use.
(1) Frequency version (we have used this in lectures)
R∞
R∞
dt
u(t) = −∞ U (f )ei2πf t df
U (f ) = −∞ u(t)e
• Used in the communications/broadcasting industry and textbooks.
• The formulae do not need scale factors of 2π anywhere.
,,,
−i2πf t
(2) Angular frequency version
e (ω) =
U
R∞
R∞
e (ω)eiωt dω
u(t) =
U
−∞
e (ω) = U (f ) = U ( ω )
Continuous spectra are unchanged: U
2π
However δ -function spectral components are multiplied by 2π so that
e (ω) = 2π × δ(ω − 2πf0 )
U (f ) = δ(f − f0 ) ⇒ U
• Used in most signal processing and control theory textbooks.
−∞
−iωt
u(t)e
1
2π
dt
(3) Angular frequency + symmetrical scale factor
b (ω) =
U
√1
2π
R∞
−iωt
u(t)e
−∞
dt
b (ω) = √1 U
e (ω)
In all cases U
u(t) =
√1
2π
R∞
b (ω)eiωt dω
U
−∞
2π
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 8 / 11
Fourier Transform Variants
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
There are three different versions of the Fourier Transform in current use.
(1) Frequency version (we have used this in lectures)
R∞
R∞
dt
u(t) = −∞ U (f )ei2πf t df
U (f ) = −∞ u(t)e
• Used in the communications/broadcasting industry and textbooks.
• The formulae do not need scale factors of 2π anywhere.
,,,
−i2πf t
(2) Angular frequency version
e (ω) =
U
R∞
R∞
e (ω)eiωt dω
u(t) =
U
−∞
e (ω) = U (f ) = U ( ω )
Continuous spectra are unchanged: U
2π
However δ -function spectral components are multiplied by 2π so that
e (ω) = 2π × δ(ω − 2πf0 )
U (f ) = δ(f − f0 ) ⇒ U
• Used in most signal processing and control theory textbooks.
−∞
−iωt
u(t)e
1
2π
dt
(3) Angular frequency + symmetrical scale factor
b (ω) =
U
√1
2π
R∞
−iωt
u(t)e
−∞
dt
b (ω) = √1 U
e (ω)
In all cases U
u(t) =
√1
2π
R∞
b (ω)eiωt dω
U
−∞
2π
• Used in many Maths textbooks (mathematicians like symmetry)
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 8 / 11
Scale Factors
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
Fourier Transform using Angular Frequency:
e (ω) =
U
R∞
E1.10 Fourier Series and Transforms (2015-5585)
−iωt
u(t)e
−∞
dt
u(t) =
1
2π
R∞
e (ω)eiωt dω
U
−∞
Fourier Transform - Correlation: 8 – 9 / 11
Scale Factors
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
Fourier Transform using Angular Frequency:
R∞
R∞
e (ω)eiωt dω
U
u(t)e
dt
u(t)
=
−∞
−∞
R
R
1
Any formula involving df will change to 2π dω
[since dω = 2πdf ]
e (ω) =
U
E1.10 Fourier Series and Transforms (2015-5585)
−iωt
1
2π
Fourier Transform - Correlation: 8 – 9 / 11
Scale Factors
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
Fourier Transform using Angular Frequency:
R∞
R∞
e (ω)eiωt dω
U
u(t)e
dt
u(t)
=
−∞
−∞
R
R
1
Any formula involving df will change to 2π dω
[since dω = 2πdf ]
e (ω) =
U
−iωt
Parseval’s Theorem:
R
∗
u (t)v(t)dt =
E1.10 Fourier Series and Transforms (2015-5585)
1
2π
R
1
2π
e ∗ (ω)Ve (ω)dω
U
Fourier Transform - Correlation: 8 – 9 / 11
Scale Factors
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
Fourier Transform using Angular Frequency:
R∞
R∞
e (ω)eiωt dω
U
u(t)e
dt
u(t)
=
−∞
−∞
R
R
1
Any formula involving df will change to 2π dω
[since dω = 2πdf ]
e (ω) =
U
−iωt
Parseval’s Theorem:
R
1
2π
R
e ∗ (ω)Ve (ω)dω
u (t)v(t)dt =
U
2
R R
2
1
e (ω) dω
Eu = |u(t)| dt = 2π U
∗
E1.10 Fourier Series and Transforms (2015-5585)
1
2π
Fourier Transform - Correlation: 8 – 9 / 11
Scale Factors
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
Fourier Transform using Angular Frequency:
R∞
R∞
e (ω)eiωt dω
U
u(t)e
dt
u(t)
=
−∞
−∞
R
R
1
Any formula involving df will change to 2π dω
[since dω = 2πdf ]
e (ω) =
U
Parseval’s Theorem:
R
1
2π
−iωt
R
e ∗ (ω)Ve (ω)dω
u (t)v(t)dt =
U
2
R R
2
1
e (ω) dω
Eu = |u(t)| dt = 2π U
∗
1
2π
Waveform Multiplication: (convolution implicitly involves integration)
f (ω) =
w(t) = u(t)v(t) ⇒ W
E1.10 Fourier Series and Transforms (2015-5585)
1 e
2π U (ω)
∗ Ve (ω)
Fourier Transform - Correlation: 8 – 9 / 11
Scale Factors
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
Fourier Transform using Angular Frequency:
R∞
R∞
e (ω)eiωt dω
U
u(t)e
dt
u(t)
=
−∞
−∞
R
R
1
Any formula involving df will change to 2π dω
[since dω = 2πdf ]
e (ω) =
U
Parseval’s Theorem:
R
1
2π
−iωt
R
e ∗ (ω)Ve (ω)dω
u (t)v(t)dt =
U
2
R R
2
1
e (ω) dω
Eu = |u(t)| dt = 2π U
∗
1
2π
Waveform Multiplication: (convolution implicitly involves integration)
f (ω) =
w(t) = u(t)v(t) ⇒ W
1 e
2π U (ω)
∗ Ve (ω)
Spectrum Multiplication: (multiplication ; integration)
f (ω) = U
e (ω)Ve (ω)
w(t) = u(t) ∗ v(t) ⇒ W
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 9 / 11
Scale Factors
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
Fourier Transform using Angular Frequency:
R∞
R∞
e (ω)eiωt dω
U
u(t)e
dt
u(t)
=
−∞
−∞
R
R
1
Any formula involving df will change to 2π dω
[since dω = 2πdf ]
e (ω) =
U
Parseval’s Theorem:
R
1
2π
−iωt
R
e ∗ (ω)Ve (ω)dω
u (t)v(t)dt =
U
2
R R
2
1
e (ω) dω
Eu = |u(t)| dt = 2π U
∗
1
2π
Waveform Multiplication: (convolution implicitly involves integration)
f (ω) =
w(t) = u(t)v(t) ⇒ W
1 e
2π U (ω)
∗ Ve (ω)
Spectrum Multiplication: (multiplication ; integration)
f (ω) = U
e (ω)Ve (ω)
w(t) = u(t) ∗ v(t) ⇒ W
b (ω),
To obtain formulae for version (3) of the Fourier Transform, U
e (ω) =
substitute into the above formulae: U
E1.10 Fourier Series and Transforms (2015-5585)
√
b (ω).
2π U
Fourier Transform - Correlation: 8 – 9 / 11
Summary
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
• Cross-Correlation: w(t) = u(t) ⊗ v(t)
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 10 / 11
Summary
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
• Cross-Correlation: w(t) = u(t) ⊗ v(t) =
E1.10 Fourier Series and Transforms (2015-5585)
R∞
−∞
u∗ (τ − t)v(τ )dτ
Fourier Transform - Correlation: 8 – 10 / 11
Summary
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
• Cross-Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
◦ Used to find similarities between v(t) and a delayed u(t)
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 10 / 11
Summary
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
• Cross-Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
◦ Used to find similarities between v(t) and a delayed u(t)
◦ Cross-correlation theorem: W (f ) = U ∗ (f )V (f )
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 10 / 11
Summary
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
• Cross-Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
◦ Used to find similarities between v(t) and a delayed u(t)
◦ Cross-correlation theorem: W (f ) = U ∗ (f )V (f )
√
◦ Cauchy-Schwarz Inequality: |u(t) ⊗ v(t)| ≤ Eu Ev
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 10 / 11
Summary
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
• Cross-Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
◦ Used to find similarities between v(t) and a delayed u(t)
◦ Cross-correlation theorem: W (f ) = U ∗ (f )V (f )
√
◦ Cauchy-Schwarz Inequality: |u(t) ⊗ v(t)| ≤ Eu Ev
u(t)⊗v(t)
⊲ Normalized cross-correlation: √E E ≤ 1
E1.10 Fourier Series and Transforms (2015-5585)
u
v
Fourier Transform - Correlation: 8 – 10 / 11
Summary
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
• Cross-Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
◦ Used to find similarities between v(t) and a delayed u(t)
◦ Cross-correlation theorem: W (f ) = U ∗ (f )V (f )
√
◦ Cauchy-Schwarz Inequality: |u(t) ⊗ v(t)| ≤ Eu Ev
u(t)⊗v(t)
⊲ Normalized cross-correlation: √E E ≤ 1
u
v
• Autocorrelation: x(t) = u(t) ⊗ u(t)
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 10 / 11
Summary
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
• Cross-Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
◦ Used to find similarities between v(t) and a delayed u(t)
◦ Cross-correlation theorem: W (f ) = U ∗ (f )V (f )
√
◦ Cauchy-Schwarz Inequality: |u(t) ⊗ v(t)| ≤ Eu Ev
u(t)⊗v(t)
⊲ Normalized cross-correlation: √E E ≤ 1
u v
R∞ ∗
• Autocorrelation: x(t) = u(t) ⊗ u(t) = −∞ u (τ − t)u(τ )dτ ≤ Eu
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 10 / 11
Summary
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
• Cross-Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
◦ Used to find similarities between v(t) and a delayed u(t)
◦ Cross-correlation theorem: W (f ) = U ∗ (f )V (f )
√
◦ Cauchy-Schwarz Inequality: |u(t) ⊗ v(t)| ≤ Eu Ev
u(t)⊗v(t)
⊲ Normalized cross-correlation: √E E ≤ 1
u v
R∞ ∗
• Autocorrelation: x(t) = u(t) ⊗ u(t) = −∞ u (τ − t)u(τ )dτ ≤ Eu
◦ Wiener-Khinchin: X(f ) = energy spectral density, |U (f )|2
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 10 / 11
Summary
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
• Cross-Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
◦ Used to find similarities between v(t) and a delayed u(t)
◦ Cross-correlation theorem: W (f ) = U ∗ (f )V (f )
√
◦ Cauchy-Schwarz Inequality: |u(t) ⊗ v(t)| ≤ Eu Ev
u(t)⊗v(t)
⊲ Normalized cross-correlation: √E E ≤ 1
u v
R∞ ∗
• Autocorrelation: x(t) = u(t) ⊗ u(t) = −∞ u (τ − t)u(τ )dτ ≤ Eu
◦ Wiener-Khinchin: X(f ) = energy spectral density, |U (f )|2
◦ Used to find periodicity in u(t)
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 10 / 11
Summary
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
• Cross-Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
◦ Used to find similarities between v(t) and a delayed u(t)
◦ Cross-correlation theorem: W (f ) = U ∗ (f )V (f )
√
◦ Cauchy-Schwarz Inequality: |u(t) ⊗ v(t)| ≤ Eu Ev
u(t)⊗v(t)
⊲ Normalized cross-correlation: √E E ≤ 1
u v
R∞ ∗
• Autocorrelation: x(t) = u(t) ⊗ u(t) = −∞ u (τ − t)u(τ )dτ ≤ Eu
◦ Wiener-Khinchin: X(f ) = energy spectral density, |U (f )|2
◦ Used to find periodicity in u(t)
• Fourier Transform using ω :
◦ Continuous spectra unchanged; spectral impulses multiplied by 2π
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 10 / 11
Summary
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
• Cross-Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
◦ Used to find similarities between v(t) and a delayed u(t)
◦ Cross-correlation theorem: W (f ) = U ∗ (f )V (f )
√
◦ Cauchy-Schwarz Inequality: |u(t) ⊗ v(t)| ≤ Eu Ev
u(t)⊗v(t)
⊲ Normalized cross-correlation: √E E ≤ 1
u v
R∞ ∗
• Autocorrelation: x(t) = u(t) ⊗ u(t) = −∞ u (τ − t)u(τ )dτ ≤ Eu
◦ Wiener-Khinchin: X(f ) = energy spectral density, |U (f )|2
◦ Used to find periodicity in u(t)
• Fourier Transform using ω :
◦ Continuous spectra unchanged; spectral impulses multiplied by 2π
R
R
1
◦ In formulae: df → 2π dω ; ω -convolution involves an integral
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 10 / 11
Summary
8: Correlation
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
R∞
• Cross-Correlation: w(t) = u(t) ⊗ v(t) = −∞ u∗ (τ − t)v(τ )dτ
◦ Used to find similarities between v(t) and a delayed u(t)
◦ Cross-correlation theorem: W (f ) = U ∗ (f )V (f )
√
◦ Cauchy-Schwarz Inequality: |u(t) ⊗ v(t)| ≤ Eu Ev
u(t)⊗v(t)
⊲ Normalized cross-correlation: √E E ≤ 1
u v
R∞ ∗
• Autocorrelation: x(t) = u(t) ⊗ u(t) = −∞ u (τ − t)u(τ )dτ ≤ Eu
◦ Wiener-Khinchin: X(f ) = energy spectral density, |U (f )|2
◦ Used to find periodicity in u(t)
• Fourier Transform using ω :
◦ Continuous spectra unchanged; spectral impulses multiplied by 2π
R
R
1
◦ In formulae: df → 2π dω ; ω -convolution involves an integral
For further details see RHB Chapter 13.1
E1.10 Fourier Series and Transforms (2015-5585)
Fourier Transform - Correlation: 8 – 10 / 11
Spectrogram
Spectrogram of “Merry Christmas” spoken by Mike Brookes ()
m e rr y ch r i s t m a
s
-10
8
7
6
5
4
3
2
-20
1
0
-40
E1.10 Fourier Series and Transforms (2015-5585)
-30
0.2
0.4
0.6
Time (s)
0.8
Power/Decade (dB)
• Cross-Correlation
• Signal Matching
• Cross-corr as Convolution
• Normalized Cross-corr
• Autocorrelation
• Autocorrelation example
• Fourier Transform Variants
• Scale Factors
• Summary
• Spectrogram
Frequency (kHz)
8: Correlation
1
Fourier Transform - Correlation: 8 – 11 / 11