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 The cross-correlation between two signals u(t) and v(t) is • Signal Matching • Cross-corr as Convolution w(t) = u(t) v(t) , ∞ u∗(τ)v(τ + t)dτ • Normalized Cross-corr ⊗ R−∞ • Autocorrelation • Autocorrelation example • Fourier Transform Variants • Scale Factors • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 2 / 11 Cross-Correlation
8: Correlation • Cross-Correlation The cross-correlation between two signals u(t) and v(t) is • Signal Matching • Cross-corr as Convolution w(t) = u(t) v(t) , ∞ u∗(τ)v(τ + t)dτ • Normalized Cross-corr ⊗ R−∞ • Autocorrelation ∞ u τ t v τ dτ [sub: τ τ t] • = ∗( ) ( ) Autocorrelation example R−∞ − → − • Fourier Transform Variants • Scale Factors • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 2 / 11 Cross-Correlation
8: Correlation • Cross-Correlation The cross-correlation between two signals u(t) and v(t) is • Signal Matching • Cross-corr as Convolution w(t) = u(t) v(t) , ∞ u∗(τ)v(τ + t)dτ • Normalized Cross-corr ⊗ R−∞ • Autocorrelation ∞ u τ t v τ dτ [sub: τ τ t] • = ∗( ) ( ) Autocorrelation example R−∞ − → − • Fourier Transform Variants • Scale Factors The complex conjugate, u∗(τ) makes no difference if u(t) is real-valued • Summary but makes the definition work even if u(t) is complex-valued. • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 2 / 11 Cross-Correlation
8: Correlation • Cross-Correlation The cross-correlation between two signals u(t) and v(t) is • Signal Matching • Cross-corr as Convolution w(t) = u(t) v(t) , ∞ u∗(τ)v(τ + t)dτ • Normalized Cross-corr ⊗ R−∞ • Autocorrelation ∞ u τ t v τ dτ [sub: τ τ t] • = ∗( ) ( ) Autocorrelation example R−∞ − → − • Fourier Transform Variants • Scale Factors The complex conjugate, u∗(τ) makes no difference if u(t) is real-valued • Summary but makes the definition work even if u(t) is complex-valued. • Spectrogram Correlation versus Convolution:
u(t) v(t) = ∞ u∗(τ)v(τ + t)dτ [correlation] ⊗ R−∞ u(t) v(t) = ∞ u(τ)v(t τ)dτ [convolution] ∗ R−∞ −
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 2 / 11 Cross-Correlation
8: Correlation • Cross-Correlation The cross-correlation between two signals u(t) and v(t) is • Signal Matching • Cross-corr as Convolution w(t) = u(t) v(t) , ∞ u∗(τ)v(τ + t)dτ • Normalized Cross-corr ⊗ R−∞ • Autocorrelation ∞ u τ t v τ dτ [sub: τ τ t] • = ∗( ) ( ) Autocorrelation example R−∞ − → − • Fourier Transform Variants • Scale Factors The complex conjugate, u∗(τ) makes no difference if u(t) is real-valued • Summary but makes the definition work even if u(t) is complex-valued. • Spectrogram Correlation versus Convolution:
u(t) v(t) = ∞ u∗(τ)v(τ + t)dτ [correlation] ⊗ R−∞ u(t) v(t) = ∞ u(τ)v(t τ)dτ [convolution] ∗ R−∞ − 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 The cross-correlation between two signals u(t) and v(t) is • Signal Matching • Cross-corr as Convolution w(t) = u(t) v(t) , ∞ u∗(τ)v(τ + t)dτ • Normalized Cross-corr ⊗ R−∞ • Autocorrelation ∞ u τ t v τ dτ [sub: τ τ t] • = ∗( ) ( ) Autocorrelation example R−∞ − → − • Fourier Transform Variants • Scale Factors The complex conjugate, u∗(τ) makes no difference if u(t) is real-valued • Summary but makes the definition work even if u(t) is complex-valued. • Spectrogram Correlation versus Convolution:
u(t) v(t) = ∞ u∗(τ)v(τ + t)dτ [correlation] ⊗ R−∞ u(t) v(t) = ∞ u(τ)v(t τ)dτ [convolution] ∗ R−∞ − 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 The cross-correlation between two signals u(t) and v(t) is • Signal Matching • Cross-corr as Convolution w(t) = u(t) v(t) , ∞ u∗(τ)v(τ + t)dτ • Normalized Cross-corr ⊗ R−∞ • Autocorrelation ∞ u τ t v τ dτ [sub: τ τ t] • = ∗( ) ( ) Autocorrelation example R−∞ − → − • Fourier Transform Variants • Scale Factors The complex conjugate, u∗(τ) makes no difference if u(t) is real-valued • Summary but makes the definition work even if u(t) is complex-valued. • Spectrogram Correlation versus Convolution:
u(t) v(t) = ∞ u∗(τ)v(τ + t)dτ [correlation] ⊗ R−∞ u(t) v(t) = ∞ u(τ)v(t τ)dτ [convolution] ∗ R−∞ − 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 The cross-correlation between two signals u(t) and v(t) is • Signal Matching • Cross-corr as Convolution w(t) = u(t) v(t) , ∞ u∗(τ)v(τ + t)dτ • Normalized Cross-corr ⊗ R−∞ • Autocorrelation ∞ u τ t v τ dτ [sub: τ τ t] • = ∗( ) ( ) Autocorrelation example R−∞ − → − • Fourier Transform Variants • Scale Factors The complex conjugate, u∗(τ) makes no difference if u(t) is real-valued • Summary but makes the definition work even if u(t) is complex-valued. • Spectrogram Correlation versus Convolution:
u(t) v(t) = ∞ u∗(τ)v(τ + t)dτ [correlation] ⊗ R−∞ u(t) v(t) = ∞ u(τ)v(t τ)dτ [convolution] ∗ R−∞ − 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 The cross-correlation between two signals u(t) and v(t) is • Signal Matching • Cross-corr as Convolution w(t) = u(t) v(t) , ∞ u∗(τ)v(τ + t)dτ • Normalized Cross-corr ⊗ R−∞ • Autocorrelation ∞ u τ t v τ dτ [sub: τ τ t] • = ∗( ) ( ) Autocorrelation example R−∞ − → − • Fourier Transform Variants • Scale Factors The complex conjugate, u∗(τ) makes no difference if u(t) is real-valued • Summary but makes the definition work even if u(t) is complex-valued. • Spectrogram Correlation versus Convolution:
u(t) v(t) = ∞ u∗(τ)v(τ + t)dτ [correlation] ⊗ R−∞ u(t) v(t) = ∞ u(τ)v(t τ)dτ [convolution] ∗ R−∞ − 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 0 0 0 0
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8: Correlation Cross correlation is used to find where two • Cross-Correlation • Signal Matching signals match • Cross-corr as Convolution • Normalized Cross-corr • Autocorrelation • Autocorrelation example • Fourier Transform Variants • Scale Factors • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 3 / 11 0 0 0 0 0
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8: Correlation Cross correlation is used to find where two 1 • Cross-Correlation 0.5 • Signal Matching u(t) signals match: u(t) is the test waveform. 0 • Cross-corr as Convolution 0 200 400 600 • Normalized Cross-corr • Autocorrelation • Autocorrelation example • Fourier Transform Variants • Scale Factors • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 3 / 11 0 0 0 0 0
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8: Correlation Cross correlation is used to find where two 1 • Cross-Correlation 0.5 • Signal Matching u(t) signals match: u(t) is the test waveform. 0 • Cross-corr as Convolution 0 200 400 600 • Normalized Cross-corr • Example 1: Autocorrelation 1 • Autocorrelation example v(t) contains u(t) with an unknown delay 0.5 v(t) • Fourier Transform Variants 0 • and added noise. Scale Factors 0 200 400 600 800 • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 3 / 11 0 0 0 0 0
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8: Correlation Cross correlation is used to find where two 1 • Cross-Correlation 0.5 • Signal Matching u(t) signals match: u(t) is the test waveform. 0 • Cross-corr as Convolution 0 200 400 600 • Normalized Cross-corr • Example 1: Autocorrelation 1 • Autocorrelation example v(t) contains u(t) with an unknown delay 0.5 v(t) • Fourier Transform Variants 0 • and added noise. Scale Factors 0 200 400 600 800 • Summary w(t) = u(t) v(t) • Spectrogram ⊗
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 3 / 11 0 0 0 0 0
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8: Correlation Cross correlation is used to find where two 1 • Cross-Correlation 0.5 • Signal Matching u(t) signals match: u(t) is the test waveform. 0 • Cross-corr as Convolution 0 200 400 600 • Normalized Cross-corr • Example 1: Autocorrelation 1 • Autocorrelation example v(t) contains u(t) with an unknown delay 0.5 v(t) • Fourier Transform Variants 0 • and added noise. Scale Factors 0 200 400 600 800 • Summary w(t) = u(t) v(t) • Spectrogram ⊗ = u∗(τ t)v(τ)dt R −
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 3 / 11 0 0 0 0 0
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Signal Matching
8: Correlation Cross correlation is used to find where two 1 • Cross-Correlation 0.5 • Signal Matching u(t) signals match: u(t) is the test waveform. 0 • Cross-corr as Convolution 0 200 400 600 • Normalized Cross-corr • Example 1: Autocorrelation 1 • Autocorrelation example v(t) contains u(t) with an unknown delay 0.5 v(t) • Fourier Transform Variants 0 • and added noise. Scale Factors 0 200 400 600 800 • Summary w(t) = u(t) v(t) 20 • Spectrogram 10
⊗ w(t) = u∗(τ t)v(τ)dt 0 R − 0 200 400 600 800
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 3 / 11 0 0 0 0 0
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Signal Matching
8: Correlation Cross correlation is used to find where two 1 • Cross-Correlation 0.5 • Signal Matching u(t) signals match: u(t) is the test waveform. 0 • Cross-corr as Convolution 0 200 400 600 • Normalized Cross-corr • Example 1: Autocorrelation 1 • Autocorrelation example v(t) contains u(t) with an unknown delay 0.5 v(t) • Fourier Transform Variants 0 • and added noise. Scale Factors 0 200 400 600 800 • Summary w(t) = u(t) v(t) 20 • Spectrogram 10
⊗ w(t) = u∗(τ t)v(τ)dt gives a peak 0 R − 0 200 400 600 800 at the time lag where u(τ t) best − matches v(τ)
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 3 / 11 0 0 0 0 0
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Signal Matching
8: Correlation Cross correlation is used to find where two 1 • Cross-Correlation 0.5 • Signal Matching u(t) signals match: u(t) is the test waveform. 0 • Cross-corr as Convolution 0 200 400 600 • Normalized Cross-corr • Example 1: Autocorrelation 1 • Autocorrelation example v(t) contains u(t) with an unknown delay 0.5 v(t) • Fourier Transform Variants 0 • and added noise. Scale Factors 0 200 400 600 800 • Summary w(t) = u(t) v(t) 20 • Spectrogram 10
⊗ w(t) = u∗(τ t)v(τ)dt gives a peak 0 R − 0 200 400 600 800 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) Fourier Transform - Correlation: 8 – 3 / 11 0 0 0 0 0
0 0 0 0 0
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Signal Matching
8: Correlation Cross correlation is used to find where two 1 • Cross-Correlation 0.5 • Signal Matching u(t) signals match: u(t) is the test waveform. 0 • Cross-corr as Convolution 0 200 400 600 • Normalized Cross-corr • Example 1: Autocorrelation 1 • Autocorrelation example v(t) contains u(t) with an unknown delay 0.5 v(t) • Fourier Transform Variants 0 • and added noise. Scale Factors 0 200 400 600 800 • Summary w(t) = u(t) v(t) 20 • Spectrogram 10
⊗ w(t) = u∗(τ t)v(τ)dt gives a peak 0 R − 0 200 400 600 800 at the time lag where u(τ t) best − matches v(τ); in this case at t = 450 2 0 y(t) -2 Example 2: 0 200 400 600 800 y(t) is the same as v(t) with more noise
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 3 / 11 0 0 0 0 0
0 0 0 0 0
Signal Matching
8: Correlation Cross correlation is used to find where two 1 • Cross-Correlation 0.5 • Signal Matching u(t) signals match: u(t) is the test waveform. 0 • Cross-corr as Convolution 0 200 400 600 • Normalized Cross-corr • Example 1: Autocorrelation 1 • Autocorrelation example v(t) contains u(t) with an unknown delay 0.5 v(t) • Fourier Transform Variants 0 • and added noise. Scale Factors 0 200 400 600 800 • Summary w(t) = u(t) v(t) 20 • Spectrogram 10
⊗ w(t) = u∗(τ t)v(τ)dt gives a peak 0 R − 0 200 400 600 800 at the time lag where u(τ t) best − matches v(τ); in this case at t = 450 2 0 y(t) -2 Example 2: 0 200 400 600 800 y t is the same as v t with more noise 20 ( ) ( ) 10 z(t) = u(t) y(t) can still detect the z(t) 0 -10 correct time⊗ delay (hard for humans) 0 200 400 600 800
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 3 / 11 0 0 0 0 0
Signal Matching
8: Correlation Cross correlation is used to find where two 1 • Cross-Correlation 0.5 • Signal Matching u(t) signals match: u(t) is the test waveform. 0 • Cross-corr as Convolution 0 200 400 600 • Normalized Cross-corr • Example 1: Autocorrelation 1 • Autocorrelation example v(t) contains u(t) with an unknown delay 0.5 v(t) • Fourier Transform Variants 0 • and added noise. Scale Factors 0 200 400 600 800 • Summary w(t) = u(t) v(t) 20 • Spectrogram 10
⊗ w(t) = u∗(τ t)v(τ)dt gives a peak 0 R − 0 200 400 600 800 at the time lag where u(τ t) best − matches v(τ); in this case at t = 450 2 0 y(t) -2 Example 2: 0 200 400 600 800 y t is the same as v t with more noise 20 ( ) ( ) 10 z(t) = u(t) y(t) can still detect the z(t) 0 -10 correct time⊗ delay (hard for humans) 0 200 400 600 800 1 0
Example 3: p(t) -1 p(t) contains u(t) 0 200 400 600 800 −
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 3 / 11 0 0 0 0 0
Signal Matching
8: Correlation Cross correlation is used to find where two 1 • Cross-Correlation 0.5 • Signal Matching u(t) signals match: u(t) is the test waveform. 0 • Cross-corr as Convolution 0 200 400 600 • Normalized Cross-corr • Example 1: Autocorrelation 1 • Autocorrelation example v(t) contains u(t) with an unknown delay 0.5 v(t) • Fourier Transform Variants 0 • and added noise. Scale Factors 0 200 400 600 800 • Summary w(t) = u(t) v(t) 20 • Spectrogram 10
⊗ w(t) = u∗(τ t)v(τ)dt gives a peak 0 R − 0 200 400 600 800 at the time lag where u(τ t) best − matches v(τ); in this case at t = 450 2 0 y(t) -2 Example 2: 0 200 400 600 800 y t is the same as v t with more noise 20 ( ) ( ) 10 z(t) = u(t) y(t) can still detect the z(t) 0 -10 correct time⊗ delay (hard for humans) 0 200 400 600 800 1 0
Example 3: p(t) -1 p(t) contains u(t) so that 0 200 400 600 800 − q(t) = u(t) p(t) has a negative peak ⊗
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 3 / 11 Signal Matching
8: Correlation Cross correlation is used to find where two 1 • Cross-Correlation 0.5 • Signal Matching u(t) signals match: u(t) is the test waveform. 0 • Cross-corr as Convolution 0 200 400 600 • Normalized Cross-corr • Example 1: Autocorrelation 1 • Autocorrelation example v(t) contains u(t) with an unknown delay 0.5 v(t) • Fourier Transform Variants 0 • and added noise. Scale Factors 0 200 400 600 800 • Summary w(t) = u(t) v(t) 20 • Spectrogram 10
⊗ w(t) = u∗(τ t)v(τ)dt gives a peak 0 R − 0 200 400 600 800 at the time lag where u(τ t) best − matches v(τ); in this case at t = 450 2 0 y(t) -2 Example 2: 0 200 400 600 800 y t is the same as v t with more noise 20 ( ) ( ) 10 z(t) = u(t) y(t) can still detect the z(t) 0 -10 correct time⊗ delay (hard for humans) 0 200 400 600 800 1 0
Example 3: p(t) -1 p(t) contains u(t) so that 0 200 400 600 800 − q(t) = u(t) p(t) has a negative peak 0 -10 ⊗ q(t) -20 0 200 400 600 800
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 3 / 11 Cross-correlation as Convolution
8: Correlation • Cross-Correlation Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution • Normalized Cross-corr • Autocorrelation • Autocorrelation example • Fourier Transform Variants • Scale Factors • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 4 / 11 Cross-correlation as Convolution
8: Correlation • Cross-Correlation Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define x(t) = u∗( t) then • Normalized Cross-corr − • Autocorrelation x(t) v(t) , ∞ x(t τ)v(τ)dτ • Autocorrelation example ∗ R−∞ − • Fourier Transform Variants • Scale Factors • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 4 / 11 Cross-correlation as Convolution
8: Correlation • Cross-Correlation Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define x(t) = u∗( t) then • Normalized Cross-corr − • Autocorrelation x(t) v(t) , ∞ x(t τ)v(τ)dτ = ∞ u∗(τ t)v(τ)dτ • Autocorrelation example ∗ R−∞ − R−∞ − • Fourier Transform Variants • Scale Factors • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 4 / 11 Cross-correlation as Convolution
8: Correlation • Cross-Correlation Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define x(t) = u∗( t) then • Normalized Cross-corr − • Autocorrelation x(t) v(t) , ∞ x(t τ)v(τ)dτ = ∞ u∗(τ t)v(τ)dτ • Autocorrelation example ∗ R−∞ − R−∞ − • Fourier Transform Variants = u(t) v(t) • Scale Factors ⊗ • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 4 / 11 Cross-correlation as Convolution
8: Correlation • Cross-Correlation Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define x(t) = u∗( t) then • Normalized Cross-corr − • Autocorrelation x(t) v(t) , ∞ x(t τ)v(τ)dτ = ∞ u∗(τ t)v(τ)dτ • Autocorrelation example ∗ R−∞ − R−∞ − • Fourier Transform Variants = u(t) v(t) • Scale Factors ⊗ • Summary • Spectrogram Fourier Transform of x(t): i2πft X(f) = ∞ x(t)e− dt R−∞
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 4 / 11 Cross-correlation as Convolution
8: Correlation • Cross-Correlation Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define x(t) = u∗( t) then • Normalized Cross-corr − • Autocorrelation x(t) v(t) , ∞ x(t τ)v(τ)dτ = ∞ u∗(τ t)v(τ)dτ • Autocorrelation example ∗ R−∞ − R−∞ − • Fourier Transform Variants = u(t) v(t) • Scale Factors ⊗ • Summary • Spectrogram Fourier Transform of x(t): i2πft i2πft X(f) = ∞ x(t)e− dt = ∞ u∗( t)e− dt R−∞ R−∞ −
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 4 / 11 Cross-correlation as Convolution
8: Correlation • Cross-Correlation Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define x(t) = u∗( t) then • Normalized Cross-corr − • Autocorrelation x(t) v(t) , ∞ x(t τ)v(τ)dτ = ∞ u∗(τ t)v(τ)dτ • Autocorrelation example ∗ R−∞ − R−∞ − • Fourier Transform Variants = u(t) v(t) • Scale Factors ⊗ • Summary • Spectrogram Fourier Transform of x(t): i2πft i2πft X(f) = ∞ x(t)e− dt = ∞ u∗( t)e− dt R−∞ R−∞ − i2πft = ∞ u∗(t)e dt R−∞
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 4 / 11 Cross-correlation as Convolution
8: Correlation • Cross-Correlation Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define x(t) = u∗( t) then • Normalized Cross-corr − • Autocorrelation x(t) v(t) , ∞ x(t τ)v(τ)dτ = ∞ u∗(τ t)v(τ)dτ • Autocorrelation example ∗ R−∞ − R−∞ − • Fourier Transform Variants = u(t) v(t) • Scale Factors ⊗ • Summary • Spectrogram Fourier Transform of x(t): i2πft i2πft X(f) = ∞ x(t)e− dt = ∞ u∗( t)e− dt R−∞ R−∞ − i2πft i2πft ∗ = ∞ u∗(t)e dt = ∞ u(t)e− dt R−∞ R−∞
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 4 / 11 Cross-correlation as Convolution
8: Correlation • Cross-Correlation Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define x(t) = u∗( t) then • Normalized Cross-corr − • Autocorrelation x(t) v(t) , ∞ x(t τ)v(τ)dτ = ∞ u∗(τ t)v(τ)dτ • Autocorrelation example ∗ R−∞ − R−∞ − • Fourier Transform Variants = u(t) v(t) • Scale Factors ⊗ • Summary • Spectrogram Fourier Transform of x(t): i2πft i2πft X(f) = ∞ x(t)e− dt = ∞ u∗( t)e− dt R−∞ R−∞ − i2πft i2πft ∗ = ∞ u∗(t)e dt = ∞ u(t)e− dt R−∞ R−∞ = U ∗(f)
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 4 / 11 Cross-correlation as Convolution
8: Correlation • Cross-Correlation Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define x(t) = u∗( t) then • Normalized Cross-corr − • Autocorrelation x(t) v(t) , ∞ x(t τ)v(τ)dτ = ∞ u∗(τ t)v(τ)dτ • Autocorrelation example ∗ R−∞ − R−∞ − • Fourier Transform Variants = u(t) v(t) • Scale Factors ⊗ • Summary • Spectrogram Fourier Transform of x(t): i2πft i2πft X(f) = ∞ x(t)e− dt = ∞ u∗( t)e− dt R−∞ R−∞ − i2πft i2πft ∗ = ∞ u∗(t)e dt = ∞ u(t)e− dt R−∞ R−∞ = 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 Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define x(t) = u∗( t) then • Normalized Cross-corr − • Autocorrelation x(t) v(t) , ∞ x(t τ)v(τ)dτ = ∞ u∗(τ t)v(τ)dτ • Autocorrelation example ∗ R−∞ − R−∞ − • Fourier Transform Variants = u(t) v(t) • Scale Factors ⊗ • Summary • Spectrogram Fourier Transform of x(t): i2πft i2πft X(f) = ∞ x(t)e− dt = ∞ u∗( t)e− dt R−∞ R−∞ − i2πft i2πft ∗ = ∞ u∗(t)e dt = ∞ u(t)e− dt R−∞ R−∞ = 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 Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define x(t) = u∗( t) then • Normalized Cross-corr − • Autocorrelation x(t) v(t) , ∞ x(t τ)v(τ)dτ = ∞ u∗(τ t)v(τ)dτ • Autocorrelation example ∗ R−∞ − R−∞ − • Fourier Transform Variants = u(t) v(t) • Scale Factors ⊗ • Summary • Spectrogram Fourier Transform of x(t): i2πft i2πft X(f) = ∞ x(t)e− dt = ∞ u∗( t)e− dt R−∞ R−∞ − i2πft i2πft ∗ = ∞ u∗(t)e dt = ∞ u(t)e− dt R−∞ R−∞ = 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 Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define x(t) = u∗( t) then • Normalized Cross-corr − • Autocorrelation x(t) v(t) , ∞ x(t τ)v(τ)dτ = ∞ u∗(τ t)v(τ)dτ • Autocorrelation example ∗ R−∞ − R−∞ − • Fourier Transform Variants = u(t) v(t) • Scale Factors ⊗ • Summary • Spectrogram Fourier Transform of x(t): i2πft i2πft X(f) = ∞ x(t)e− dt = ∞ u∗( t)e− dt R−∞ R−∞ − i2πft i2πft ∗ = ∞ u∗(t)e dt = ∞ u(t)e− dt R−∞ R−∞ = 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 Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define x(t) = u∗( t) then • Normalized Cross-corr − • Autocorrelation x(t) v(t) , ∞ x(t τ)v(τ)dτ = ∞ u∗(τ t)v(τ)dτ • Autocorrelation example ∗ R−∞ − R−∞ − • Fourier Transform Variants = u(t) v(t) • Scale Factors ⊗ • Summary • Spectrogram Fourier Transform of x(t): i2πft i2πft X(f) = ∞ x(t)e− dt = ∞ u∗( t)e− dt R−∞ R−∞ − i2πft i2πft ∗ = ∞ u∗(t)e dt = ∞ u(t)e− dt R−∞ R−∞ = 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) W (f) = U ∗(f)V (f) ⊗ ⇔ = 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 Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define x(t) = u∗( t) then • Normalized Cross-corr − • Autocorrelation x(t) v(t) , ∞ x(t τ)v(τ)dτ = ∞ u∗(τ t)v(τ)dτ • Autocorrelation example ∗ R−∞ − R−∞ − • Fourier Transform Variants = u(t) v(t) • Scale Factors ⊗ • Summary • Spectrogram Fourier Transform of x(t): i2πft i2πft X(f) = ∞ x(t)e− dt = ∞ u∗( t)e− dt R−∞ R−∞ − i2πft i2πft ∗ = ∞ u∗(t)e dt = ∞ u(t)e− dt R−∞ R−∞ = 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) W (f) = U ∗(f)V (f) ⊗ ⇔ = u∗( t) v(t) − ∗ 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 Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define x(t) = u∗( t) then • Normalized Cross-corr − • Autocorrelation x(t) v(t) , ∞ x(t τ)v(τ)dτ = ∞ u∗(τ t)v(τ)dτ • Autocorrelation example ∗ R−∞ − R−∞ − • Fourier Transform Variants = u(t) v(t) • Scale Factors ⊗ • Summary • Spectrogram Fourier Transform of x(t): i2πft i2πft X(f) = ∞ x(t)e− dt = ∞ u∗( t)e− dt R−∞ R−∞ − i2πft i2πft ∗ = ∞ u∗(t)e dt = ∞ u(t)e− dt R−∞ R−∞ = 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) W (f) = U ∗(f)V (f) ⊗ ⇔ = u∗( t) v(t) − ∗ 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 Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define x(t) = u∗( t) then • Normalized Cross-corr − • Autocorrelation x(t) v(t) , ∞ x(t τ)v(τ)dτ = ∞ u∗(τ t)v(τ)dτ • Autocorrelation example ∗ R−∞ − R−∞ − • Fourier Transform Variants = u(t) v(t) • Scale Factors ⊗ • Summary • Spectrogram Fourier Transform of x(t): i2πft i2πft X(f) = ∞ x(t)e− dt = ∞ u∗( t)e− dt R−∞ R−∞ − i2πft i2πft ∗ = ∞ u∗(t)e dt = ∞ u(t)e− dt R−∞ R−∞ = 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) W (f) = U ∗(f)V (f) ⊗ ⇔ = u∗( t) v(t) − ∗ 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 Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution • Normalized Cross-corr • Autocorrelation • Autocorrelation example • Fourier Transform Variants • Scale Factors • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 5 / 11 Normalized Cross-correlation
8: Correlation • Cross-Correlation Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define y(t) = u(t t0) for some fixed t0, then Ey = Eu: • Normalized Cross-corr − • Autocorrelation • Autocorrelation example • Fourier Transform Variants • Scale Factors • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 5 / 11 Normalized Cross-correlation
8: Correlation • Cross-Correlation Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define y(t) = u(t t0) for some fixed t0, then Ey = Eu: • Normalized Cross-corr − • Autocorrelation 2 Ey = ∞ y(t) dt • Autocorrelation example R−∞ | | • Fourier Transform Variants • Scale Factors • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 5 / 11 Normalized Cross-correlation
8: Correlation • Cross-Correlation Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define y(t) = u(t t0) for some fixed t0, then Ey = Eu: • Normalized Cross-corr − • Autocorrelation 2 2 Ey = ∞ y(t) dt = ∞ u(t t ) dt • Autocorrelation example 0 R−∞ | | R−∞ | − | • Fourier Transform Variants • Scale Factors • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 5 / 11 Normalized Cross-correlation
8: Correlation • Cross-Correlation Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define y(t) = u(t t0) for some fixed t0, then Ey = Eu: • Normalized Cross-corr − • Autocorrelation 2 2 Ey = ∞ y(t) dt = ∞ u(t t ) dt • Autocorrelation example 0 R−∞ | | R−∞ | − | • Fourier Transform Variants 2 • Scale Factors = ∞ u(τ) dτ = Eu [t τ + t0] • Summary R−∞ | | → • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 5 / 11 Normalized Cross-correlation
8: Correlation • Cross-Correlation Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define y(t) = u(t t0) for some fixed t0, then Ey = Eu: • Normalized Cross-corr − • Autocorrelation 2 2 Ey = ∞ y(t) dt = ∞ u(t t ) dt • Autocorrelation example 0 R−∞ | | R−∞ | − | • Fourier Transform Variants 2 • Scale Factors = ∞ u(τ) dτ = Eu [t τ + t0] • Summary R−∞ | | → • Spectrogram 2 Cauchy-Schwarz inequality: ∞ y∗(τ)v(τ)dτ EyEv R−∞ ≤
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 5 / 11 Normalized Cross-correlation
8: Correlation • Cross-Correlation Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define y(t) = u(t t0) for some fixed t0, then Ey = Eu: • Normalized Cross-corr − • Autocorrelation 2 2 Ey = ∞ y(t) dt = ∞ u(t t ) dt • Autocorrelation example 0 R−∞ | | R−∞ | − | • Fourier Transform Variants 2 • Scale Factors = ∞ u(τ) dτ = Eu [t τ + t0] • Summary R−∞ | | → • Spectrogram 2 Cauchy-Schwarz inequality: ∞ y∗(τ)v(τ)dτ EyEv R−∞ ≤ 2 2 w(t0) = ∞ u∗(τ t0)v(τ)dτ ⇒ | | R−∞ −
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 5 / 11 Normalized Cross-correlation
8: Correlation • Cross-Correlation Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define y(t) = u(t t0) for some fixed t0, then Ey = Eu: • Normalized Cross-corr − • Autocorrelation 2 2 Ey = ∞ y(t) dt = ∞ u(t t ) dt • Autocorrelation example 0 R−∞ | | R−∞ | − | • Fourier Transform Variants 2 • Scale Factors = ∞ u(τ) dτ = Eu [t τ + t0] • Summary R−∞ | | → • Spectrogram 2 Cauchy-Schwarz inequality: ∞ y∗(τ)v(τ)dτ EyEv R−∞ ≤ 2 2 w(t0) = ∞ u∗(τ t0)v(τ)dτ EyEv ⇒ | | R−∞ − ≤
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 5 / 11 Normalized Cross-correlation
8: Correlation • Cross-Correlation Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define y(t) = u(t t0) for some fixed t0, then Ey = Eu: • Normalized Cross-corr − • Autocorrelation 2 2 Ey = ∞ y(t) dt = ∞ u(t t ) dt • Autocorrelation example 0 R−∞ | | R−∞ | − | • Fourier Transform Variants 2 • Scale Factors = ∞ u(τ) dτ = Eu [t τ + t0] • Summary R−∞ | | → • Spectrogram 2 Cauchy-Schwarz inequality: ∞ y∗(τ)v(τ)dτ EyEv R−∞ ≤ 2 2 w(t0) = ∞ u∗(τ t0)v(τ)dτ EyEv = EuEv ⇒ | | R−∞ − ≤
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 5 / 11 Normalized Cross-correlation
8: Correlation • Cross-Correlation Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define y(t) = u(t t0) for some fixed t0, then Ey = Eu: • Normalized Cross-corr − • Autocorrelation 2 2 Ey = ∞ y(t) dt = ∞ u(t t ) dt • Autocorrelation example 0 R−∞ | | R−∞ | − | • Fourier Transform Variants 2 • Scale Factors = ∞ u(τ) dτ = Eu [t τ + t0] • Summary R−∞ | | → • Spectrogram 2 Cauchy-Schwarz inequality: ∞ y∗(τ)v(τ)dτ EyEv R−∞ ≤ 2 2 w(t0) = ∞ u∗(τ t0)v(τ)dτ EyEv = EuEv ⇒ | | R−∞ − ≤
but t was arbitrary, so we must have w(t) √EuEv for all t 0 | | ≤
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 5 / 11 Normalized Cross-correlation
8: Correlation • Cross-Correlation Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define y(t) = u(t t0) for some fixed t0, then Ey = Eu: • Normalized Cross-corr − • Autocorrelation 2 2 Ey = ∞ y(t) dt = ∞ u(t t ) dt • Autocorrelation example 0 R−∞ | | R−∞ | − | • Fourier Transform Variants 2 • Scale Factors = ∞ u(τ) dτ = Eu [t τ + t0] • Summary R−∞ | | → • Spectrogram 2 Cauchy-Schwarz inequality: ∞ y∗(τ)v(τ)dτ EyEv R−∞ ≤ 2 2 w(t0) = ∞ u∗(τ t0)v(τ)dτ EyEv = EuEv ⇒ | | R−∞ − ≤
but t was arbitrary, so we must have w(t) √EuEv for all t 0 | | ≤ We can define the normalized cross-correlation u(t) v(t) z(t) = ⊗ √EuEv
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 5 / 11 Normalized Cross-correlation
8: Correlation • Cross-Correlation Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define y(t) = u(t t0) for some fixed t0, then Ey = Eu: • Normalized Cross-corr − • Autocorrelation 2 2 Ey = ∞ y(t) dt = ∞ u(t t ) dt • Autocorrelation example 0 R−∞ | | R−∞ | − | • Fourier Transform Variants 2 • Scale Factors = ∞ u(τ) dτ = Eu [t τ + t0] • Summary R−∞ | | → • Spectrogram 2 Cauchy-Schwarz inequality: ∞ y∗(τ)v(τ)dτ EyEv R−∞ ≤ 2 2 w(t0) = ∞ u∗(τ t0)v(τ)dτ EyEv = EuEv ⇒ | | R−∞ − ≤
but t was arbitrary, so we must have w(t) √EuEv for all t 0 | | ≤ We can define the normalized cross-correlation u(t) v(t) z(t) = ⊗ √EuEv 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 Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching ⊗ R−∞ − • Cross-corr as Convolution If we define y(t) = u(t t0) for some fixed t0, then Ey = Eu: • Normalized Cross-corr − • Autocorrelation 2 2 Ey = ∞ y(t) dt = ∞ u(t t ) dt • Autocorrelation example 0 R−∞ | | R−∞ | − | • Fourier Transform Variants 2 • Scale Factors = ∞ u(τ) dτ = Eu [t τ + t0] • Summary R−∞ | | → • Spectrogram 2 Cauchy-Schwarz inequality: ∞ y∗(τ)v(τ)dτ EyEv R−∞ ≤ 2 2 w(t0) = ∞ u∗(τ t0)v(τ)dτ EyEv = EuEv ⇒ | | R−∞ − ≤
but t was arbitrary, so we must have w(t) √EuEv for all t 0 | | ≤ We can define the normalized cross-correlation u(t) v(t) z(t) = ⊗ √EuEv with properties: (1) z(t) 1 for all t | | ≤ (2) z(t ) = 1 v(τ) = αu(τ t ) with α constant | 0 | ⇔ − 0
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 5 / 11 Autocorrelation
8: Correlation The correlation of a signal with itself is its autocorrelation: • Cross-Correlation • Signal Matching w(t) = u(t) u(t) • Cross-corr as Convolution ⊗ • Normalized Cross-corr • Autocorrelation • Autocorrelation example • Fourier Transform Variants • Scale Factors • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 6 / 11 Autocorrelation
8: Correlation The correlation of a signal with itself is its autocorrelation: • Cross-Correlation • Signal Matching w(t) = u(t) u(t) = ∞ u∗(τ t)u(τ)dτ • Cross-corr as Convolution ⊗ R − • Normalized Cross-corr −∞ • Autocorrelation • Autocorrelation example • Fourier Transform Variants • Scale Factors • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 6 / 11 Autocorrelation
8: Correlation The correlation of a signal with itself is its autocorrelation: • Cross-Correlation • Signal Matching w(t) = u(t) u(t) = ∞ u∗(τ t)u(τ)dτ • Cross-corr as Convolution ⊗ R − • Normalized Cross-corr −∞ • Autocorrelation The autocorrelation at zero lag: • Autocorrelation example • Fourier Transform Variants w(0) = ∞ u∗(τ 0)u(τ)dτ • Scale Factors R−∞ − • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 6 / 11 Autocorrelation
8: Correlation The correlation of a signal with itself is its autocorrelation: • Cross-Correlation • Signal Matching w(t) = u(t) u(t) = ∞ u∗(τ t)u(τ)dτ • Cross-corr as Convolution ⊗ R − • Normalized Cross-corr −∞ • Autocorrelation The autocorrelation at zero lag: • Autocorrelation example • Fourier Transform Variants w(0) = ∞ u∗(τ 0)u(τ)dτ • Scale Factors R−∞ − • Summary = ∞ u (τ)u(τ)dτ • ∗ Spectrogram R−∞
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 6 / 11 Autocorrelation
8: Correlation The correlation of a signal with itself is its autocorrelation: • Cross-Correlation • Signal Matching w(t) = u(t) u(t) = ∞ u∗(τ t)u(τ)dτ • Cross-corr as Convolution ⊗ R − • Normalized Cross-corr −∞ • Autocorrelation The autocorrelation at zero lag: • Autocorrelation example • Fourier Transform Variants w(0) = ∞ u∗(τ 0)u(τ)dτ • Scale Factors R−∞ − • Summary = ∞ u (τ)u(τ)dτ • ∗ Spectrogram R−∞ 2 = ∞ u(τ) dτ = Eu R−∞ | |
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 6 / 11 Autocorrelation
8: Correlation The correlation of a signal with itself is its autocorrelation: • Cross-Correlation • Signal Matching w(t) = u(t) u(t) = ∞ u∗(τ t)u(τ)dτ • Cross-corr as Convolution ⊗ R − • Normalized Cross-corr −∞ • Autocorrelation The autocorrelation at zero lag: • Autocorrelation example • Fourier Transform Variants w(0) = ∞ u∗(τ 0)u(τ)dτ • Scale Factors R−∞ − • Summary = ∞ u (τ)u(τ)dτ • ∗ Spectrogram R−∞ 2 = ∞ u(τ) dτ = Eu R−∞ | | The autocorrelation at zero lag, w(0), is the energy of the signal.
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 6 / 11 Autocorrelation
8: Correlation The correlation of a signal with itself is its autocorrelation: • Cross-Correlation • Signal Matching w(t) = u(t) u(t) = ∞ u∗(τ t)u(τ)dτ • Cross-corr as Convolution ⊗ R − • Normalized Cross-corr −∞ • Autocorrelation The autocorrelation at zero lag: • Autocorrelation example • Fourier Transform Variants w(0) = ∞ u∗(τ 0)u(τ)dτ • Scale Factors R−∞ − • Summary = ∞ u (τ)u(τ)dτ • ∗ Spectrogram R−∞ 2 = ∞ u(τ) dτ = Eu R−∞ | | The autocorrelation at zero lag, w(0), is the energy of the signal.
u(t) u(t) The normalized autocorrelation: z t ⊗ ( ) = Eu
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 6 / 11 Autocorrelation
8: Correlation The correlation of a signal with itself is its autocorrelation: • Cross-Correlation • Signal Matching w(t) = u(t) u(t) = ∞ u∗(τ t)u(τ)dτ • Cross-corr as Convolution ⊗ R − • Normalized Cross-corr −∞ • Autocorrelation The autocorrelation at zero lag: • Autocorrelation example • Fourier Transform Variants w(0) = ∞ u∗(τ 0)u(τ)dτ • Scale Factors R−∞ − • Summary = ∞ u (τ)u(τ)dτ • ∗ Spectrogram R−∞ 2 = ∞ u(τ) dτ = Eu R−∞ | | The autocorrelation at zero lag, w(0), is the energy of the signal.
u(t) u(t) The normalized autocorrelation: z t ⊗ ( ) = Eu satisfies z(0) = 1 and z(t) 1 for any t. | | ≤
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 6 / 11 Autocorrelation
8: Correlation The correlation of a signal with itself is its autocorrelation: • Cross-Correlation • Signal Matching w(t) = u(t) u(t) = ∞ u∗(τ t)u(τ)dτ • Cross-corr as Convolution ⊗ R − • Normalized Cross-corr −∞ • Autocorrelation The autocorrelation at zero lag: • Autocorrelation example • Fourier Transform Variants w(0) = ∞ u∗(τ 0)u(τ)dτ • Scale Factors R−∞ − • Summary = ∞ u (τ)u(τ)dτ • ∗ Spectrogram R−∞ 2 = ∞ u(τ) dτ = Eu R−∞ | | The autocorrelation at zero lag, w(0), is the energy of the signal.
u(t) u(t) The normalized autocorrelation: z t ⊗ ( ) = Eu satisfies z(0) = 1 and z(t) 1 for any t. | | ≤
Wiener-Khinchin Theorem: [Cross-correlation theorem when v(t) = u(t)]
w(t) = u(t) u(t) W (f) = U ∗(f)U(f) ⊗ ⇔
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 6 / 11 Autocorrelation
8: Correlation The correlation of a signal with itself is its autocorrelation: • Cross-Correlation • Signal Matching w(t) = u(t) u(t) = ∞ u∗(τ t)u(τ)dτ • Cross-corr as Convolution ⊗ R − • Normalized Cross-corr −∞ • Autocorrelation The autocorrelation at zero lag: • Autocorrelation example • Fourier Transform Variants w(0) = ∞ u∗(τ 0)u(τ)dτ • Scale Factors R−∞ − • Summary = ∞ u (τ)u(τ)dτ • ∗ Spectrogram R−∞ 2 = ∞ u(τ) dτ = Eu R−∞ | | The autocorrelation at zero lag, w(0), is the energy of the signal.
u(t) u(t) The normalized autocorrelation: z t ⊗ ( ) = Eu satisfies z(0) = 1 and z(t) 1 for any t. | | ≤
Wiener-Khinchin Theorem: [Cross-correlation theorem when v(t) = u(t)] 2 w(t) = u(t) u(t) W (f) = U ∗(f)U(f) = U(f) ⊗ ⇔ | |
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 6 / 11 Autocorrelation
8: Correlation The correlation of a signal with itself is its autocorrelation: • Cross-Correlation • Signal Matching w(t) = u(t) u(t) = ∞ u∗(τ t)u(τ)dτ • Cross-corr as Convolution ⊗ R − • Normalized Cross-corr −∞ • Autocorrelation The autocorrelation at zero lag: • Autocorrelation example • Fourier Transform Variants w(0) = ∞ u∗(τ 0)u(τ)dτ • Scale Factors R−∞ − • Summary = ∞ u (τ)u(τ)dτ • ∗ Spectrogram R−∞ 2 = ∞ u(τ) dτ = Eu R−∞ | | The autocorrelation at zero lag, w(0), is the energy of the signal.
u(t) u(t) The normalized autocorrelation: z t ⊗ ( ) = Eu satisfies z(0) = 1 and z(t) 1 for any t. | | ≤
Wiener-Khinchin Theorem: [Cross-correlation theorem when v(t) = u(t)] 2 w(t) = u(t) u(t) W (f) = U ∗(f)U(f) = U(f) ⊗ ⇔ | | 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 is used to find when two different signals are similar. • Cross-Correlation • Signal Matching • Cross-corr as Convolution • Normalized Cross-corr • Autocorrelation • Autocorrelation example • Fourier Transform Variants • Scale Factors • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 7 / 11 Autocorrelation example
8: Correlation Cross-correlation is used to find when two different signals are similar. • Cross-Correlation • Signal Matching Autocorrelation is used to find when a signal is similar to itself delayed. • Cross-corr as Convolution • Normalized Cross-corr • Autocorrelation • Autocorrelation example • Fourier Transform Variants • Scale Factors • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 7 / 11 Autocorrelation example
8: Correlation Cross-correlation is used to find when two different signals are similar. • Cross-Correlation • Signal Matching Autocorrelation is used to find when a signal is similar to itself delayed. • Cross-corr as Convolution • Normalized Cross-corr First graph shows s t a segment of the microphone signal from the initial • Autocorrelation ( ) • Autocorrelation example vowel of “early” spoken by me. • Fourier Transform Variants • Scale Factors • Summary • Spectrogram
1
0.5
0
Speech s(t) Speech -0.5
-1 10 20 30 40 50 Time (ms)
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 7 / 11 Autocorrelation example
8: Correlation Cross-correlation is used to find when two different signals are similar. • Cross-Correlation • Signal Matching Autocorrelation is used to find when a signal is similar to itself delayed. • Cross-corr as Convolution • Normalized Cross-corr First graph shows s t a segment of the microphone signal from the initial • Autocorrelation ( ) • Autocorrelation example vowel of “early” spoken by me. The waveform is “quasi-periodic” = • Fourier Transform Variants • Scale Factors “almost periodic but not quite”. • Summary • Spectrogram
1
0.5
0
Speech s(t) Speech -0.5
-1 10 20 30 40 50 Time (ms)
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 7 / 11 Autocorrelation example
8: Correlation Cross-correlation is used to find when two different signals are similar. • Cross-Correlation • Signal Matching Autocorrelation is used to find when a signal is similar to itself delayed. • Cross-corr as Convolution • Normalized Cross-corr First graph shows s t a segment of the microphone signal from the initial • Autocorrelation ( ) • Autocorrelation example vowel of “early” spoken by me. The waveform is “quasi-periodic” = • Fourier Transform Variants • Scale Factors “almost periodic but not quite”. • Summary s(t) s(t) • Spectrogram Second graph shows normalized autocorrelation, z t ⊗ . ( ) = Es
1 1 0.82 Lag = 6.2 ms (161 Hz) 0.5 0.5 0
Speech s(t) Speech -0.5 0
-1 z(t) Autocorr, Norm 10 20 30 40 50 0 5 10 15 20 Time (ms) Lag: t (ms)
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 7 / 11 Autocorrelation example
8: Correlation Cross-correlation is used to find when two different signals are similar. • Cross-Correlation • Signal Matching Autocorrelation is used to find when a signal is similar to itself delayed. • Cross-corr as Convolution • Normalized Cross-corr First graph shows s t a segment of the microphone signal from the initial • Autocorrelation ( ) • Autocorrelation example vowel of “early” spoken by me. The waveform is “quasi-periodic” = • Fourier Transform Variants • Scale Factors “almost periodic but not quite”. • Summary s(t) s(t) • Spectrogram Second graph shows normalized autocorrelation, z t ⊗ . ( ) = Es z(0) = 1 for t = 0 since a signal always matches itself exactly.
1 1 0.82 Lag = 6.2 ms (161 Hz) 0.5 0.5 0
Speech s(t) Speech -0.5 0
-1 z(t) Autocorr, Norm 10 20 30 40 50 0 5 10 15 20 Time (ms) Lag: t (ms)
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 7 / 11 Autocorrelation example
8: Correlation Cross-correlation is used to find when two different signals are similar. • Cross-Correlation • Signal Matching Autocorrelation is used to find when a signal is similar to itself delayed. • Cross-corr as Convolution • Normalized Cross-corr First graph shows s t a segment of the microphone signal from the initial • Autocorrelation ( ) • Autocorrelation example vowel of “early” spoken by me. The waveform is “quasi-periodic” = • Fourier Transform Variants • Scale Factors “almost periodic but not quite”. • Summary s(t) s(t) • Spectrogram Second graph shows normalized autocorrelation, z 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).
1 1 0.82 Lag = 6.2 ms (161 Hz) 0.5 0.5 0
Speech s(t) Speech -0.5 0
-1 z(t) Autocorr, Norm 10 20 30 40 50 0 5 10 15 20 Time (ms) Lag: t (ms)
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 7 / 11 Autocorrelation example
8: Correlation Cross-correlation is used to find when two different signals are similar. • Cross-Correlation • Signal Matching Autocorrelation is used to find when a signal is similar to itself delayed. • Cross-corr as Convolution • Normalized Cross-corr First graph shows s t a segment of the microphone signal from the initial • Autocorrelation ( ) • Autocorrelation example vowel of “early” spoken by me. The waveform is “quasi-periodic” = • Fourier Transform Variants • Scale Factors “almost periodic but not quite”. • Summary s(t) s(t) • Spectrogram Second graph shows normalized autocorrelation, z 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).
1 1 0.82 Lag = 6.2 ms (161 Hz) 0.5 0.5 0
Speech s(t) Speech -0.5 0
-1 z(t) Autocorr, Norm 10 20 30 40 50 0 5 10 15 20 Time (ms) Lag: t (ms)
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 7 / 11 Fourier Transform Variants
8: Correlation There are three different versions of the Fourier Transform in current use. • Cross-Correlation • Signal Matching • Cross-corr as Convolution • Normalized Cross-corr • Autocorrelation • Autocorrelation example • Fourier Transform Variants • Scale Factors • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 8 / 11 Fourier Transform Variants
8: Correlation There are three different versions of the Fourier Transform in current use. • Cross-Correlation • Signal Matching • Cross-corr as Convolution (1) Frequency version (we have used this in lectures) • Normalized Cross-corr i2πft i2πft • Autocorrelation U(f) = ∞ u(t)e− dt u(t) = ∞ U(f)e df • Autocorrelation example R−∞ R−∞ • Fourier Transform Variants • Scale Factors • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 8 / 11 Fourier Transform Variants
8: Correlation There are three different versions of the Fourier Transform in current use. • Cross-Correlation • Signal Matching • Cross-corr as Convolution (1) Frequency version (we have used this in lectures) • Normalized Cross-corr i2πft i2πft • Autocorrelation U(f) = ∞ u(t)e− dt u(t) = ∞ U(f)e df • Autocorrelation example R−∞ R−∞ • Fourier Transform Variants Used in the communications/broadcasting industry and textbooks. • Scale Factors • ,,, • Summary The formulae do not need scale factors of 2π anywhere. • Spectrogram •
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 8 / 11 Fourier Transform Variants
8: Correlation There are three different versions of the Fourier Transform in current use. • Cross-Correlation • Signal Matching • Cross-corr as Convolution (1) Frequency version (we have used this in lectures) • Normalized Cross-corr i2πft i2πft • Autocorrelation U(f) = ∞ u(t)e− dt u(t) = ∞ U(f)e df • Autocorrelation example R−∞ R−∞ • Fourier Transform Variants Used in the communications/broadcasting industry and textbooks. • Scale Factors • ,,, • Summary The formulae do not need scale factors of 2π anywhere. • Spectrogram • (2) Angular frequency version iωt 1 iωt U(ω) = ∞ u(t)e− dt u(t) = ∞ U(ω)e dω R 2π R e −∞ −∞ e
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 8 / 11 Fourier Transform Variants
8: Correlation There are three different versions of the Fourier Transform in current use. • Cross-Correlation • Signal Matching • Cross-corr as Convolution (1) Frequency version (we have used this in lectures) • Normalized Cross-corr i2πft i2πft • Autocorrelation U(f) = ∞ u(t)e− dt u(t) = ∞ U(f)e df • Autocorrelation example R−∞ R−∞ • Fourier Transform Variants Used in the communications/broadcasting industry and textbooks. • Scale Factors • ,,, • Summary The formulae do not need scale factors of 2π anywhere. • Spectrogram • (2) Angular frequency version iωt 1 iωt U(ω) = ∞ u(t)e− dt u(t) = ∞ U(ω)e dω R 2π R e −∞ −∞ e ω Continuous spectra are unchanged: U(ω) = U(f) = U( 2π ) e
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 8 / 11 Fourier Transform Variants
8: Correlation There are three different versions of the Fourier Transform in current use. • Cross-Correlation • Signal Matching • Cross-corr as Convolution (1) Frequency version (we have used this in lectures) • Normalized Cross-corr i2πft i2πft • Autocorrelation U(f) = ∞ u(t)e− dt u(t) = ∞ U(f)e df • Autocorrelation example R−∞ R−∞ • Fourier Transform Variants Used in the communications/broadcasting industry and textbooks. • Scale Factors • ,,, • Summary The formulae do not need scale factors of 2π anywhere. • Spectrogram • (2) Angular frequency version iωt 1 iωt U(ω) = ∞ u(t)e− dt u(t) = ∞ U(ω)e dω R 2π R e −∞ −∞ e ω Continuous spectra are unchanged: U(ω) = U(f) = U( 2π ) However δ-function spectral componentse are multiplied by 2π so that U(f) = δ(f f ) U(ω) = 2π δ(ω 2πf ) − 0 ⇒ × − 0 e
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 8 / 11 Fourier Transform Variants
8: Correlation There are three different versions of the Fourier Transform in current use. • Cross-Correlation • Signal Matching • Cross-corr as Convolution (1) Frequency version (we have used this in lectures) • Normalized Cross-corr i2πft i2πft • Autocorrelation U(f) = ∞ u(t)e− dt u(t) = ∞ U(f)e df • Autocorrelation example R−∞ R−∞ • Fourier Transform Variants Used in the communications/broadcasting industry and textbooks. • Scale Factors • ,,, • Summary The formulae do not need scale factors of 2π anywhere. • Spectrogram • (2) Angular frequency version iωt 1 iωt U(ω) = ∞ u(t)e− dt u(t) = ∞ U(ω)e dω R 2π R e −∞ −∞ e ω Continuous spectra are unchanged: U(ω) = U(f) = U( 2π ) However δ-function spectral componentse are multiplied by 2π so that U(f) = δ(f f ) U(ω) = 2π δ(ω 2πf ) − 0 ⇒ × − 0 Used in most signal processing ande control theory textbooks. •
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 8 / 11 Fourier Transform Variants
8: Correlation There are three different versions of the Fourier Transform in current use. • Cross-Correlation • Signal Matching • Cross-corr as Convolution (1) Frequency version (we have used this in lectures) • Normalized Cross-corr i2πft i2πft • Autocorrelation U(f) = ∞ u(t)e− dt u(t) = ∞ U(f)e df • Autocorrelation example R−∞ R−∞ • Fourier Transform Variants Used in the communications/broadcasting industry and textbooks. • Scale Factors • ,,, • Summary The formulae do not need scale factors of 2π anywhere. • Spectrogram • (2) Angular frequency version iωt 1 iωt U(ω) = ∞ u(t)e− dt u(t) = ∞ U(ω)e dω R 2π R e −∞ −∞ e ω Continuous spectra are unchanged: U(ω) = U(f) = U( 2π ) However δ-function spectral componentse are multiplied by 2π so that U(f) = δ(f f ) U(ω) = 2π δ(ω 2πf ) − 0 ⇒ × − 0 Used in most signal processing ande control theory textbooks. • (3) Angular frequency + symmetrical scale factor U(ω) = 1 ∞ u(t)e iωtdt u(t) = 1 ∞ U(ω)eiωtdω √2π R − √2π R b −∞ −∞ b
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 8 / 11 Fourier Transform Variants
8: Correlation There are three different versions of the Fourier Transform in current use. • Cross-Correlation • Signal Matching • Cross-corr as Convolution (1) Frequency version (we have used this in lectures) • Normalized Cross-corr i2πft i2πft • Autocorrelation U(f) = ∞ u(t)e− dt u(t) = ∞ U(f)e df • Autocorrelation example R−∞ R−∞ • Fourier Transform Variants Used in the communications/broadcasting industry and textbooks. • Scale Factors • ,,, • Summary The formulae do not need scale factors of 2π anywhere. • Spectrogram • (2) Angular frequency version iωt 1 iωt U(ω) = ∞ u(t)e− dt u(t) = ∞ U(ω)e dω R 2π R e −∞ −∞ e ω Continuous spectra are unchanged: U(ω) = U(f) = U( 2π ) However δ-function spectral componentse are multiplied by 2π so that U(f) = δ(f f ) U(ω) = 2π δ(ω 2πf ) − 0 ⇒ × − 0 Used in most signal processing ande control theory textbooks. • (3) Angular frequency + symmetrical scale factor U(ω) = 1 ∞ u(t)e iωtdt u(t) = 1 ∞ U(ω)eiωtdω √2π − √2π R−∞ R−∞ Inb all cases U(ω) = 1 U(ω) b √2π b e
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 8 / 11 Fourier Transform Variants
8: Correlation There are three different versions of the Fourier Transform in current use. • Cross-Correlation • Signal Matching • Cross-corr as Convolution (1) Frequency version (we have used this in lectures) • Normalized Cross-corr i2πft i2πft • Autocorrelation U(f) = ∞ u(t)e− dt u(t) = ∞ U(f)e df • Autocorrelation example R−∞ R−∞ • Fourier Transform Variants Used in the communications/broadcasting industry and textbooks. • Scale Factors • ,,, • Summary The formulae do not need scale factors of 2π anywhere. • Spectrogram • (2) Angular frequency version iωt 1 iωt U(ω) = ∞ u(t)e− dt u(t) = ∞ U(ω)e dω R 2π R e −∞ −∞ e ω Continuous spectra are unchanged: U(ω) = U(f) = U( 2π ) However δ-function spectral componentse are multiplied by 2π so that U(f) = δ(f f ) U(ω) = 2π δ(ω 2πf ) − 0 ⇒ × − 0 Used in most signal processing ande control theory textbooks. • (3) Angular frequency + symmetrical scale factor U(ω) = 1 ∞ u(t)e iωtdt u(t) = 1 ∞ U(ω)eiωtdω √2π − √2π R−∞ R−∞ Inb all cases U(ω) = 1 U(ω) b √2π Used in manyb Maths textbookse (mathematicians like symmetry) • E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 8 / 11 Scale Factors
8: Correlation Fourier Transform using Angular Frequency: • Cross-Correlation • Signal Matching U(ω) = ∞ u(t)e iωtdt u(t) = 1 ∞ U(ω)eiωtdω • Cross-corr as Convolution − 2π R−∞ R−∞ • Normalized Cross-corr e e • Autocorrelation • Autocorrelation example • Fourier Transform Variants • Scale Factors • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 9 / 11 Scale Factors
8: Correlation Fourier Transform using Angular Frequency: • Cross-Correlation • Signal Matching U(ω) = ∞ u(t)e iωtdt u(t) = 1 ∞ U(ω)eiωtdω • Cross-corr as Convolution − 2π R−∞ R−∞ • Normalized Cross-corr e e • Autocorrelation 1 Any formula involving df will change to 2π dω [since dω = 2πdf] • Autocorrelation example R R • Fourier Transform Variants • Scale Factors • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 9 / 11 Scale Factors
8: Correlation Fourier Transform using Angular Frequency: • Cross-Correlation • Signal Matching U(ω) = ∞ u(t)e iωtdt u(t) = 1 ∞ U(ω)eiωtdω • Cross-corr as Convolution − 2π R−∞ R−∞ • Normalized Cross-corr e e • Autocorrelation 1 Any formula involving df will change to 2π dω [since dω = 2πdf] • Autocorrelation example R R • Fourier Transform Variants • Scale Factors Parseval’s Theorem: • Summary • Spectrogram u (t)v(t)dt = 1 U (ω)V (ω)dω R ∗ 2π R ∗ e e
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 9 / 11 Scale Factors
8: Correlation Fourier Transform using Angular Frequency: • Cross-Correlation • Signal Matching U(ω) = ∞ u(t)e iωtdt u(t) = 1 ∞ U(ω)eiωtdω • Cross-corr as Convolution − 2π R−∞ R−∞ • Normalized Cross-corr e e • Autocorrelation 1 Any formula involving df will change to 2π dω [since dω = 2πdf] • Autocorrelation example R R • Fourier Transform Variants • Scale Factors Parseval’s Theorem: • Summary • Spectrogram u (t)v(t)dt = 1 U (ω)V (ω)dω R ∗ 2π R ∗ e e 2 E u t 2 dt 1 U ω dω u = ( ) = 2π ( ) R | | R e
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 9 / 11 Scale Factors
8: Correlation Fourier Transform using Angular Frequency: • Cross-Correlation • Signal Matching U(ω) = ∞ u(t)e iωtdt u(t) = 1 ∞ U(ω)eiωtdω • Cross-corr as Convolution − 2π R−∞ R−∞ • Normalized Cross-corr e e • Autocorrelation 1 Any formula involving df will change to 2π dω [since dω = 2πdf] • Autocorrelation example R R • Fourier Transform Variants • Scale Factors Parseval’s Theorem: • Summary • Spectrogram u (t)v(t)dt = 1 U (ω)V (ω)dω R ∗ 2π R ∗ e e 2 E u t 2 dt 1 U ω dω u = ( ) = 2π ( ) R | | R e Waveform Multiplication: (convolution implicitly involves integration) w(t) = u(t)v(t) W (ω) = 1 U(ω) V (ω) ⇒ 2π ∗ f e e
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 9 / 11 Scale Factors
8: Correlation Fourier Transform using Angular Frequency: • Cross-Correlation • Signal Matching U(ω) = ∞ u(t)e iωtdt u(t) = 1 ∞ U(ω)eiωtdω • Cross-corr as Convolution − 2π R−∞ R−∞ • Normalized Cross-corr e e • Autocorrelation 1 Any formula involving df will change to 2π dω [since dω = 2πdf] • Autocorrelation example R R • Fourier Transform Variants • Scale Factors Parseval’s Theorem: • Summary • Spectrogram u (t)v(t)dt = 1 U (ω)V (ω)dω R ∗ 2π R ∗ e e 2 E u t 2 dt 1 U ω dω u = ( ) = 2π ( ) R | | R e Waveform Multiplication: (convolution implicitly involves integration) w(t) = u(t)v(t) W (ω) = 1 U(ω) V (ω) ⇒ 2π ∗ f e e Spectrum Multiplication: (multiplication ; integration) w(t) = u(t) v(t) W (ω) = U(ω)V (ω) ∗ ⇒ f e e
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 9 / 11 Scale Factors
8: Correlation Fourier Transform using Angular Frequency: • Cross-Correlation • Signal Matching U(ω) = ∞ u(t)e iωtdt u(t) = 1 ∞ U(ω)eiωtdω • Cross-corr as Convolution − 2π R−∞ R−∞ • Normalized Cross-corr e e • Autocorrelation 1 Any formula involving df will change to 2π dω [since dω = 2πdf] • Autocorrelation example R R • Fourier Transform Variants • Scale Factors Parseval’s Theorem: • Summary • Spectrogram u (t)v(t)dt = 1 U (ω)V (ω)dω R ∗ 2π R ∗ e e 2 E u t 2 dt 1 U ω dω u = ( ) = 2π ( ) R | | R e Waveform Multiplication: (convolution implicitly involves integration) w(t) = u(t)v(t) W (ω) = 1 U(ω) V (ω) ⇒ 2π ∗ f e e Spectrum Multiplication: (multiplication ; integration) w(t) = u(t) v(t) W (ω) = U(ω)V (ω) ∗ ⇒ f e e To obtain formulae for version (3) of the Fourier Transform, U(ω), substitute into the above formulae: U(ω) = √2πU(ω). b e b E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 9 / 11 Summary
8: Correlation • Cross-Correlation Cross-Correlation: w(t) = u(t) v(t) • Signal Matching • ⊗ • Cross-corr as Convolution • Normalized Cross-corr • Autocorrelation • Autocorrelation example • Fourier Transform Variants • Scale Factors • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 10 / 11 Summary
8: Correlation • Cross-Correlation Cross-Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching • ⊗ R−∞ − • Cross-corr as Convolution • Normalized Cross-corr • Autocorrelation • Autocorrelation example • Fourier Transform Variants • Scale Factors • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 10 / 11 Summary
8: Correlation • Cross-Correlation Cross-Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching • ⊗ R−∞ − • Cross-corr as Convolution Used to find similarities between v(t) and a delayed u(t) • Normalized Cross-corr ◦ • Autocorrelation • Autocorrelation example • Fourier Transform Variants • Scale Factors • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 10 / 11 Summary
8: Correlation • Cross-Correlation Cross-Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching • ⊗ R−∞ − • Cross-corr as Convolution Used to find similarities between v(t) and a delayed u(t) • Normalized Cross-corr ◦ • Autocorrelation Cross-correlation theorem: W (f) = U ∗(f)V (f) • Autocorrelation example ◦ • Fourier Transform Variants • Scale Factors • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 10 / 11 Summary
8: Correlation • Cross-Correlation Cross-Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching • ⊗ R−∞ − • Cross-corr as Convolution Used to find similarities between v(t) and a delayed u(t) • Normalized Cross-corr ◦ • Autocorrelation Cross-correlation theorem: W (f) = U ∗(f)V (f) • Autocorrelation example ◦ • Fourier Transform Variants Cauchy-Schwarz Inequality: u(t) v(t) √EuEv • Scale Factors ◦ | ⊗ | ≤ • Summary • Spectrogram
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 10 / 11 Summary
8: Correlation • Cross-Correlation Cross-Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching • ⊗ R−∞ − • Cross-corr as Convolution Used to find similarities between v(t) and a delayed u(t) • Normalized Cross-corr ◦ • Autocorrelation Cross-correlation theorem: W (f) = U ∗(f)V (f) • Autocorrelation example ◦ • Fourier Transform Variants Cauchy-Schwarz Inequality: u(t) v(t) √EuEv • Scale Factors ◦ | ⊗ | ≤ u(t) v(t) • Summary ⊲ Normalized cross-correlation: ⊗ 1 • Spectrogram √EuEv ≤
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 10 / 11 Summary
8: Correlation • Cross-Correlation Cross-Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching • ⊗ R−∞ − • Cross-corr as Convolution Used to find similarities between v(t) and a delayed u(t) • Normalized Cross-corr ◦ • Autocorrelation Cross-correlation theorem: W (f) = U ∗(f)V (f) • Autocorrelation example ◦ • Fourier Transform Variants Cauchy-Schwarz Inequality: u(t) v(t) √EuEv • Scale Factors ◦ | ⊗ | ≤ u(t) v(t) • Summary ⊲ Normalized cross-correlation: ⊗ 1 • Spectrogram √EuEv ≤
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 Cross-Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching • ⊗ R−∞ − • Cross-corr as Convolution Used to find similarities between v(t) and a delayed u(t) • Normalized Cross-corr ◦ • Autocorrelation Cross-correlation theorem: W (f) = U ∗(f)V (f) • Autocorrelation example ◦ • Fourier Transform Variants Cauchy-Schwarz Inequality: u(t) v(t) √EuEv • Scale Factors ◦ | ⊗ | ≤ u(t) v(t) • Summary ⊲ Normalized cross-correlation: ⊗ 1 • Spectrogram √EuEv ≤
Autocorrelation: x(t) = u(t) u(t) = ∞ u∗(τ t)u(τ)dτ Eu • ⊗ R−∞ − ≤
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 10 / 11 Summary
8: Correlation • Cross-Correlation Cross-Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching • ⊗ R−∞ − • Cross-corr as Convolution Used to find similarities between v(t) and a delayed u(t) • Normalized Cross-corr ◦ • Autocorrelation Cross-correlation theorem: W (f) = U ∗(f)V (f) • Autocorrelation example ◦ • Fourier Transform Variants Cauchy-Schwarz Inequality: u(t) v(t) √EuEv • Scale Factors ◦ | ⊗ | ≤ u(t) v(t) • Summary ⊲ Normalized cross-correlation: ⊗ 1 • Spectrogram √EuEv ≤
Autocorrelation: x(t) = u(t) u(t) = ∞ u∗(τ t)u(τ)dτ Eu • ⊗ R−∞ − ≤ 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 Cross-Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching • ⊗ R−∞ − • Cross-corr as Convolution Used to find similarities between v(t) and a delayed u(t) • Normalized Cross-corr ◦ • Autocorrelation Cross-correlation theorem: W (f) = U ∗(f)V (f) • Autocorrelation example ◦ • Fourier Transform Variants Cauchy-Schwarz Inequality: u(t) v(t) √EuEv • Scale Factors ◦ | ⊗ | ≤ u(t) v(t) • Summary ⊲ Normalized cross-correlation: ⊗ 1 • Spectrogram √EuEv ≤
Autocorrelation: x(t) = u(t) u(t) = ∞ u∗(τ t)u(τ)dτ Eu • ⊗ R−∞ − ≤ 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 Cross-Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching • ⊗ R−∞ − • Cross-corr as Convolution Used to find similarities between v(t) and a delayed u(t) • Normalized Cross-corr ◦ • Autocorrelation Cross-correlation theorem: W (f) = U ∗(f)V (f) • Autocorrelation example ◦ • Fourier Transform Variants Cauchy-Schwarz Inequality: u(t) v(t) √EuEv • Scale Factors ◦ | ⊗ | ≤ u(t) v(t) • Summary ⊲ Normalized cross-correlation: ⊗ 1 • Spectrogram √EuEv ≤
Autocorrelation: x(t) = u(t) u(t) = ∞ u∗(τ t)u(τ)dτ Eu • ⊗ R−∞ − ≤ 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 Cross-Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching • ⊗ R−∞ − • Cross-corr as Convolution Used to find similarities between v(t) and a delayed u(t) • Normalized Cross-corr ◦ • Autocorrelation Cross-correlation theorem: W (f) = U ∗(f)V (f) • Autocorrelation example ◦ • Fourier Transform Variants Cauchy-Schwarz Inequality: u(t) v(t) √EuEv • Scale Factors ◦ | ⊗ | ≤ u(t) v(t) • Summary ⊲ Normalized cross-correlation: ⊗ 1 • Spectrogram √EuEv ≤
Autocorrelation: x(t) = u(t) u(t) = ∞ u∗(τ t)u(τ)dτ Eu • ⊗ R−∞ − ≤ 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 ◦ In formulae: df 1 dω; ω-convolution involves an integral ◦ R → 2π R
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 10 / 11 Summary
8: Correlation • Cross-Correlation Cross-Correlation: w(t) = u(t) v(t) = ∞ u∗(τ t)v(τ)dτ • Signal Matching • ⊗ R−∞ − • Cross-corr as Convolution Used to find similarities between v(t) and a delayed u(t) • Normalized Cross-corr ◦ • Autocorrelation Cross-correlation theorem: W (f) = U ∗(f)V (f) • Autocorrelation example ◦ • Fourier Transform Variants Cauchy-Schwarz Inequality: u(t) v(t) √EuEv • Scale Factors ◦ | ⊗ | ≤ u(t) v(t) • Summary ⊲ Normalized cross-correlation: ⊗ 1 • Spectrogram √EuEv ≤
Autocorrelation: x(t) = u(t) u(t) = ∞ u∗(τ t)u(τ)dτ Eu • ⊗ R−∞ − ≤ 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 ◦ In formulae: df 1 dω; ω-convolution involves an integral ◦ R → 2π R For further details see RHB Chapter 13.1
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 10 / 11 Spectrogram
8: Correlation Spectrogram of “Merry Christmas” spoken by Mike Brookes () • Cross-Correlation • Signal Matching • Cross-corr as Convolution • Normalized Cross-corr • Autocorrelation • Autocorrelation example me rry christm a s -10 • Fourier Transform Variants 8 • Scale Factors • Summary • Spectrogram 7 6 -20 5 4 -30 3 Frequency (kHz) Frequency
2 (dB) Power/Decade 1 -40 0 0.2 0.4 0.6 0.8 1 Time (s)
E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 – 11 / 11