8: Correlation

8: Correlation

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 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

E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 2 / 11

The cross-correlation between two signals u(t) and v(t) is

w(t) = u(t) v(t) , u

( )v( + t)d

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

E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 2 / 11

The cross-correlation between two signals u(t) and v(t) is

w(t) = u(t) v(t) , u

( )v( + t)d

= u

( t)v( )d [sub: t]

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

E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 2 / 11

The cross-correlation between two signals u(t) and v(t) is

w(t) = u(t) v(t) , u

( )v( + t)d

= u

( t)v( )d [sub: t]The complex conjugate, u( ) makes no difference if u(t) is real-valuedbut makes the definition work even if u(t) is complex-valued.

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

E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 2 / 11

The cross-correlation between two signals u(t) and v(t) is

w(t) = u(t) v(t) , u

( )v( + t)d

= u

( t)v( )d [sub: t]The complex conjugate, u( ) makes no difference if u(t) is real-valuedbut makes the definition work even if u(t) is complex-valued.

Correlation versus Convolution:

u(t) v(t) = u

( )v( + t)d [correlation]

u(t) v(t) = u( )v(t )d [convolution]

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

E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 2 / 11

The cross-correlation between two signals u(t) and v(t) is

w(t) = u(t) v(t) , u

( )v( + t)d

= u

( t)v( )d [sub: t]The complex conjugate, u( ) makes no difference if u(t) is real-valuedbut makes the definition work even if u(t) is complex-valued.

Correlation versus Convolution:

u(t) v(t) = u

( )v( + t)d [correlation]

u(t) v(t) = u( )v(t )d [convolution]

Unlike convolution, the integration variable, , has the same sign in thearguments of u( ) and v( ) so the arguments have a constantdifference instead of a constant sum (i.e. v(t) is not time-flipped).

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

E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 2 / 11

The cross-correlation between two signals u(t) and v(t) is

w(t) = u(t) v(t) , u

( )v( + t)d

= u

( t)v( )d [sub: t]The complex conjugate, u( ) makes no difference if u(t) is real-valuedbut makes the definition work even if u(t) is complex-valued.

Correlation versus Convolution:

u(t) v(t) = u

( )v( + t)d [correlation]

u(t) v(t) = u( )v(t )d [convolution]

Unlike convolution, the integration variable, , has the same sign in thearguments of u( ) and v( ) so the arguments have a constantdifference 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).

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

E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 2 / 11

The cross-correlation between two signals u(t) and v(t) is

w(t) = u(t) v(t) , u

( )v( + t)d

= u

( t)v( )d [sub: t]The complex conjugate, u( ) makes no difference if u(t) is real-valuedbut makes the definition work even if u(t) is complex-valued.

Correlation versus Convolution:

u(t) v(t) = u

( )v( + t)d [correlation]

u(t) v(t) = u( )v(t )d [convolution]

Unlike convolution, the integration variable, , has the same sign in thearguments of u( ) and v( ) so the arguments have a constantdifference 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).

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

E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 2 / 11

The cross-correlation between two signals u(t) and v(t) is

w(t) = u(t) v(t) , u

( )v( + t)d

= u

( t)v( )d [sub: t]The complex conjugate, u( ) makes no difference if u(t) is real-valuedbut makes the definition work even if u(t) is complex-valued.

Correlation versus Convolution:

u(t) v(t) = u

( )v( + t)d [correlation]

u(t) v(t) = u( )v(t )d [convolution]

Unlike convolution, the integration variable, , has the same sign in thearguments of u( ) and v( ) so the arguments have a constantdifference 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.

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

E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 2 / 11

The cross-correlation between two signals u(t) and v(t) is

w(t) = u(t) v(t) , u

( )v( + t)d

= u

( t)v( )d [sub: t]The complex conjugate, u( ) makes no difference if u(t) is real-valuedbut makes the definition work even if u(t) is complex-valued.

Correlation versus Convolution:

u(t) v(t) = u

( )v( + t)d [correlation]

u(t) v(t) = u( )v(t )d [convolution]

Unlike convolution, the integration variable, , has the same sign in thearguments of u( ) and v( ) so the arguments have a constantdifference 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 /.

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

E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 3 / 11

Cross correlation is used to find where twosignals match

0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

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

E1.10 Fourier Series and Transforms (2015-5585) Fourier Transform - Correlation: 8 3 / 11

Cross correlation is used to find where twosignals match: u(t) is the test waveform.

0 200 400 600

0

0.5

1

u(t)

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

S