1d-correlation.md

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Imports

The tutorial below imports NumPy, Pandas, SciPy and Plotly.

importplotly.plotlyaspyimportplotly.graph_objsasgoimportnumpyasnpimportpandasaspdimportscipyfromscipyimportsignal

Correlation Between Saw and Square Wave

Similar to convolution, the 1D Cross-Correlation between two functions $f$ and $g$ is a measure of their similarity in terms of the lag of one to another (source). Since we are dealing with arrays of data rather than continuous functions, the cross-correlation is mathematically defined as:

$$ \begin{align*} (fg)[n] = \sum_{m=-\infty}^{\infty} f^{}[m]g[n+m] \end{align*} $$

where $f^*$ is the complex conjugate of $f$.

To give us an intuitive glance at what this does, let us look at the cross-correlation between a saw wave and a square wave.

x=np.linspace(0, 20, 50) saw_y=signal.sawtooth(t=x) square_y=signal.square(t=x) convolve_y=signal.convolve(saw_y, square_y) trace_saw=go.Scatter( x=x, y=saw_y, mode='lines', name='Saw', marker=dict( color='#57D1C9' ) ) trace_square=go.Scatter( x=x, y=square_y, mode='lines', name='Square', marker=dict( color='#ED5485' ) ) trace_convolution=go.Scatter( x=x, y=convolve_y, mode='lines', name='Convolution', marker=dict( color='#FFE869' ) ) data= [trace_saw, trace_square, trace_convolution] py.iplot(data, filename='1d-convolution-of-saw-and-square')

Correlation Between Saw and Shifted Saw Wave

To compare with the plot above, we can plot a saw wave, a phase shifted saw wave and the convolution between the two to see how they correlate along the axis.

x=np.linspace(0, 20, 50) saw_y=signal.sawtooth(t=x) square_y=signal.square(t=x) shifted_saw_y=signal.sawtooth(t=np.linspace(10, 30, 50)) convolve_y=signal.convolve(saw_y, shifted_saw_y) trace_saw=go.Scatter( x=x, y=saw_y, mode='lines', name='Saw', marker=dict( color='#FF7844' ), opacity=0.8 ) trace_shifted_saw=go.Scatter( x=x, y=shifted_saw_y, mode='lines', name='Shifted Saw', marker=dict( color='#A64942' ), opacity=0.8 ) trace_convolution=go.Scatter( x=x, y=convolve_y, mode='lines', name='Convolution', marker=dict( color='#53354A' ) ) data= [trace_saw, trace_shifted_saw, trace_convolution] py.iplot(data, filename='1d-convolution-of-saw-and-shifted-saw')

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