Filtering and Convolution¶

Convolution is a process of blending two functions together to create a third. The convolution of two functions $f$ and $g$ is symbolized mathematically as $(f * g)$.

Convolution steps¶

For each point in $f$:

Flip $g$ around
Take product of $f$ and $g$ at all overlapping location times
Add the resulting values together
Move on to next point in $f$, repeat for different lag between $f$ and $g$

This process is very similar to the procedures for cross-covariance of two functions $f$ and $g$, and auto-covariance of a single function $g$ with itself.

images/convolution2.png

source: https://en.wikipedia.org/wiki/Convolution

Expressing this procedure mathematically for two time series $f(t)$ and $g(t)$:

$f$ has M samples, $f_k$ where k = 1,2,3…M
$g$ has N samples, $g_i$ where i = 1,2,3…N

$ (f * g)_j = \sum_{k=1}^M g_{j-k} f_k $

The resulting function has N+M-1 points.

Filtering¶

One of the common applications of convolution is smoothing data. Smoothing is used in time series analysis and image analysis to suppress noise or high-frequency variability. Another term for smoothing is low-pass filter becuase it suppresses high-frequency variability and allows low-frequency variability to pass through.

One of the most common and intuitive smoothing operations is the running Average (or Box Car filter). This involves taking the average over a certain block (30 hrs, 5 days … etc), then moving forward in time and repeating the same process for the next segment of data.

Filtering is a type of convolution where

$f$ is a time series (observations)
$g$ is a set of weights (filter)

For a running average, the weights in $g$ are all identical and sum to one. If you plot the weights, the look like a rectangle, or “box car” shape.

Another common filtering method is the Hanning filter, which is a raised cosine function. All the values are positive (and sum to 1). This weights nearby points more than those that are farther away. Choosing the best filter for a certain situation is a major topic in digital signal processing.

What makes an ideal low pass filter?¶

Note: Images in this section come from the excellent source at http://www.labbookpages.co.uk/audio/firWindowing.html

For white noise (equal energy at all frequencies)

all energy in low frequencies (pass band) retrained
all energy in high frequencies (stop band) removed

images/filtering2.png

Image source: http://www.labbookpages.co.uk/audio/firWindowing.html

Ideal filter
Assuming an infinite time series, the filter would look like a sinc function:

sinc function is the FFT of a step function
convolving the sinc function with an infinitely long time series of white noise would give the step function in frequency content shown above
A sinc function would require an infinite number of weights (this is NOT practical)