DSP for Data Analysis

Brief introduction into digital signal processing

Mykola Pavlov
Founder at Biomech

Topics

  • What is DSP and where it can be applied?
  • Linear systems
  • Fourier analysis
  • Filter design in R

What is Signal?

Anything that carries information is a signal

Signals

What is DSP and where it can be applied?

The mathematics, the algorithms, and the techniques used to manipulate signals after they have been converted into a digital form.

Application of DSP

Contineous and discrete

  • Contineous f(t)
  • Discrete (digital) f[t]

Discrete signal

Linear system

Linear system

Examples: Wave propagation, resistors, capacitors, inductors, amplifiers, filters, mechanical interaction of masses, springs, echoes, resonances, image blurring...

Linear system properties: Homogenity

Amplitude change in the input results in an identical amplitude change in the output.

Homogenity of signal

Linear system properties: Additivity

Signals added at the input produce signals that are added at the output.

Additivity of signal

Linear system properties: Shift invariance

Shift in the input signal causes an identical shift in the output signal.

Shift invariance of signal

Superposition strategy: divide-and-conquer

Synthesis and decomposition

Synthesis: 5 + 10 = 15
Decomposition: 15 = 3 + 12 or 1 + 5 + 9 or ...

Superposition strategy: divide-and-conquer

Superposition

Superposition strategy: divide-and-conquer

Divide-and-conquer: 1027 * 4 = 1000 * 4 + 20 * 4 + 7 * 4

Intuition: No more complex signals! All we need to know is how simple signals are modified by a system.

Two decompositions: Impulse and Fourier

Impulse decomposition

Impulse decomposition

Delta function and Impulse Response

Impulse decomposition

Same as filter kernel, convolution kernel, kernel, point spread function (image processing).

Let's summarize

  1. Input signal can be decomposed into a set of impulses
  2. Each impulse can be viewed as a scaled and shifted delta function
  3. Output signal can be found by adding these scaled and shifted impulse responses (This operations is known as Convolution).

Intuition: If we know impulse response, then we can calculate the output for any possible input signal!

Fourier decomposition

Fourier decomposition

Fourier decomposition

Fourier decomposition animated

DFT, IDFT, FFT

signal <- rnorm(100)
fourier_transform <- fft(signal)

Re(fourier_transform)[1:5]

## [1] -4.874 -3.616 -9.085  5.431 -2.516

Im(fourier_transform)[1:5]

## [1]  0.0000 -0.9916 10.2477 -1.5758 -0.3789
  • Re[] - Cos, Index - Freq, Value - Amplitude
  • Im[] - Sin, Index - Freq, Value - Amplitude

Rectangular vs Polar

Fourier decomposition animated

Mod(fourier_transform)[1:5]

## [1]  4.874  3.749 13.695  5.655  2.545

Furier analysis

ChemCam

Intuition: How much information/signal is presented on a given frequency

Filter design

Filter design

Filter types

Filter types

Filter types

Filter types

Window Sinc Filter

library(signal)
t <- as.vector(as.matrix(read.csv("data/data.csv", row.names = NULL, header = F)))  # 20 Hz 
plot.ts(t, ylab = "g", main = "User Acceleration")

plot of chunk unnamed-chunk-4

Window Sinc Filter

filt <- fir1(15, 0.15, type = "low")  # 3 Hz low-pass filter
z <- filter(filt, t)  # apply filter
plot(z, col = "red", lwd = 3, type = "l", ylab = "g", main = "User Acceleration")

plot of chunk unnamed-chunk-5

Credits

Questions?