Digital Signal Processing

This section of the user guide explores functions that are commonly used in the field of Digital Signal Processing (DSP).

Dot Product

The dotProduct function is used to calculate the dot product of two numeric arrays. The dot product is a fundamental calculation for the DSP functions discussed in this section. Before diving into the more advanced DSP functions its useful to develop a deeper intuition of the dot product.

The dot product operation is performed in two steps:

  1. Element-by-element multiplication of two vectors which produces a vector of products.

  2. Sum the vector of products to produce a scalar result.

This simple bit of math has a number of important applications.

Representing Linear Combinations

The dotProduct performs the math of a linear combination. A linear combination has the following form:

(a1*v1)+(a2*v2)...

In the above example a1 and a2 are random variables that change. v1 and v2 are constant values.

When computing the dot product the elements of two vectors are multiplied together and the results are added. If the first vector contains random variables and the second vector contains constant values then the dot product is performing a linear combination.

This scenario comes up again and again in machine learning. For example both linear and logistic regression solve for a vector of constant weights. In order to perform a prediction, a dot product is calculated between a random observation vector and the constant weight vector. That dot product is a linear combination because one of the vectors holds constant weights.

Lets look at simple example of how a linear combination can be used to find the mean of a vector of numbers.

In the example below two arrays are set to variables a and b and then operated on by the dotProduct function. The output of the dotProduct function is set to variable c.

The mean function is then used to compute the mean of the first array which is set to the variable d.

Both the dot product and the mean are included in the output.

When we look at the output of this expression we see that the dot product and the mean of the first array are both 30.

The dotProduct function calculated the mean of the first array.

let(echo="c, d",
    a=array(10, 20, 30, 40, 50),
    b=array(.2, .2, .2, .2, .2),
    c=dotProduct(a, b),
    d=mean(a))

When this expression is sent to the /stream handler it responds with:

{
  "result-set": {
    "docs": [
      {
        "c": 30,
        "d": 30
      },
      {
        "EOF": true,
        "RESPONSE_TIME": 0
      }
    ]
  }
}

To get a better understanding of how the dot product calculated the mean we can perform the steps of the calculation using vector math and look at the output of each step.

In the example below the ebeMultiply function performs an element-by-element multiplication of two arrays. This is the first step of the dot product calculation. The result of the element-by-element multiplication is assigned to variable c.

In the next step the add function adds all the elements of the array in variable c.

Notice that multiplying each element of the first array by .2 and then adding the results is equivalent to the formula for computing the mean of the first array. The formula for computing the mean of an array is to add all the elements and divide by the number of elements.

The output includes the output of both the ebeMultiply function and the add function.

let(echo="c, d",
    a=array(10, 20, 30, 40, 50),
    b=array(.2, .2, .2, .2, .2),
    c=ebeMultiply(a, b),
    d=add(c))

When this expression is sent to the /stream handler it responds with:

{
  "result-set": {
    "docs": [
      {
        "c": [
          2,
          4,
          6,
          8,
          10
        ],
        "d": 30
      },
      {
        "EOF": true,
        "RESPONSE_TIME": 0
      }
    ]
  }
}

In the example above two arrays were combined in a way that produced the mean of the first. In the second array each value was set to .2. Another way of looking at this is that each value in the second array is applying the same weight to the values in the first array. By varying the weights in the second array we can produce a different result. For example if the first array represents a time series, the weights in the second array can be set to add more weight to a particular element in the first array.

The example below creates a weighted average with the weight decreasing from right to left. Notice that the weighted mean of 36.666 is larger than the previous mean which was 30. This is because more weight was given to last element in the array.

let(echo="c, d",
    a=array(10, 20, 30, 40, 50),
    b=array(.066666666666666,.133333333333333,.2, .266666666666666, .33333333333333),
    c=ebeMultiply(a, b),
    d=add(c))

When this expression is sent to the /stream handler it responds with:

{
  "result-set": {
    "docs": [
      {
        "c": [
          0.66666666666666,
          2.66666666666666,
          6,
          10.66666666666664,
          16.6666666666665
        ],
        "d": 36.66666666666646
      },
      {
        "EOF": true,
        "RESPONSE_TIME": 0
      }
    ]
  }
}

Representing Correlation

Often when we think of correlation, we are thinking of Pearson correlation in the field of statistics. But the definition of correlation is actually more general: a mutual relationship or connection between two or more things. In the field of digital signal processing the dot product is used to represent correlation. The examples below demonstrates how the dot product can be used to represent correlation.

In the example below the dot product is computed for two vectors. Notice that the vectors have different values that fluctuate together. The output of the dot product is 190, which is hard to reason about because it’s not scaled.

let(echo="c, d",
    a=array(10, 20, 30, 20, 10),
    b=array(1, 2, 3, 2, 1),
    c=dotProduct(a, b))

When this expression is sent to the /stream handler it responds with:

{
  "result-set": {
    "docs": [
      {
        "c": 190
      },
      {
        "EOF": true,
        "RESPONSE_TIME": 0
      }
    ]
  }
}

One approach to scaling the dot product is to first scale the vectors so that both vectors have a magnitude of 1. Vectors with a magnitude of 1, also called unit vectors, are used when comparing only the angle between vectors rather then the magnitude. The unitize function can be used to unitize the vectors before calculating the dot product.

Notice in the example below the dot product result, set to variable e, is effectively 1. When applied to unit vectors the dot product will be scaled between 1 and -1. Also notice in the example cosineSimilarity is calculated on the unscaled vectors and the answer is also effectively 1. This is because cosine similarity is a scaled dot product.

let(echo="e, f",
    a=array(10, 20, 30, 20, 10),
    b=array(1, 2, 3, 2, 1),
    c=unitize(a),
    d=unitize(b),
    e=dotProduct(c, d),
    f=cosineSimilarity(a, b))

When this expression is sent to the /stream handler it responds with:

{
  "result-set": {
    "docs": [
      {
        "e": 0.9999999999999998,
        "f": 0.9999999999999999
      },
      {
        "EOF": true,
        "RESPONSE_TIME": 0
      }
    ]
  }
}

If we transpose the first two numbers in the first array, so that the vectors are not perfectly correlated, we see that the cosine similarity drops. This illustrates how the dot product represents correlation.

let(echo="c, d",
    a=array(20, 10, 30, 20, 10),
    b=array(1, 2, 3, 2, 1),
    c=cosineSimilarity(a, b))

When this expression is sent to the /stream handler it responds with:

{
  "result-set": {
    "docs": [
      {
        "c": 0.9473684210526314
      },
      {
        "EOF": true,
        "RESPONSE_TIME": 0
      }
    ]
  }
}

Convolution

The conv function calculates the convolution of two vectors. The convolution is calculated by reversing the second vector and sliding it across the first vector. The dot product of the two vectors is calculated at each point as the second vector is slid across the first vector. The dot products are collected in a third vector which is the convolution of the two vectors.

Moving Average Function

Before looking at an example of convolution its useful to review the movingAvg function. The moving average function computes a moving average by sliding a window across a vector and computing the average of the window at each shift. If that sounds similar to convolution, that’s because the movingAvg function is syntactic sugar for convolution.

Below is an example of a moving average with a window size of 5. Notice that original vector has 13 elements but the result of the moving average has only 9 elements. This is because the movingAvg function only begins generating results when it has a full window. In this case because the window size is 5 so the moving average starts generating results from the 4th index of the original array.

let(a=array(1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1),
    b=movingAvg(a, 5))

When this expression is sent to the /stream handler it responds with:

{
  "result-set": {
    "docs": [
      {
        "b": [
          3,
          4,
          5,
          5.6,
          5.8,
          5.6,
          5,
          4,
          3
        ]
      },
      {
        "EOF": true,
        "RESPONSE_TIME": 0
      }
    ]
  }
}

Convolutional Smoothing

The moving average can also be computed using convolution. In the example below the conv function is used to compute the moving average of the first array by applying the second array as the filter.

Looking at the result, we see that it is not exactly the same as the result of the movingAvg function. That is because the conv pads zeros to the front and back of the first vector so that the window size is always full.

let(a=array(1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1),
    b=array(.2, .2, .2, .2, .2),
    c=conv(a, b))

When this expression is sent to the /stream handler it responds with:

{
  "result-set": {
    "docs": [
      {
        "c": [
          0.2,
          0.6000000000000001,
          1.2,
          2.0000000000000004,
          3.0000000000000004,
          4,
          5,
          5.6000000000000005,
          5.800000000000001,
          5.6000000000000005,
          5.000000000000001,
          4,
          3,
          2,
          1.2000000000000002,
          0.6000000000000001,
          0.2
        ]
      },
      {
        "EOF": true,
        "RESPONSE_TIME": 0
      }
    ]
  }
}

We achieve the same result as the movingAvg function by using the copyOfRange function to copy a range of the result that drops the first and last 4 values of the convolution result. In the example below the precision function is also also used to remove floating point errors from the convolution result. When this is added the output is exactly the same as the movingAvg function.

let(a=array(1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1),
    b=array(.2, .2, .2, .2, .2),
    c=conv(a, b),
    d=copyOfRange(c, 4, 13),
    e=precision(d, 2))

When this expression is sent to the /stream handler it responds with:

{
  "result-set": {
    "docs": [
      {
        "e": [
          3,
          4,
          5,
          5.6,
          5.8,
          5.6,
          5,
          4,
          3
        ]
      },
      {
        "EOF": true,
        "RESPONSE_TIME": 0
      }
    ]
  }
}

Cross-Correlation

Cross-correlation is used to determine the delay between two signals. This is accomplished by sliding one signal across another and calculating the dot product at each shift. The dot products are collected into a vector which represents the correlation at each shift. The highest dot product in the cross-correlation vector is the point where the two signals are most closely correlated.

The sliding dot product used in convolution can also be used to represent cross-correlation between two vectors. The only difference in the formula when representing correlation is that the second vector is not reversed.

Notice in the example below that the second vector is reversed by the rev function before it is operated on by the conv function. The conv function reverses the second vector so it will be flipped back to its original order to perform the correlation calculation rather then the convolution calculation.

Notice in the result the highest value is 217. This is the point where the two vectors have the highest correlation.

let(a=array(1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1),
    b=array(4, 5, 6, 7, 6, 5, 4, 3, 2, 1),
    c=conv(a, rev(b)))

When this expression is sent to the /stream handler it responds with:

{
  "result-set": {
    "docs": [
      {
        "c": [
          1,
          4,
          10,
          20,
          35,
          56,
          84,
          116,
          149,
          180,
          203,
          216,
          217,
          204,
          180,
          148,
          111,
          78,
          50,
          28,
          13,
          4
        ]
      },
      {
        "EOF": true,
        "RESPONSE_TIME": 0
      }
    ]
  }
}

Find Delay

It is fairly simple to compute the delay from the cross-correlation result, but a convenience function called finddelay can be used to find the delay directly. Under the covers finddelay uses convolutional math to compute the cross-correlation vector and then computes the delay between the two signals.

Below is an example of the finddelay function. Notice that the finddelay function reports a 3 period delay between the first and second signal.

let(a=array(1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1),
    b=array(4, 5, 6, 7, 6, 5, 4, 3, 2, 1),
    c=finddelay(a, b))

When this expression is sent to the /stream handler it responds with:

{
  "result-set": {
    "docs": [
      {
        "c": 3
      },
      {
        "EOF": true,
        "RESPONSE_TIME": 0
      }
    ]
  }
}

Oscillate (Sine Wave)

The oscillate function generates a periodic oscillating signal which can be used to model and study sine waves.

The oscillate function takes three parameters: amplitude, angular frequency and phase and returns a vector containing the y-axis points of a sine wave.

The y-axis points were generated from an x-axis sequence of 0-127.

Below is an example of the oscillate function called with an amplitude of 1, and angular frequency of .28 and phase of 1.57.

oscillate(1, 0.28, 1.57)

The result of the oscillate function is plotted below:

sinewave

Sine Wave Interpolation, Extrapolation

The oscillate function returns a function which can be used by the predict function to interpolate or extrapolate a sine wave. The example below extrapolates the sine wave to an x-axis sequence of 0-256.

let(a=oscillate(1, 0.28, 1.57),
    b=predict(a, sequence(256, 0, 1)))

The extrapolated sine wave is plotted below:

sinewave256

Autocorrelation

Autocorrelation measures the degree to which a signal is correlated with itself. Autocorrelation is used to determine if a vector contains a signal or is purely random.

A few examples, with plots, will help to understand the concepts.

The first example simply revisits the example above of an extrapolated sine wave. The result of this is plotted in the image below. Notice that there is a structure to the plot that is clearly not random.

let(a=oscillate(1, 0.28, 1.57),
    b=predict(a, sequence(256, 0, 1)))
sinewave256

In the next example the sample function is used to draw 256 samples from a uniformDistribution to create a vector of random data. The result of this is plotted in the image below. Notice that there is no clear structure to the data and the data appears to be random.

sample(uniformDistribution(-1.5, 1.5), 256)
noise

In the next example the random noise is added to the sine wave using the ebeAdd function. The result of this is plotted in the image below. Notice that the sine wave has been hidden somewhat within the noise. Its difficult to say for sure if there is structure. As plots becomes more dense it can become harder to see a pattern hidden within noise.

let(a=oscillate(1, 0.28, 1.57),
    b=predict(a, sequence(256, 0, 1)),
    c=sample(uniformDistribution(-1.5, 1.5), 256),
    d=ebeAdd(b,c))
hidden signal

In the next examples autocorrelation is performed with each of the vectors shown above to see what the autocorrelation plots look like.

In the example below the conv function is used to autocorrelate the first vector which is the sine wave. Notice that the conv function is simply correlating the sine wave with itself.

The plot has a very distinct structure to it. As the sine wave is slid across a copy of itself the correlation moves up and down in increasing intensity until it reaches a peak. This peak is directly in the center and is the the point where the sine waves are directly lined up. Following the peak the correlation moves up and down in decreasing intensity as the sine wave slides farther away from being directly lined up.

This is the autocorrelation plot of a pure signal.

let(a=oscillate(1, 0.28, 1.57),
    b=predict(a, sequence(256, 0, 1)),
    c=conv(b, rev(b)))
signal autocorrelation

In the example below autocorrelation is performed with the vector of pure noise. Notice that the autocorrelation plot has a very different plot then the sine wave. In this plot there is long period of low intensity correlation that appears to be random. Then in the center a peak of high intensity correlation where the vectors are directly lined up. This is followed by another long period of low intensity correlation.

This is the autocorrelation plot of pure noise.

let(a=sample(uniformDistribution(-1.5, 1.5), 256),
    b=conv(a, rev(a)),
noise autocorrelation

In the example below autocorrelation is performed on the vector with the sine wave hidden within the noise. Notice that this plot shows very clear signs of structure which is similar to autocorrelation plot of the pure signal. The correlation is less intense due to noise but the shape of the correlation plot suggests strongly that there is an underlying signal hidden within the noise.

let(a=oscillate(1, 0.28, 1.57),
    b=predict(a, sequence(256, 0, 1)),
    c=sample(uniformDistribution(-1.5, 1.5), 256),
    d=ebeAdd(b, c),
    e=conv(d, rev(d)))
hidden signal autocorrelation

Discrete Fourier Transform

The convolution based functions described above are operating on signals in the time domain. In the time domain the X axis is time and the Y axis is the quantity of some value at a specific point in time.

The discrete Fourier Transform translates a time domain signal into the frequency domain. In the frequency domain the X axis is frequency, and Y axis is the accumulated power at a specific frequency.

The basic principle is that every time domain signal is composed of one or more signals (sine waves) at different frequencies. The discrete Fourier transform decomposes a time domain signal into its component frequencies and measures the power at each frequency.

The discrete Fourier transform has many important uses. In the example below, the discrete Fourier transform is used to determine if a signal has structure or if it is purely random.

Complex Result

The fft function performs the discrete Fourier Transform on a vector of real data. The result of the fft function is returned as complex numbers. A complex number has two parts, real and imaginary. The imaginary part of the complex number is ignored in the examples below, but there are many tutorials on the FFT and that include complex numbers available online.

But before diving into the examples it is important to understand how the fft function formats the complex numbers in the result.

The fft function returns a matrix with two rows. The first row in the matrix is the real part of the complex result. The second row in the matrix is the imaginary part of the complex result.

The rowAt function can be used to access the rows so they can be processed as vectors. This approach was taken because all of the vector math functions operate on vectors of real numbers. Rather then introducing a complex number abstraction into the expression language, the fft result is represented as two vectors of real numbers.

Fast Fourier Transform Examples

In the first example the fft function is called on the sine wave used in the autocorrelation example.

The results of the fft function is a matrix. The rowAt function is used to return the first row of the matrix which is a vector containing the real values of the fft response.

The plot of the real values of the fft response is shown below. Notice there are two peaks on opposite sides of the plot. The plot is actually showing a mirrored response. The right side of the plot is an exact mirror of the left side. This is expected when the fft is run on real rather then complex data.

Also notice that the fft has accumulated significant power in a single peak. This is the power associated with the specific frequency of the sine wave. The vast majority of frequencies in the plot have close to 0 power associated with them. This fft shows a clear signal with very low levels of noise.

let(a=oscillate(1, 0.28, 1.57),
    b=predict(a, sequence(256, 0, 1)),
    c=fft(b),
    d=rowAt(c, 0))
signal fft

In the second example the fft function is called on a vector of random data similar to one used in the autocorrelation example. The plot of the real values of the fft response is shown below.

Notice that in is this response there is no clear peak. Instead all frequencies have accumulated a random level of power. This fft shows no clear sign of signal and appears to be noise.

let(a=sample(uniformDistribution(-1.5, 1.5), 256),
    b=fft(a),
    c=rowAt(b, 0))
noise fft

In the third example the fft function is called on the same signal hidden within noise that was used for the autocorrelation example. The plot of the real values of the fft response is shown below.

Notice that there are two clear mirrored peaks, at the same locations as the fft of the pure signal. But there is also now considerable noise on the frequencies. The fft has found the signal and but also shows that there is considerable noise along with the signal.

let(a=oscillate(1, 0.28, 1.57),
    b=predict(a, sequence(256, 0, 1)),
    c=sample(uniformDistribution(-1.5, 1.5), 256),
    d=ebeAdd(b, c),
    e=fft(d),
    f=rowAt(e, 0))
hidden signal fft