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Subject: [16/39] lucene-solr:jira/solr-11779: Squashed commit of the following:
Squashed commit of the following:
commit e5074c3223e394af17f686294a67a1dd3ecdd147
Author: Joel Bernstein
Date: Wed May 9 13:16:34 2018 -0400
SOLR-12280: WIP
commit 69cdeccf161177d10f4d2407542392aaee3fcfe8
Author: Joel Bernstein
Date: Wed May 9 13:08:02 2018 -0400
SOLR-12280: WIP
commit c94f0c87c3e57c023d622ad2411e522c4aac491c
Author: Joel Bernstein
Date: Wed May 9 11:54:58 2018 -0400
SOLR-12280: WIP
commit 68dd1e73355cb84410f2d0ff3a51797ed6194a10
Author: Joel Bernstein
Date: Wed May 9 10:54:32 2018 -0400
SOLR-12280: WIP
commit 04a010543418a469100fa299c606a7b1eed452e1
Author: Joel Bernstein
Date: Wed May 9 10:47:27 2018 -0400
SOLR-12280: WIP
commit a6bbfbadaafe33fcdf93d5c72755e30dadadf017
Author: Joel Bernstein
Date: Wed May 9 09:40:08 2018 -0400
SOLR-12280: WIP
commit 5d27961aa291bcd71527337632981bcdf62369b4
Author: Joel Bernstein
Date: Tue May 8 20:43:33 2018 -0400
SOLR-12280: WIP
commit e982cf939f429c05b736f6292c68dd96d7ebc027
Author: Joel Bernstein
Date: Tue May 8 13:27:29 2018 -0400
SOLR-12280: WIP
commit aae78ab6f387c28a080021bc919ef51864540be2
Author: Joel Bernstein
Date: Tue May 8 12:23:52 2018 -0400
SOLR-12280: WIP
commit 0787ad76f0f4c62c860784b15490d8a988939997
Author: Joel Bernstein
Date: Tue May 8 12:20:38 2018 -0400
SOLR-12280: WIP
commit 4df098376ba05188702cca8582959c3fe18066f5
Author: Joel Bernstein
Date: Tue May 8 12:12:11 2018 -0400
SOLR-12280: WIP
commit 5c0be5136bbab7e0c33b3b8a7b0395b1b330e96d
Author: Joel Bernstein
Date: Tue May 8 12:04:57 2018 -0400
SOLR-12280: WIP
commit 6c6feac4c2e5a49a5eab87a228713d1b93c8fc70
Author: Joel Bernstein
Date: Tue May 8 11:57:49 2018 -0400
SOLR-12280: WIP
commit 7d46d11c9dd3a51b68600c2c889f586147545294
Author: Joel Bernstein
Date: Tue May 8 11:50:51 2018 -0400
SOLR-12280: WIP
commit 8b6bf19d0091203ed63b39d070dd02a9bece6a61
Author: Joel Bernstein
Date: Mon May 7 10:53:14 2018 -0400
SOLR-12280: WIP
commit 5466591999816eaacde6ce18d824d7688e5f4fe8
Author: Joel Bernstein
Date: Fri May 4 15:12:43 2018 -0400
SOLR-12280: WIP
commit d7fff7d557a7fd26011c21445b7969b2cd81036f
Author: Joel Bernstein
Date: Fri Apr 27 12:50:27 2018 -0400
SOLR-12280: Initial commit
Project: http://git-wip-us.apache.org/repos/asf/lucene-solr/repo
Commit: http://git-wip-us.apache.org/repos/asf/lucene-solr/commit/144f00a1
Tree: http://git-wip-us.apache.org/repos/asf/lucene-solr/tree/144f00a1
Diff: http://git-wip-us.apache.org/repos/asf/lucene-solr/diff/144f00a1
Branch: refs/heads/jira/solr-11779
Commit: 144f00a1e315541a28f526f4cdf1e55eb60c862b
Parents: 4177252
Author: Joel Bernstein
Authored: Wed May 9 13:24:08 2018 -0400
Committer: Joel Bernstein
Committed: Wed May 9 13:24:08 2018 -0400
----------------------------------------------------------------------
solr/solr-ref-guide/src/dsp.adoc | 719 +++++++++++++++++++
.../hidden-signal-autocorrelation.png | Bin 0 -> 258831 bytes
.../math-expressions/hidden-signal-fft.png | Bin 0 -> 215981 bytes
.../images/math-expressions/hidden-signal.png | Bin 0 -> 319100 bytes
.../math-expressions/noise-autocorrelation.png | Bin 0 -> 204511 bytes
.../src/images/math-expressions/noise-fft.png | Bin 0 -> 319551 bytes
.../src/images/math-expressions/noise.png | Bin 0 -> 375565 bytes
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.../src/images/math-expressions/signal-fft.png | Bin 0 -> 140111 bytes
.../src/images/math-expressions/signal.png | Bin 0 -> 365018 bytes
solr/solr-ref-guide/src/math-expressions.adoc | 2 +
11 files changed, 721 insertions(+)
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http://git-wip-us.apache.org/repos/asf/lucene-solr/blob/144f00a1/solr/solr-ref-guide/src/dsp.adoc
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diff --git a/solr/solr-ref-guide/src/dsp.adoc b/solr/solr-ref-guide/src/dsp.adoc
new file mode 100644
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--- /dev/null
+++ b/solr/solr-ref-guide/src/dsp.adoc
@@ -0,0 +1,719 @@
+= Digital Signal Processing
+// Licensed to the Apache Software Foundation (ASF) under one
+// or more contributor license agreements. See the NOTICE file
+// distributed with this work for additional information
+// regarding copyright ownership. The ASF licenses this file
+// to you under the Apache License, Version 2.0 (the
+// "License"); you may not use this file except in compliance
+// with the License. You may obtain a copy of the License at
+//
+// http://www.apache.org/licenses/LICENSE-2.0
+//
+// Unless required by applicable law or agreed to in writing,
+// software distributed under the License is distributed on an
+// "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
+// KIND, either express or implied. See the License for the
+// specific language governing permissions and limitations
+// under the License.
+
+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 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 get a better understanding of how the dot product calculation works.
+
+=== Combining two arrays
+
+The `dotProduct` function can be used to combine two arrays into a single product. A simple example can help
+illustrate this concept.
+
+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*.
+
+Then 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 dot product function *calculated the mean* of the first array.
+
+[source,text]
+----
+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:
+
+[source,json]
+----
+{
+ "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.
+
+[source,text]
+----
+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:
+
+[source,json]
+----
+{
+ "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 has the same weight.
+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.
+
+[source,text]
+----
+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:
+
+[source,json]
+----
+{
+ "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 *Pearsons* 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 because its not scaled.
+
+[source,text]
+----
+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:
+
+[source,json]
+----
+{
+ "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*.
+
+
+[source,text]
+----
+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:
+
+[source,json]
+----
+{
+ "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.
+
+[source,text]
+----
+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:
+
+[source,json]
+----
+{
+ "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
+
+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.
+
+[source,text]
+----
+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:
+
+[source,json]
+----
+{
+ "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.
+
+[source,text]
+----
+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:
+
+[source,json]
+----
+{
+ "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` gunction 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.
+
+[source,text]
+----
+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:
+
+[source,json]
+----
+{
+ "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.
+
+[source,text]
+----
+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:
+
+[source,json]
+----
+{
+ "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.
+
+[source,text]
+----
+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:
+
+[source,json]
+----
+{
+ "result-set": {
+ "docs": [
+ {
+ "c": 3
+ },
+ {
+ "EOF": true,
+ "RESPONSE_TIME": 0
+ }
+ ]
+ }
+}
+----
+
+== 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.
+
+In the first example the `sin` function is wrapped around a `sequence` function to generate a 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.
+
+[source,text]
+----
+sin(sequence(256, 0, 6))
+----
+
+image::images/math-expressions/signal.png[]
+
+
+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.
+
+[source,text]
+----
+sample(uniformDistribution(-1.5, 1.5), 256)
+----
+
+image::images/math-expressions/noise.png[]
+
+
+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.
+
+[source,text]
+----
+let(a=sin(sequence(256, 0, 6)),
+ b=sample(uniformDistribution(-1.5, 1.5), 256),
+ c=ebeAdd(a, b))
+----
+
+image::images/math-expressions/hidden-signal.png[]
+
+
+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.
+
+[source,text]
+----
+let(a=sin(sequence(256, 0, 6)),
+ b=conv(a, rev(a)),
+----
+
+image::images/math-expressions/signal-autocorrelation.png[]
+
+
+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.
+
+[source,text]
+----
+let(a=sample(uniformDistribution(-1.5, 1.5), 256),
+ b=conv(a, rev(a)),
+----
+
+image::images/math-expressions/noise-autocorrelation.png[]
+
+
+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.
+
+[source,text]
+----
+let(a=sin(sequence(256, 0, 6)),
+ b=sample(uniformDistribution(-1.5, 1.5), 256),
+ c=ebeAdd(a, b),
+ d=conv(c, rev(c))
+----
+
+image::images/math-expressions/hidden-signal-autocorrelation.png[]
+
+
+== 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.
+
+[source,text]
+----
+let(a=sin(sequence(256, 0, 6)),
+ b=fft(a),
+ c=rowAt(b, 0))
+----
+
+
+image::images/math-expressions/signal-fft.png[]
+
+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.
+
+
+[source,text]
+----
+let(a=sample(uniformDistribution(-1.5, 1.5), 256),
+ b=fft(a),
+ c=rowAt(b, 0))
+----
+
+image::images/math-expressions/noise-fft.png[]
+
+
+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.
+
+[source,text]
+----
+let(a=sin(sequence(256, 0, 6)),
+ b=sample(uniformDistribution(-1.5, 1.5), 256),
+ c=ebeAdd(a, b),
+ d=fft(c),
+ e=rowAt(d, 0))
+----
+
+image::images/math-expressions/hidden-signal-fft.png[]
+
+
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+++ b/solr/solr-ref-guide/src/math-expressions.adoc
@@ -56,4 +56,6 @@ record in your Solr Cloud cluster computable.
== <>
+== <>
+
== <>