Convolution is a mathematical method of combining two signals to form a third signal. Passing the Dirac delta function (unit impulse) $\delta[n]$ through a linear system results in the impulse response $h[n]$. The impulse response is simply the signal resulting from passing the unit impulse (Dirac delta function) through a linear system.

## Principle

The properties of homogeneity and shift-invariance in Linear Time-Invariant System Theory hold that scaling and shifting an input signal in a linear system results in the same scaling and shifting in the output signal. Because of these properties, we can represent any impulse as a shifted and scaled delta function and consequently know what the impulse response, i.e. output signal in response to an impulse input signal, will be for that scaled and shifted impulse.

An impulse of $-3$ at the $8^{th}$ sample would be represented as a unit impulse $h$ by scaling the delta function $\delta$ by $-3$ and shifting it to the right by $8$ samples: $-3\delta[n-8]$, where $n-8$ means the $8^{th}$ sample is now the $0^{th}$. Due to homogeneity and shift invariance, we can determine the impulse response of this impulse by simply scaling and shifting the unit impulse response in the same manner. In other words:

$$-3\delta[n-8] \mapsto -3h[n-8]$$

What this means is that if we know the unit impulse response of a system, we consequently know how the system will react to any impulse, not just a unit impulse. These impulse responses can then be synthesized to form the output signal that would result from running the input signal through the actual system. An example of the powerful implications of this property is convolution reverb, in which an impulse response of a physical or virtual space is generated and then convolved with any input signal to simulate the effect of reverberation in that space.

In short, the input signal convolved with the unit impulse response results in the output signal. Convolution of input signal $x[n]$ with unit impulse $h[n]$ to generate output signal $y[n]$ is denoted as:

$$x[n] * h[n] = y[n]$$

Since convolution allows us to go from input signal $x[n]$ to output signal $y[n]$, we can conclude that convolution involves the generation of the impulse response for each impulse in the input signal as decomposed by impulse decomposition, as well as the subsequent synthesis of each impulse response, to generate the output signal.

## Definition

Convolution can be described by the so-called convolution summation. The convolution summation is pretty simple, and is defined as follows:

$$y[i] = \sum_{j=0}^{M-1} h[j]\ x[i-j]$$

Where the length of the output signal $y[n]$ is defined as $M + N - 1$ where $M$ is the length of the unit impulse response and $N$ is the length of the input signal.

All this says is that a given sample $y[i]$ in the output signal $y[n]$ is determined by the summation of every $i^{th}$ sample in every resultant impulse response. In effect, the summation above encodes how different samples in the resulting impulse responses contribute to a single output sample.

Natural imperative instinct might lead you to conclude that this can be easily implemented using nested iterations and arrays:

const int outputLength = M + N - 1;
int *y = new int[outputLength]();

for (int i = 0; i < outputLength; ++i) {
for (int j = 0; j < M; ++j) {
if (i - j >= 0)
y[i] += x[i - j] * h[j];
}
}


But wait up! We are using Haskell, a functional programming language which typically does without both arrays and iteration. This means that to implement convolution in Haskell without the use of Arrays or imperative iteration loops, we need to really understand the operation occurring in the convolution summation.

### Convolution Machine

The book The Scientist and Engineer’s Guide to Digital Signal Processing uses a metaphor known as the Convolution Machine to help conceptualize the convolution operation at a granular level. The convolution machine is simply a theoretical machine in which the unit impulse response is:

1. wrapped onto a roller/cylinder
2. rolled over the input signal such that each sample lines up with one on the reversed impulse response
3. each lined-up pair of samples from input signal and impulse response is multiplied and each product is summed

If you’re wondering why step 3 mentions a reversed impulse response, imagine that you have a roller and that the impulse response is on a strip of tape. Now imagine that you apply the impulse response tape over and around the roller, such that the numbers are facing you and are in the correct order. Now, when you roll this roller over and across the input signal, from left to right, the numbers on the impulse response tape will make contact with the input signal in reverse order.

See this page for an illustration of the convolution machine in Figure 6-8.

## Implementation

Implementing the convolution machine is pretty straightforward once we are able to conceptualize what it is actually doing.

Let’s start with the type signature. Since we’re not using arrays, we’ll represent the signals as lists of numbers. Convolution does something with two signals to produce a third signal, so the type signature is pretty straightforward:

convolve :: (Num a) => [a] -> [a] -> [a]
convolve hs xs = undefined

In the signature, xs refers to the input signal and hs refers to the impulse response.

Now for the implementation of convolve. First, consider this component of the convolution summation:

$$x[i-j]$$

When we are computing the first sample, such that $i = 0$, in the output signal $y[n]$, then at one point we need to refer to the $x[-(M-1)]$ sample where $M$ is length of impulse response. However, there are no samples to the left of the first sample.

So what we have to do is prepad the input signal with $M-1$ samples of value $0$. This padding has the added benefit of allowing us to simply map over the padded input signal to generate the output signal. This is because the convolution operation’s output signal length is $M + N - 1$ where $M$ is the length of the impulse response and $N$ is the length of the input signal. The padding can be achieved with:

let pad = replicate ((length hs) - 1) 0
ts  = pad ++ xs

Once we prepad the input signal with enough zero samples, we can pass the padded input signal and impulse response to a function which simulates the rolling of the convolution machine. This function will be nested within convolve and will simply be used as a recursive helper function.

### Let’s Roll

roll :: (Num a) => [a] -> [a] -> [a]
roll _  [] = []
roll hs ts = undefined

The roll function is recursive and will simulate the actual rolling of the convolution machine over the input signal. As it rolls, it will consume the head of the input signal ts. Think of the consumption of the head as if the input signal is being wrapped around the roller as it rolls. The input signal ts will therefore eventually be empty, meaning the convolution machine has finished rolling over the entire input signal.

The roll function is run for every sample in the output signal. This is where the bulk of the implementation comes in. At any given sample in the input signal, we simulate the roll by zipping the input signal from that sample forward along with the impulse response. This generates a list of pairs each consisting of the input signal sample with its corresponding impulse response sample (which is being rolled over it).

If you have trouble conceptualizing this, imagine that the impulse response on the roller is tape, so that when you roll it over the input signal, the impulse response—which, remember, makes contact with the input signal in reverse—sticks to the input signal and is lined up such that each sample in the impulse response is directly over a sample of the input signal.

We then need to multiply the components of each pair with each other, i.e. the input sample multiplied by its corresponding impulse response sample. The act of zipping and multiplying the zipped up pairs can be done in one go with zipWith (*). We then gather all of these products and sum them up. This sum is the latest computed sample in the output signal.

We construct the complete output signal by cons’ing the sample with a recursive call to roll, however this roll will concern only the next sample forward, thereby simulating rolling across the input signal.

With this information, we can finish the definition of roll:

roll :: (Num a) => [a] -> [a] -> [a]
roll _  [] = []
roll hs ts = let sample = sum $zipWith (*) ts hs in sample : roll hs (tail ts) Here is the whole convolution function convolve put together: convolve :: (Num a) => [a] -> [a] -> [a] convolve hs xs = let pad = replicate ((length hs) - 1) 0 ts = pad ++ xs in roll ts (reverse hs) where roll :: (Num a) => [a] -> [a] -> [a] roll _ [] = [] roll hs ts = let sample = sum$ zipWith (*) ts hs
in sample : roll hs (tail ts)

## Reduction

Now that we understand the concept behind convolution, we can reduce the above implementation a bit further.

The observation we should make is that the roll function acts like map, specifically over ts. The only detail is that on every element mapped over, the result of that element’s mapping concerns the list ts from that element forward. If we are on the third element of ts, we only act on the third element forward. In other words, we are mapping over every tail of ts. Knowing this, we can change the roll function to a straight up map over tails ts.

However, tails considers [] to be a tail of any list—which is technically correct—so we’ll always have a trailing 0 element if we do it this way. That’s why we simply take the init of the result of tails, which returns every element in a list except the last one. We also still need to prepad the signal, so those lines remain:

convolve :: (Num a) => [a] -> [a] -> [a]
convolve hs xs =
let pad = replicate ((length hs) - 1) 0

### Benchmark

The criterion Haskell package provides tools for benchmarking and analyzing code. The synthetic benchmark we will conduct will run each implementation with an impulse response of length 100 and an input signal of length 1,000.

In the following code, conv is the naive implementation, conv' is the reduced naive implementation, and parConv is the parallel implementation:

data ConvType = Naive | Reduced | Parallel deriving (Eq, Ord)
convTypes = Data.Map.fromList [(Naive, conv), (Reduced, conv'), (Parallel, parConv)]

main = defaultMain [
bench "Naive Convolution" (runConv Naive),
bench "Reduced Convolution" (runConv Reduced),
bench "Parallelized Convolution" (runConv Parallel) ]
where runConv ctype =
let hs = [1..100 :: Int]
ts = [1..1000 :: Int]
convfn = fromJust $Data.Map.lookup ctype convTypes in nf (convfn hs) ts Compile the benchmark with: $ ghc --make -O2 -threaded -o conv conv.hs

Run it with:

## Conclusion

I’m new to Digital Signal Processing, so if you notice any glaring errors please feel free to correct me; I would appreciate it. If you are interested in this subject and would like to read a book to learn more, I wholeheartedly recommend The Scientist and Engineer’s Guide to Digital Signal Processing. If you would like to learn more about Convolution, you can check the relevant chapters in that freely available book.

### Imperative Approach

You can also check out this page as well, which also covers multidimensional convolution with a concrete example of a Gaussian filter applied to an image for the purposes of blurring it. This specific application of the Gaussian filter is known as the Gaussian Blur. The Gaussian Blur is pretty popular in realtime image rendering, such as in video games, because of a property it has which allows it be applied in two dimensions, e.g. in an image, as two independent one-dimensional operations. This makes it dramatically faster and more efficient, and is trivial to implement in modern GPU Compute shaders 1. Such shaders can then be used to implement effects such as motion blur in games 2. The page also provides imperative implementations of convolution in C++.

### Optimizations

Haskell is known for having many ways of doing any one thing, so if you come up with a better solution feel free to gist it and post it in the comments.

Of course, this post concerns a naive implementation of convolution. There are other more optimized implementations of convolution, such as FFT convolution which exploits the Fast Fourier Transform and the principle of duality—convolution in the time domain is equivalent to multiplication in the frequency domain—to perform convolution a lot faster in some cases.

1. As described in 3D Game Programming with DirectX 11 by Frank D. Luna in Chapter 12, page 450, § 12.7
2. Despite this optimization of Gaussian Blurring, many implementations optimize further. Blurring typically involves rendering the scene to a separate buffer (e.g. Render-to-Texture) at a scaled-down resolution. This speeds up the blurring operation as there are less pixels to operate on. Then the result is rendered to the actual screen. Since the point is to blur, the upscaling is usually hardly noticeable.
January 4, 2013
787aeda — October 11, 2018