Convolution Calculator
Our free sequences calculator solves convolution problems. Get worked examples, visual aids, and downloadable results.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Convolution Calculator
Calculator
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Formula: (a*b)[n] = Sum of a[k] x b[n-k] for all valid k
Worked example — Output: [0.667, 2.000, 4.000, 6.000, 8.000, 6.000, 3.333]
Formula
(a*b)[n] = Sum of a[k] x b[n-k] for all valid k
For each output position n, the convolution sums the products of overlapping elements from sequence a and the reversed, shifted sequence b. The output length equals len(a) + len(b) - 1. This operation is equivalent to polynomial multiplication when sequences represent polynomial coefficients.
Worked Examples
Example 1: Moving Average Filter
Problem:Convolve the signal [2, 4, 6, 8, 10] with the averaging kernel [1/3, 1/3, 1/3] to smooth the signal.
Solution:Signal a = [2, 4, 6, 8, 10], Kernel b = [0.333, 0.333, 0.333] Output length = 5 + 3 - 1 = 7 y[0] = 2 x 0.333 = 0.667 y[1] = 2 x 0.333 + 4 x 0.333 = 2.000 y[2] = 2 x 0.333 + 4 x 0.333 + 6 x 0.333 = 4.000 y[3] = 4 x 0.333 + 6 x 0.333 + 8 x 0.333 = 6.000 y[4] = 6 x 0.333 + 8 x 0.333 + 10 x 0.333 = 8.000 y[5] = 8 x 0.333 + 10 x 0.333 = 6.000 y[6] = 10 x 0.333 = 3.333
Result:Output: [0.667, 2.000, 4.000, 6.000, 8.000, 6.000, 3.333]
Example 2: Polynomial Multiplication
Problem:Multiply (1 + 2x + 3x^2) by (1 + x) using convolution of coefficients [1,2,3] and [1,1].
Solution:a = [1, 2, 3] (coefficients of 1 + 2x + 3x^2) b = [1, 1] (coefficients of 1 + x) Output length = 3 + 2 - 1 = 4 y[0] = 1 x 1 = 1 y[1] = 1 x 1 + 2 x 1 = 3 y[2] = 2 x 1 + 3 x 1 = 5 y[3] = 3 x 1 = 3 Result polynomial: 1 + 3x + 5x^2 + 3x^3
Result:Convolution: [1, 3, 5, 3] = polynomial 1 + 3x + 5x^2 + 3x^3
Frequently Asked Questions
What is convolution and what does it represent mathematically?
Convolution is a mathematical operation that combines two sequences (or functions) to produce a third sequence expressing how the shape of one is modified by the other. For discrete sequences, the convolution of a[n] and b[n] is defined as (a*b)[n] = sum of a[k] x b[n-k] for all valid k. Intuitively, convolution slides one sequence across the other, multiplying overlapping elements and summing the products at each position. It measures the overlap between one function and a reversed, shifted copy of another. Convolution is commutative (a*b = b*a), associative, and distributive over addition, making it algebraically well-behaved.
How is discrete convolution computed step by step?
Discrete convolution follows a systematic slide-multiply-sum process. Given sequences a = [a0, a1, ..., am] and b = [b0, b1, ..., bn], the output has length m+n+1. For each output index k, reverse sequence b, shift it by k positions, multiply element-wise with a where they overlap, and sum all products. At position k=0, only a[0]*b[0] contributes. At position k=1, both a[0]*b[1] and a[1]*b[0] contribute. This continues until the final position where only the last elements overlap. The process is equivalent to polynomial multiplication when sequences represent polynomial coefficients.
What is the relationship between convolution and polynomial multiplication?
Convolution of two sequences is mathematically identical to multiplying two polynomials whose coefficients are those sequences. If a = [1, 2, 3] represents the polynomial 1 + 2x + 3x^2, and b = [1, 1] represents 1 + x, then their convolution [1, 3, 5, 3] represents the product polynomial 1 + 3x + 5x^2 + 3x^3. This connection makes convolution fundamental in algebra and computer science. The Fast Fourier Transform (FFT) exploits this relationship by converting sequences to frequency domain representations where convolution becomes simple element-wise multiplication, reducing complexity from O(n^2) to O(n log n).
How is convolution used in signal processing?
In signal processing, convolution is the fundamental operation for applying filters to signals. When a digital signal passes through a Linear Time-Invariant (LTI) system, the output equals the convolution of the input signal with the system impulse response. Low-pass filters smooth signals by convolving with averaging kernels like [1,1,1]/3. High-pass filters detect edges by convolving with difference kernels like [-1,2,-1]. Audio effects like reverb simulate room acoustics by convolving dry audio with a recorded room impulse response. Equalization, noise reduction, and echo cancellation all rely on convolution operations.
What is the convolution theorem and why is it important?
The convolution theorem states that convolution in the time domain equals multiplication in the frequency domain, and vice versa. Mathematically: FFT(a*b) = FFT(a) x FFT(b), where FFT is the Fourier Transform and x represents element-wise multiplication. This theorem is enormously important because direct convolution requires O(n^2) operations, while FFT-based convolution requires only O(n log n) operations. For large signals, this speedup is dramatic. A convolution of two signals with 1 million samples each would take roughly 10^12 operations directly but only about 4 x 10^7 operations using FFT, a speedup factor of about 25,000.
How does convolution work in image processing and convolutional neural networks?
In image processing, 2D convolution slides a small kernel (filter matrix) across an image, computing weighted sums at each position. Common kernels include blur (averaging nearby pixels), sharpen (emphasizing center pixel), edge detection (Sobel, Prewitt operators), and emboss filters. Convolutional Neural Networks (CNNs) learn optimal kernel values from training data rather than using hand-designed filters. A CNN might have dozens of layers, each with multiple learned kernels that detect progressively more abstract features, from edges and textures in early layers to complex shapes and objects in deeper layers. This hierarchical feature learning makes CNNs exceptionally powerful for image recognition.
What are the properties of convolution that make it useful?
Convolution has several elegant mathematical properties. Commutativity: a*b = b*a, meaning the order of operands does not matter. Associativity: (a*b)*c = a*(b*c), allowing cascaded operations to be combined. Distributivity: a*(b+c) = a*b + a*c, enabling linear decomposition. The identity element is the unit impulse [1,0,0,...], since convolving any signal with it returns the original signal. The sum of the convolution output equals the product of the input sums. These properties enable powerful system analysis techniques, such as decomposing complex filters into cascades of simpler ones.
What is the difference between convolution and correlation?
Convolution and correlation are closely related but differ in one key aspect: convolution reverses one signal before sliding, while correlation does not reverse either signal. Mathematically, correlation of a and b equals convolution of a with the time-reversed b. For symmetric signals (palindromes), convolution and correlation produce identical results. Correlation measures similarity between signals and is used in pattern matching, radar signal detection, and statistical analysis. Convolution describes how a system transforms an input signal. Cross-correlation is commonly used to find time delays between two signals, such as determining the direction of a sound source from microphone array recordings.
How do I interpret the length and values of the convolution output?
The output length equals len(a) + len(b) - 1. This makes intuitive sense: the output starts when the first elements begin overlapping and ends when the last elements stop overlapping. The output values represent the total weighted overlap at each shift position. Large output values indicate strong alignment between the sequences at that position. The output sum always equals the product of the input sums (a key verification property). The output is typically widest (most non-zero elements) in the middle and tapers at the edges where fewer elements overlap. For finite sequences, the convolution acts like a smoothing or spreading operation.
What are some practical applications of convolution beyond engineering?
Convolution appears in many fields beyond traditional engineering. In probability theory, the probability distribution of the sum of two independent random variables is the convolution of their individual distributions. In economics, moving average calculations are convolutions with uniform kernels. In pharmacokinetics, drug concentration over time involves convolving the absorption rate with the elimination rate. In optics, image formation by a lens is a convolution of the object with the point spread function. Even daily life involves implicit convolution: the blurriness of a photograph results from convolving a sharp image with the cameras blur kernel.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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