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Convolution Calculator

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Example
Created on 2024-06-20Asked by Ava Young (Solvelet student)
Compute the convolution of the functions f(x)=e−x f(x) = e^{-x} and g(x)=cos⁡x g(x) = \cos x .

Solution

Step 1: Identify the functions f(x)=e−x f(x) = e^{-x} and g(x)=cos⁡x g(x) = \cos x . Step 2: Write down the convolution integral: (f∗g)(x)=∫−∞∞f(t)g(x−t) dt. (f * g)(x) = \int_{-\infty}^{\infty} f(t) g(x - t) \, dt. Step 3: Substitute the given functions: (f∗g)(x)=∫−∞∞e−tcos⁡(x−t) dt. (f * g)(x) = \int_{-\infty}^{\infty} e^{-t} \cos(x - t) \, dt. Step 4: Use the convolution property of cosine functions: cos⁡(x−t)=cos⁡xcos⁡t+sin⁡xsin⁡t. \cos(x - t) = \cos x \cos t + \sin x \sin t. Step 5: Perform the integration: (f∗g)(x)=cos⁡x∫−∞∞e−tcos⁡t dt+sin⁡x∫−∞∞e−tsin⁡t dt. (f * g)(x) = \cos x \int_{-\infty}^{\infty} e^{-t} \cos t \, dt + \sin x \int_{-\infty}^{\infty} e^{-t} \sin t \, dt. Step 6: Evaluate the integrals, noting that ∫−∞∞e−tcos⁡t dt \int_{-\infty}^{\infty} e^{-t} \cos t \, dt and ∫−∞∞e−tsin⁡t dt \int_{-\infty}^{\infty} e^{-t} \sin t \, dt are constants. Step 7: Conclusion. The convolution of f(x)=e−x f(x) = e^{-x} and g(x)=cos⁡x g(x) = \cos x is a linear combination of cosine and sine functions. Solved on Solvelet with Basic AI Model
Some of the related questions asked by Jackson Robinson on Solvelet
1. Compute the convolution of the functions f(x)=e−x f(x) = e^{-x} and g(x)=x2 g(x) = x^2 over the interval [0,∞) [0, \infty) 2. Determine the convolution of the functions f(x)=u(x) f(x) = u(x) and g(x)=e−x g(x) = e^{-x} , where u(x) u(x) is the unit step function.,
DefinitionThis leads us to the concept of convolution which is apparently a mathematical operation that takes two functions h and x, and to create a third function, that in a sense represents the integral of the pointwise product of the different functions of one (t) reversed and shifted by the fixed number of time units (t). Signals processing, probability, and image processing all use convolution to model linear systems, calculate running averages, and compute probability distributions respectively. This provides it a versatile ability to analyse & manipulate signals and data sets in a wide array of applications Signal processing; example: In signal processing, we need to filter signals so we convolve the signal with the filter kernel in the time-domain and obtain the smoothed or modified output signal.
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