Enter the first input sequence and the second input sequence, shifted by k units, into the calculator to determine the output sequence.

## Discrete Time Convolution Formula

The following formula is used to calculate the discrete time convolution of two sequences.

y[n] = Σ (x[k] * h[n-k])

Variables:

- y[n] is the output sequence x[k] is the first input sequence h[n-k] is the second input sequence, shifted by k units The summation Σ is over all k

To calculate the discrete time convolution, for each value of n, multiply each element x[k] of the first sequence by the corresponding element h[n-k] of the second sequence, shifted by k units. Then sum all these products to get the value y[n] of the output sequence at that point. Repeat this process for all values of n.

## What is a Discrete Time Convolution?

Discrete Time Convolution is a mathematical operation used primarily in signal processing and control systems. It is a method to combine two sequences to produce a third sequence, representing the area under the product of the two original sequences as a function of displacement. In simpler terms, it provides a way to calculate the output of a system based on the input and the system's impulse response. It is a fundamental tool in digital signal processing, used to implement filters and other signal processing operations.

## How to Calculate Discrete Time Convolution?

The following steps outline how to calculate the Discrete Time Convolution using the given formula:

- First, determine the values of the input sequences, x[k] and h[n-k].
- Next, calculate the product of each corresponding pair of values from x[k] and h[n-k].
- Then, sum up all the products obtained in the previous step.
- Finally, assign the resulting sum to the corresponding value of the output sequence, y[n].

**Example Problem:**

Use the following variables as an example problem to test your knowledge:

x[k] = [2, 3, 1]

h[n-k] = [1, 0, -1]

Find the value of y[n] for a specific value of n.