To calculate the Discrete Time Convolution:
\[ y[n] = \sum_{k} (x[k] \cdot h[n-k]) \]
Where:
Discrete Time Convolution is a mathematical operation used in signal processing and control systems to combine two sequences. It produces a third sequence that represents the area under the product of the two original sequences as a function of displacement. This operation is used to calculate the output of a system based on the input and the system's impulse response. It is essential in digital signal processing for implementing filters and other signal processing operations.
Let's assume the following sequences:
Using the formula:
\[ y[0] = (1 \cdot 0) + (0 \cdot 1) = 0 \]
\[ y[1] = (1 \cdot 1) + (2 \cdot 0) = 1 \]
\[ y[2] = (1 \cdot 0.5) + (2 \cdot 1) + (3 \cdot 0) = 0.5 + 2 = 2.5 \]
\[ y[3] = (2 \cdot 0.5) + (3 \cdot 1) = 1 + 3 = 4 \]
\[ y[4] = (3 \cdot 0.5) = 1.5 \]
The Convolution Result (\(y[n]\)) is [0, 1, 2.5, 4, 1.5].
