Which of the following is NOT a method of computation of linear convolution?

Explanation:

Correct Option Analysis:

The correct option for the question "Which of the following is NOT a method of computation of linear convolution?" is:

Option 4: Substitution method

Linear convolution is a fundamental operation in signal processing used to combine two sequences to form a third sequence. The methods typically employed to compute the linear convolution of two sequences include the graphical method, multiplication method, and tabulation method. These methods provide different approaches to achieve the same result, each with its own advantages and practical applications.

The substitution method, on the other hand, is not recognized as a standard technique for computing linear convolution. The term "substitution method" does not pertain to the process of convolution in signal processing. Therefore, it is the correct answer to the question as it does not belong to the list of methods used for computing linear convolution.

Important Information:

To further understand the analysis, let’s evaluate the other options:

Option 1: Multiplication method

The multiplication method, also known as the direct method or the formula method, involves directly applying the definition of convolution. For two discrete-time signals \(x[n]\) and \(h[n]\), their linear convolution \(y[n]\) is given by:

\[ y[n] = \sum_{k=-\infty}^{\infty} x[k] * h[n-k] \]

In this approach, the signals are multiplied and summed according to the convolution sum formula. This method is straightforward and directly implements the mathematical definition of convolution.

Option 2: Graphical method

The graphical method is a visual approach to compute the convolution of two sequences. This method involves the following steps:

1. Plotting the sequences \(x[n]\) and \(h[n]\) on a graph.
2. Flipping one of the sequences, usually \(h[n]\), to get \(h[-n]\).
3. Shifting \(h[-n]\) by \(n\) units to get \(h[n-k]\).
4. Multiplying the overlapping values of \(x[k]\) and \(h[n-k]\) for each shift.
5. Summing the products to obtain the convolution result \(y[n]\).

This method provides an intuitive understanding of the convolution process, especially useful for educational purposes and for gaining insights into how the sequences interact.

Option 3: Tabulation method

The tabulation method involves arranging the sequences in a tabular form to facilitate the computation of convolution. This method typically involves the following steps:

1. Listing the values of the sequences \(x[n]\) and \(h[n]\) in a table format.
2. Performing element-wise multiplication for each possible shift of the sequences.
3. Summing the products for each shift to obtain the convolution result \(y[n]\).

This method is systematic and helps in organizing the calculations, making it easier to handle longer sequences and reducing the chances of errors.

Conclusion:

Understanding the methods used for computing linear convolution is crucial in signal processing. The multiplication method, graphical method, and tabulation method are standard techniques that offer different approaches to achieve the same result. The substitution method, however, is not a recognized method for convolution computation, making it the correct option as it does not belong to the list of methods used for this purpose.