(a) If y(n) = x(n) * h (n) , show that (b) Compute the convolution y (n) = x(n) * h (n...

Simplify further.

Here,

Therefore,

Therefore, the expression is proved.

(b)

(1)

Write the sequences for the signals.

The procedure gives the required convolution between the two sequences.

1. Write down the sequence and as shown.

2. Multiply each sample in with the samples of and tabulate the values.

3. Divide the elements in the table by drawing diagonals, lines as shown.

4. Starting from the left sum all the elements in each strip and write down in same order.

Calculate the output sequence, .

The output sequence is,

Therefore, the convolution of the two signals is, .

The sum of the elements of the signal is,

The sum of the elements of the signal is,

The sum of the elements of the signal is,

Check the correctness of the result:

Therefore, the results are verified with the part (a) theorem.

Write the sequences for the signals.

Calculate the output sequence, .

The output sequence is,

Therefore, the convolution of the two signals is, .

The sum of the elements of the signal is,

The sum of the elements of the signal is,

The sum of the elements of the signal is,

Check the correctness of the result:

Therefore, the results are verified with the part (a) theorem.

(3)

Write the sequences for the signals.

Calculate the output sequence, .

The output sequence is,

Therefore, the convolution of the two signals is .

The sum of the elements of the signal is,

The sum of the elements of the signal is,

The sum of the elements of the signal is,

Check the correctness of the result:

Therefore, the results are verified with the part (a) theorem.

Write the sequences for the signals.

Calculate the output sequence, .

The output sequence is,

Therefore, the convolution of the two signals is, .

The sum of the elements of the signal is,

The sum of the elements of the signal is,

The sum of the elements of the signal is,

Check the correctness of the result:

Therefore, the results are verified with the part (a) theorem.

(5)

Write the sequences for the signals.

Calculate the output sequence, .

The output sequence is,

Therefore, the convolution of the two signals is .

The sum of the elements of the signal is,

The sum of the elements of the signal is,

The sum of the elements of the signal is,

Check the correctness of the result:

Therefore, the results are verified with the part (a) theorem.

Write the sequences for the signals.

Calculate the output sequence, .

The output sequence is,

Therefore, the convolution of the two signals is, .

The sum of the elements of the signal is,

The sum of the elements of the signal is,

The sum of the elements of the signal is,

Check the correctness of the result:

Therefore, the results are verified with the part (a) theorem.

(7)

Write the sequences for the signals.

Calculate the output sequence, .

The output sequence is,

Therefore, the convolution of the two signals is, .

The sum of the elements of the signal is,

The sum of the elements of the signal is,

The sum of the elements of the signal is,

Check the correctness of the result:

Therefore, the results are verified with the part (a) theorem.

Write the sequences for the signals.

Calculate the output sequence, .

Here,

The output sequence is,

Therefore, the convolution of the two signals is, .

The sum of the elements of the signal is,

The sum of the elements of the signal is,

The sum of the elements of the signal is,

Check the correctness of the result:

Therefore, the results are verified with the part (a) theorem.

(9)

Write the sequences for the signals.

Calculate the output sequence, .

The output sequence is,

Therefore, the convolution of the two signals is, .

The sum of the elements of the signal is,

The sum of the elements of the signal is,

The sum of the elements of the signal is,

Check the correctness of the result:

Therefore, the results are verified with the part (a) theorem.

Write the sequences for the signals.

Calculate the output sequence, .

The output sequence is,

Therefore, the convolution of the two signals is, .

The sum of the elements of the signal is,

The sum of the elements of the signal is,

The sum of the elements of the signal is,

Check the correctness of the result:

Therefore, the results are verified with the part (a) theorem.

(11)

Write the sequences for the signals.

The impulse response for . So, the system is causal and for , hence the sequence is causal sequence.

Calculate the expression for the sequence, .

Therefore, the convolution of the two signals is, .

The sum of the elements of the signal is,

The sum of the elements of the signal is,

Check the correctness of the result:

Therefore, the results are verified with the part (a) theorem.