Problem on Convolution Sum and Its Properties || EC Academy

Summary

This video lecture provides a comprehensive explanation of the convolution sum in Linear Time-Invariant (LTI) systems. It covers key concepts like impulse response, time-shifting, and the formula for convolution sum. The lecture walks through the steps of computing convolution, including folding, shifting, and summing, and provides graphical examples for better understanding. It also explains properties of convolution such as commutative, associative, and distributive properties. The video concludes with practical examples to solidify the understanding of convolution in signal processing.

Takeaways

• 😀 Convolution is a fundamental concept in linear time-invariant (LTI) systems, used to calculate the output signal by combining the input sequence and the system's impulse response.
• 😀 The convolution sum formula is given by y(n) = Σ x(k) * h(n - k), where x(k) is the input sequence and h(n-k) is the time-shifted impulse response.
• 😀 To calculate the convolution, first perform a folding operation on the impulse response h(n) to get h(-k), then shift it by 'n' units, and finally multiply the sequences element-wise and sum the results.
• 😀 The length of the output sequence y(n) is determined by the lengths of the input sequences, with the formula being L1 + L2 - 1, where L1 and L2 are the lengths of the sequences x(n) and h(n).
• 😀 Graphical representation of convolution involves plotting the input sequence x(k) and impulse response h(k), folding h(k), shifting it, and then computing the output sequence step-by-step.
• 😀 The folding operation in convolution reverses the impulse response h(k) around k=0, and the shifting operation moves it to align with the input sequence at different time steps.
• 😀 The output sequence y(n) is computed for each value of n, and the result is the sum of the element-wise multiplication of the shifted impulse response and the input sequence.
• 😀 In the example provided, the sequence values for the output were calculated for n = -1, 0, 1, 2, 3, 4, and 5, resulting in the final output sequence: y(n) = [1, 4, 8, 8, 3, -2, -1].
• 😀 Convolution can be performed either by folding and shifting the input sequence x(n) or the impulse response h(n), depending on the specific calculation method used.
• 😀 Convolution follows key properties such as commutativity, associativity, and distributivity, meaning the order and grouping of sequences in the convolution operation do not affect the final result.

Q & A

What is convolution in the context of linear time-invariant (LTI) systems?

Convolution is a mathematical operation used to calculate the output of an LTI system when the input and impulse response are known. It involves summing the products of the input sequence and a time-shifted version of the impulse response.

What does the impulse response, H(n), represent in an LTI system?

The impulse response, H(n), represents the system's response to a unit impulse input (Delta(n)). It characterizes how the system reacts over time to this impulse, providing essential information about the system's behavior.

What is the significance of the time shift operation in the convolution sum?

The time shift operation in convolution is used to align the impulse response with the current value of the input sequence. It helps to compute the output by adjusting the position of the impulse response at each point in time.

What does the formula for convolution sum look like?

The convolution sum is mathematically represented as: y(n) = Σ [x(k) * h(n - k)], where x(k) is the input sequence, h(n - k) is the time-shifted impulse response, and the sum is taken over all values of k.

What is the first step in performing a convolution sum?

The first step is folding, which means reflecting the impulse response, H(k), around k = 0 to obtain H(-k). This is followed by shifting the reflected impulse response by n units.

Why is folding important in convolution?

Folding is crucial because it performs a time-reversal operation on the impulse response. This reflection allows the impulse response to be aligned properly with the input signal during the convolution process.

How do you calculate the length of the output sequence in a convolution operation?

The length of the output sequence, y(n), is calculated as L1 + L2 - 1, where L1 is the length of the impulse response, H(n), and L2 is the length of the input sequence, X(n).

In the example with finite sequences, what are the values of n for which the output sequence will be calculated?

For the finite sequences, the range of n for the output sequence will be from -1 to 5, covering 7 values. This is because the length of both the input sequence and the impulse response is 4, leading to an output sequence length of 7.

What is the graphical representation of the convolution process?

The graphical representation involves plotting the input sequence X(k) and the impulse response H(k), then performing the folding, shifting, and multiplying operations step-by-step. The result is the output sequence y(n) obtained by summing the products.

What properties of convolution are discussed in the transcript?

The transcript discusses three key properties of convolution: commutative property (x(n) * h(n) = h(n) * x(n)), associative property (x(n) * [h1(n) * h2(n)] = [x(n) * h1(n)] * h2(n)), and distributive property (x(n) * [h1(n) + h2(n)] = [x(n) * h1(n)] + [x(n) * h2(n)]).