Convolution Basics: Step-by-Step Practice Quiz Quiz

Enhance your understanding of convolution by tackling practical, scenario-based questions that explore its core concepts, operations, and common pitfalls. Designed for learners familiar with basic signal processing, this quiz helps reinforce foundational convolution knowledge and step-by-step techniques.

1. Fundamental Operation

   Which statement best describes the meaning of convolution between two discrete-time signals, such as x[n] and h[n]?

   1. Convolution always involves multiplying both signals by the same constant value.
   2. Convolution combines two signals by flipping and shifting one over the other and summing their products at each step.
   3. Convolution subtracts one signal from another at each time sample.
   4. Convolution multiplies corresponding points of two signals without any shifting or flipping.

   Explanation: The correct answer reflects the core of the convolution process: one signal (usually the impulse response) is flipped, shifted, and multiplied pointwise with the other before summing over all overlaps. The second option describes pointwise multiplication, which is not convolution. Multiplying by a constant (third option) is unrelated, and subtraction (fourth option) is not related to convolution at all.

2. Output Length Determination

   If a finite-length input signal x[n] has 4 samples and the filter h[n] has 3 samples, how many samples will the output of their linear convolution have?

   1. 3
   2. 7
   3. 5
   4. 6

   Explanation: The output length of the linear convolution of lengths N and M is always N + M - 1, so 4 + 3 - 1 equals 6. Choosing 5 or 7 does not match the correct formula, and 3 would only be true for circular convolution with certain constraints. Therefore, 6 is the only accurate answer.

3. Impulse Response Application

   What is the result of convolving any discrete signal x[n] with a unit impulse signal δ[n]?

   1. The signal x[n] becomes reversed.
   2. The signal x[n] is shifted by one sample.
   3. The original signal x[n] remains unchanged.
   4. All values in x[n] become zero.

   Explanation: Convolution with a unit impulse, δ[n], leaves the original signal unchanged because the impulse acts as the identity element in this context. Reversal occurs with the unit impulse flip, not with straightforward convolution. Setting all values to zero only happens if convolving with a zero signal, and a one-sample shift is not a default effect of δ[n] convolution.

4. Order in Convolution

   If y[n] is found by convolving x[n] with h[n], which property ensures that switching the order of the inputs gives the same output?

   1. Associativity
   2. Distributivity
   3. Commutativity
   4. Causality

   Explanation: Commutativity in convolution means that x[n] * h[n] yields the same result as h[n] * x[n], ensuring the output remains unchanged regardless of input order. Distributivity involves distributing convolution over addition, associativity relates to grouping when convolving three signals, and causality refers to system behavior, not convolution order.

5. Graphical Convolution Steps

   When using the step-by-step graphical method to compute the convolution of two finite signals, which operation is performed immediately before summing the overlapping products?

   1. Taking the derivative of one signal
   2. Shifting one of the signals by n samples
   3. Integrating both signals over time
   4. Multiplying both signals by zero

   Explanation: Immediately before summing the overlaps, graphical convolution requires shifting one of the flipped signals by n samples, aligning it with the other. Taking the derivative or integrating (second and fourth options) are operations from calculus, not pertinent to discrete convolution. Multiplying by zero would erase the signals' effect, which is not correct here.