With respect to convolution in discrete-time systems, consider the following statements:1.Convolution represents the output of an LTI system as a weighted sum of shifted impulse responses.2.The graphical method of convolution involves time reversal and shifting of one signal.3.Convolution operation always reduces the length of the output sequence.4.Convolution is valid for both continuous-time and discrete-time signals.Which of the above statements is/are correct?
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Convolution is the bridge between an input signal and the impulse response of an LTI system. The time-reversal step arises because the kernel is h[n−k] or h(t−τ), not h[k−n]. Key relation: DT: y[n]=Σ_{k=−∞}^{∞}x[k]h[n−k]. CT: y(t)=∫x(τ)h(t−τ)dτ. Finite DT length: Ly=Lx+Lh−1.
Exam focus: Graphical convolution sequence: fold → shift → multiply → sum. Common trap: Confusing convolution with element-wise multiplication or assuming output length always decreases.
Correct Answer: A - 1, 2 and 4 only Quick Concept Explanation Convolution is the bridge between an input signal and the impulse response of an LTI system. The time-reversal step arises because the kernel is h[n−k] or h(t−τ), not h[k−n]. Key relation: DT: y[n]=Σ_{k=−∞}^{∞}x[k]h[n−k]. CT: y(t)=∫x(τ)h(t−τ)dτ. Finite DT length: Ly=Lx+Lh−1.Exam focus: Graphical convolution sequence: fold → shift → multiply → sum. Common trap: Confusing convolution with element-wise multiplication or assuming output length always decreases. Statement-wise Verification Statement 1 - Correct.Convolution represents the output of an LTI system as a weighted sum of shifted impulse responses.Y[n]=Σ x[k]h[n−k] is literally a…
Correct Answer: A - 1, 2 and 4 only
Quick Concept Explanation
Convolution is the bridge between an input signal and the impulse response of an LTI system. The time-reversal step arises because the kernel is h[n−k] or h(t−τ), not h[k−n]. Key relation: DT: y[n]=Σ_{k=−∞}^{∞}x[k]h[n−k]. CT: y(t)=∫x(τ)h(t−τ)dτ. Finite DT length: Ly=Lx+Lh−1.
Exam focus: Graphical convolution sequence: fold → shift → multiply → sum. Common trap: Confusing convolution with element-wise multiplication or assuming output length always decreases.
Statement-wise Verification
Statement 1 - Correct. Convolution represents the output of an LTI system as a weighted sum of shifted impulse responses. Y[n]=Σ x[k]h[n−k] is literally a weighted sum of shifted impulse responses h[n−k].
Statement 2 - Correct. The graphical method of convolution involves time reversal and shifting of one signal. The graphical procedure reverses one signal, shifts it, multiplies overlap values and sums/integrates.
Statement 3 - Incorrect. Convolution operation always reduces the length of the output sequence. For finite sequences of lengths Lx and Lh, full linear convolution normally has length Lx+Lh−1, which is often longer than either input.
Statement 4 - Correct. Convolution is valid for both continuous-time and discrete-time signals. Convolution is defined as an integral in continuous time and a sum in discrete time.
Core Concept
Convolution is the bridge between an input signal and the impulse response of an LTI system. The time-reversal step arises because the kernel is h[n−k] or h(t−τ), not h[k−n].
Definitions verify 1,2,4; the finite-length formula gives a direct counterexample to 3. Option A.
Why the Other Options Are Wrong
Option B is incorrect because it includes incorrect statement(s) 3 and omits correct statement(s) 1, 4. Option C is incorrect because it includes incorrect statement(s) 3 and omits correct statement(s) 2, 4. Option D is incorrect because it includes incorrect statement(s) 3.
Answer-Key Verification Note
The uploaded provisional key marks C . After independent technical verification, the defensible answer is A . This discrepancy is stated explicitly rather than silently copying the provisional key.
Confusing convolution with element-wise multiplication or assuming output length always decreases.
Quick Revision
Convolution Procedure and Output Length is linked with Discrete-Time Convolution, LTI Systems and Signals and Systems. Graphical convolution sequence: fold → shift → multiply → sum.
Option B is incorrect because it includes incorrect statement(s) 3 and omits correct statement(s) 1, 4. Option C is incorrect because it includes incorrect statement(s) 3 and omits correct statement(s) 2, 4. Option D is incorrect because it includes incorrect statement(s) 3.
Common Mistake
Confusing convolution with element-wise multiplication or assuming output length always decreases.
