Every LTI Systems & Convolution question asked in GPSC DEE Electrical, GPSC AE Electrical — solve, check instantly, then read the formula and exam tip.
Topic practice0/7✓ 0✗ 0
View answer & solution
h(t)=e^{-|t|} is nonzero for t<0, so the system is not causal. But ∫|h(t)|dt over all time is finite (=2), so the LTI system is BIBO stable. Therefore it is stable but not causal.
View answer & solutionFor an LTI system, output is the convolution of input and impulse response. Convolution is commutative, associative and distributive. Key relation: y=x*h; DT: y[n]=Σ x[k]h[n−k].
Exam focus: Flip, shift, multiply, sum. Common trap: Convolution usually increases finite-sequence length; it does not always reduce it.
View answer & solutionThe autocorrelation of a rectangular pulse is triangular. As one pulse slides across the other, their overlap length decreases linearly with shift; for an original pulse duration T, nonzero overlap exists from −T to +T, giving total triangular duration 2T.
View answer & solutionConvolution 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.
View answer & solutionA damped sinusoid is the imaginary part of a complex exponential e^{(-a+jω)t}. For an LTI system, a complex exponential is an eigenfunction: the system changes only its complex gain, not the exponential's pole. Therefore the output retains the same decay rate a and angular frequency ω; only magnitude and phase change.
View answer & solutionThe impulse response uniquely characterizes an LTI system for a specified input-output relationship because any input can be decomposed into shifted impulses and the output follows by convolution. This concept applies in both continuous and discrete time. Key relation: y(t)=x(t)*h(t); y[n]=x[n]*h[n].
Exam focus: For LTI questions, think 'impulse response + convolution'. Common trap: Impulse response is not a discrete-time-only concept.
View answer & solutionThe impulse response uniquely characterizes an LTI system for a specified input-output relationship because any input can be decomposed into shifted impulses and the output follows by convolution. This concept applies in both continuous and discrete time. Key relation: y(t)=x(t)*h(t); y[n]=x[n]*h[n].
Exam focus: For LTI questions, think 'impulse response + convolution'. Common trap: Impulse response is not a discrete-time-only concept.
