What is the relation between the unit impulse function and the unit ramp function?
Answer & step-by-step solution
The unit ramp r(t) differentiates to the unit step u(t), and differentiating once more gives the unit impulse δ(t). Therefore δ(t) = d²r(t)/dt².
Concept & Reasoning
The basic continuous-time singularity chain is obtained by integration/differentiation. Starting from δ(t), integration gives u(t); integrating u(t) gives r(t) = t u(t). Reversing the chain, dr/dt = u(t) and du/dt = δ(t). Hence the second derivative of the ramp is the impulse. This is understood in the generalized/distribution sense at t = 0.
How to Solve It in the Exam
Ramp → derivative → step → derivative → impulse.
Important Exam Point
Remember the derivative chain rather than memorizing isolated formulas.
Quick Trick
Ramp → derivative → step → derivative → impulse.
Common Mistake
The impulse is not the first derivative of the ramp; that derivative is the step.
इकाई आवेग फलन और इकाई रैम्प फलन के बीच संबंध क्या है?
उत्तर और चरणबद्ध हल
C — δ(t) = d²r(t)/dt²। इकाई रैम्प r(t) का अवकलज इकाई स्टेप u(t) है और उसका अवकलज इकाई आवेग δ(t)।
विस्तृत हल (मूल भाषा में)
Concept & Reasoning
The basic continuous-time singularity chain is obtained by integration/differentiation. Starting from δ(t), integration gives u(t); integrating u(t) gives r(t) = t u(t). Reversing the chain, dr/dt = u(t) and du/dt = δ(t). Hence the second derivative of the ramp is the impulse. This is understood in the generalized/distribution sense at t = 0.
How to Solve It in the Exam
Ramp → derivative → step → derivative → impulse.
Important Exam Point
Remember the derivative chain rather than memorizing isolated formulas.
Quick Trick
Ramp → derivative → step → derivative → impulse.
Common Mistake
The impulse is not the first derivative of the ramp; that derivative is the step.
એકમ આવેગ વિધેય અને એકમ રેમ્પ વિધેય વચ્ચેનો સંબંધ શું છે?
જવાબ અને પગલાવાર ઉકેલ
C — δ(t) = d²r(t)/dt². એકમ રેમ્પ r(t)નું વિકલજ એકમ સ્ટેપ u(t) છે અને તેનું વિકલજ એકમ આવેગ δ(t).
વિગતવાર ઉકેલ (મૂળ ભાષામાં)
Concept & Reasoning
The basic continuous-time singularity chain is obtained by integration/differentiation. Starting from δ(t), integration gives u(t); integrating u(t) gives r(t) = t u(t). Reversing the chain, dr/dt = u(t) and du/dt = δ(t). Hence the second derivative of the ramp is the impulse. This is understood in the generalized/distribution sense at t = 0.
How to Solve It in the Exam
Ramp → derivative → step → derivative → impulse.
Important Exam Point
Remember the derivative chain rather than memorizing isolated formulas.
Quick Trick
Ramp → derivative → step → derivative → impulse.
Common Mistake
The impulse is not the first derivative of the ramp; that derivative is the step.
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