With reference to impulse response of LTI systems, consider the following statements:
1. The impulse response completely characterizes an LTI system.
2. The output of an LTI system can be obtained by convolving input with impulse response.
3. For a given system, the impulse response is unique.
4. Impulse response concept is applicable only to discrete-time systems.
Which of the above statements is/are correct?
Answer & step-by-step solution
The 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.
Statement-wise Verification
- Statement 1 - Correct.
The impulse response completely characterizes an LTI system.
The 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. - Statement 2 - Correct.
The output of an LTI system can be obtained by convolving input with impulse response.
The 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. - Statement 3 - Correct.
For a given system, the impulse response is unique.
For LTI questions, think 'impulse response + convolution'. - Statement 4 - Incorrect.
Impulse response concept is applicable only to discrete-time systems.
Impulse-response characterization applies to continuous- and discrete-time LTI systems
Core Concept
The 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.
Formula / Key Relationship
y(t)=x(t)*h(t); y[n]=x[n]*h[n].
Why the Other Options Are Wrong
Option A is incorrect because it omits correct statement(s) 3. Option C is incorrect because it includes incorrect statement(s) 4 and omits correct statement(s) 1, 3. Option D is incorrect because it includes incorrect statement(s) 4.
Exam Shortcut / Approach
For LTI questions, think 'impulse response + convolution'.
Common Mistake
Impulse response is not a discrete-time-only concept.
Quick Revision
LTI Characterization by Impulse Response is linked with Impulse Response, LTI Systems and Signals and Systems. For LTI questions, think 'impulse response + convolution'.
LTI प्रणालियों की आवेग अनुक्रिया (impulse response) के संदर्भ में निम्नलिखित कथनों पर विचार कीजिए:
1. आवेग अनुक्रिया LTI प्रणाली को पूर्णतः अभिलक्षित करती है।
2. LTI प्रणाली का आउटपुट इनपुट को आवेग अनुक्रिया के साथ कनवॉल्व करके प्राप्त किया जा सकता है।
3. किसी दी गई प्रणाली के लिए आवेग अनुक्रिया अद्वितीय है।
4. आवेग अनुक्रिया की अवधारणा केवल असतत-समय प्रणालियों पर लागू होती है।
उपरोक्त में से कौन-सा/से कथन सही है/हैं?
उत्तर और चरणबद्ध हल
B — केवल 1, 2 और 3। कोई भी इनपुट स्थानांतरित आवेगों में विभाजित किया जा सकता है, इसलिए आवेग अनुक्रिया LTI प्रणाली को अद्वितीय रूप से अभिलक्षित करती है और y = x*h। यह सतत और असतत दोनों समय में लागू है।
विस्तृत हल (मूल भाषा में)
Statement-wise Verification
- Statement 1 - Correct.
The impulse response completely characterizes an LTI system.
The 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. - Statement 2 - Correct.
The output of an LTI system can be obtained by convolving input with impulse response.
The 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. - Statement 3 - Correct.
For a given system, the impulse response is unique.
For LTI questions, think 'impulse response + convolution'. - Statement 4 - Incorrect.
Impulse response concept is applicable only to discrete-time systems.
Impulse-response characterization applies to continuous- and discrete-time LTI systems
Core Concept
The 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.
Formula / Key Relationship
y(t)=x(t)*h(t); y[n]=x[n]*h[n].
Why the Other Options Are Wrong
Option A is incorrect because it omits correct statement(s) 3. Option C is incorrect because it includes incorrect statement(s) 4 and omits correct statement(s) 1, 3. Option D is incorrect because it includes incorrect statement(s) 4.
Exam Shortcut / Approach
For LTI questions, think 'impulse response + convolution'.
Common Mistake
Impulse response is not a discrete-time-only concept.
Quick Revision
LTI Characterization by Impulse Response is linked with Impulse Response, LTI Systems and Signals and Systems. For LTI questions, think 'impulse response + convolution'.
LTI પ્રણાલીઓના આવેગ પ્રતિભાવ (impulse response)ના સંદર્ભમાં નીચેનાં વિધાનો ધ્યાનમાં લો:
1. આવેગ પ્રતિભાવ LTI પ્રણાલીને સંપૂર્ણ રીતે દર્શાવે છે.
2. LTI પ્રણાલીનો આઉટપુટ ઇનપુટને આવેગ પ્રતિભાવ સાથે કન્વોલ્વ કરીને મેળવી શકાય છે.
3. આપેલી પ્રણાલી માટે આવેગ પ્રતિભાવ અનન્ય છે.
4. આવેગ પ્રતિભાવનો ખ્યાલ ફક્ત અસતત-સમય પ્રણાલીઓને લાગુ પડે છે.
ઉપરોક્તમાંથી કયું/કયાં વિધાન સાચું/સાચાં છે?
જવાબ અને પગલાવાર ઉકેલ
B — ફક્ત 1, 2 અને 3. કોઈ પણ ઇનપુટને ખસેડેલા આવેગોમાં વિભાજિત કરી શકાય છે, તેથી આવેગ પ્રતિભાવ LTI પ્રણાલીને અનન્ય રીતે દર્શાવે છે અને y = x*h. તે સતત અને અસતત બંને સમયમાં લાગુ પડે છે.
વિગતવાર ઉકેલ (મૂળ ભાષામાં)
Statement-wise Verification
- Statement 1 - Correct.
The impulse response completely characterizes an LTI system.
The 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. - Statement 2 - Correct.
The output of an LTI system can be obtained by convolving input with impulse response.
The 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. - Statement 3 - Correct.
For a given system, the impulse response is unique.
For LTI questions, think 'impulse response + convolution'. - Statement 4 - Incorrect.
Impulse response concept is applicable only to discrete-time systems.
Impulse-response characterization applies to continuous- and discrete-time LTI systems
Core Concept
The 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.
Formula / Key Relationship
y(t)=x(t)*h(t); y[n]=x[n]*h[n].
Why the Other Options Are Wrong
Option A is incorrect because it omits correct statement(s) 3. Option C is incorrect because it includes incorrect statement(s) 4 and omits correct statement(s) 1, 3. Option D is incorrect because it includes incorrect statement(s) 4.
Exam Shortcut / Approach
For LTI questions, think 'impulse response + convolution'.
Common Mistake
Impulse response is not a discrete-time-only concept.
Quick Revision
LTI Characterization by Impulse Response is linked with Impulse Response, LTI Systems and Signals and Systems. For LTI questions, think 'impulse response + convolution'.
More LTI Systems & Convolution questions