संकेतों के सैंपलिंग और पुनर्निर्माण के संदर्भ में निम्नलिखित कथनों पर विचार कीजिए:
1. सैंपलिंग आवृत्ति संकेत में उपस्थित अधिकतम आवृत्ति घटक की कम से कम दोगुनी होनी चाहिए।
2. जब सैंपलिंग आवृत्ति संकेत बैंडविड्थ को पकड़ने के लिए अपर्याप्त हो तब एलियासिंग होती है।
3. सैंपल्स से सतत-समय संकेत के पुनर्निर्माण के लिए आदर्श लो-पास फ़िल्टर आवश्यक है।
4. आवश्यकता से अधिक सैंपलिंग आवृत्ति बढ़ाने से एलियासिंग प्रभाव होते हैं।
उपरोक्त में से कौन-सा/से कथन सही है/हैं?
उत्तर और चरणबद्ध हल
D — केवल 1, 2 और 3। सैंपलिंग प्रमेय: fs ≥ 2fmax; कम fs पर स्पेक्ट्रम अतिव्यापन (एलियासिंग) होता है और आदर्श पुनर्निर्माण लो-पास इंटरपोलेशन फ़िल्टर से होता है। ओवरसैंपलिंग एलियासिंग जोखिम घटाती है, बढ़ाती नहीं।
विस्तृत हल (मूल भाषा में)
Statement-wise Verification
- Statement 1 - Correct.
The sampling frequency must be at least twice the maximum frequency component present in the signal.
The sampling theorem requires a band-limited signal to be sampled at at least twice its highest frequency for ideal reconstruction. - Statement 2 - Correct.
Aliasing occurs when the sampling frequency is insufficient to capture the signal bandwidth.
The sampling theorem requires a band-limited signal to be sampled at at least twice its highest frequency for ideal reconstruction. - Statement 3 - Correct.
Reconstruction of a continuous-time signal from its samples requires an ideal low-pass filter.
Anti-aliasing filters limit input bandwidth before sampling, and ideal reconstruction uses a low-pass interpolation filter. - Statement 4 - Incorrect.
Increasing the sampling frequency beyond requirement leads to aliasing effects.
Oversampling increases separation of spectral replicas and reduces aliasing risk
Core Concept
The sampling theorem requires a band-limited signal to be sampled at at least twice its highest frequency for ideal reconstruction. Sampling below this Nyquist rate folds spectral replicas onto one another, causing aliasing that cannot be undone from samples alone. Anti-aliasing filters limit input bandwidth before sampling, and ideal reconstruction uses a low-pass interpolation filter.
Formula / Key Relationship
f_s≥2f_max; Nyquist rate=2f_max.
Why the Other Options Are Wrong
Option A is incorrect because it omits correct statement(s) 3. Option B is incorrect because it includes incorrect statement(s) 4 and omits correct statement(s) 1, 3. Option C is incorrect because it includes incorrect statement(s) 4 and omits correct statement(s) 3.
Exam Shortcut / Approach
Filter before the ADC, not after aliasing has already occurred.
Common Mistake
Oversampling reduces aliasing risk; it does not create aliasing.
Quick Revision
Nyquist Rate Aliasing and Reconstruction is linked with Sampling Theorem, Sampling and Signals and Systems. Filter before the ADC, not after aliasing has already occurred.
With reference to sampling and reconstruction of signals, consider the following statements:
1. The sampling frequency must be at least twice the maximum frequency component present in the signal.
2. Aliasing occurs when the sampling frequency is insufficient to capture the signal bandwidth.
3. Reconstruction of a continuous-time signal from its samples requires an ideal low-pass filter.
4. Increasing the sampling frequency beyond requirement leads to aliasing effects.
Which of the above statements is/are correct?
Answer & step-by-step solution
The sampling theorem requires a band-limited signal to be sampled at at least twice its highest frequency for ideal reconstruction. Anti-aliasing filters limit input bandwidth before sampling, and ideal reconstruction uses a low-pass interpolation filter. Key relation: f_s≥2f_max; Nyquist rate=2f_max.
Exam focus: Filter before the ADC, not after aliasing has already occurred. Common trap: Oversampling reduces aliasing risk; it does not create aliasing.
Statement-wise Verification
- Statement 1 - Correct.
The sampling frequency must be at least twice the maximum frequency component present in the signal.
The sampling theorem requires a band-limited signal to be sampled at at least twice its highest frequency for ideal reconstruction. - Statement 2 - Correct.
Aliasing occurs when the sampling frequency is insufficient to capture the signal bandwidth.
The sampling theorem requires a band-limited signal to be sampled at at least twice its highest frequency for ideal reconstruction. - Statement 3 - Correct.
Reconstruction of a continuous-time signal from its samples requires an ideal low-pass filter.
Anti-aliasing filters limit input bandwidth before sampling, and ideal reconstruction uses a low-pass interpolation filter. - Statement 4 - Incorrect.
Increasing the sampling frequency beyond requirement leads to aliasing effects.
Oversampling increases separation of spectral replicas and reduces aliasing risk
Core Concept
The sampling theorem requires a band-limited signal to be sampled at at least twice its highest frequency for ideal reconstruction. Sampling below this Nyquist rate folds spectral replicas onto one another, causing aliasing that cannot be undone from samples alone. Anti-aliasing filters limit input bandwidth before sampling, and ideal reconstruction uses a low-pass interpolation filter.
Formula / Key Relationship
f_s≥2f_max; Nyquist rate=2f_max.
Why the Other Options Are Wrong
Option A is incorrect because it omits correct statement(s) 3. Option B is incorrect because it includes incorrect statement(s) 4 and omits correct statement(s) 1, 3. Option C is incorrect because it includes incorrect statement(s) 4 and omits correct statement(s) 3.
Exam Shortcut / Approach
Filter before the ADC, not after aliasing has already occurred.
Common Mistake
Oversampling reduces aliasing risk; it does not create aliasing.
Quick Revision
Nyquist Rate Aliasing and Reconstruction is linked with Sampling Theorem, Sampling and Signals and Systems. Filter before the ADC, not after aliasing has already occurred.
સંકેતોના સેમ્પલિંગ અને પુનઃનિર્માણના સંદર્ભમાં નીચેનાં વિધાનો ધ્યાનમાં લો:
1. સેમ્પલિંગ આવૃત્તિ સંકેતમાં હાજર મહત્તમ આવૃત્તિ ઘટકના ઓછામાં ઓછા બમણી હોવી જોઈએ.
2. જ્યારે સેમ્પલિંગ આવૃત્તિ સંકેત બેન્ડવિડ્થને પકડવા અપૂરતી હોય ત્યારે એલિયાસિંગ થાય છે.
3. સેમ્પલ્સમાંથી સતત-સમય સંકેતના પુનઃનિર્માણ માટે આદર્શ લો-પાસ ફિલ્ટર જરૂરી છે.
4. જરૂરિયાત કરતાં વધુ સેમ્પલિંગ આવૃત્તિ વધારવાથી એલિયાસિંગ અસરો થાય છે.
ઉપરોક્તમાંથી કયું/કયાં વિધાન સાચું/સાચાં છે?
જવાબ અને પગલાવાર ઉકેલ
D — ફક્ત 1, 2 અને 3. સેમ્પલિંગ પ્રમેય: fs ≥ 2fmax; ઓછી fs પર સ્પેક્ટ્રમ ઓવરલેપ (એલિયાસિંગ) થાય છે અને આદર્શ પુનઃનિર્માણ લો-પાસ ઇન્ટરપોલેશન ફિલ્ટરથી થાય છે. ઓવરસેમ્પલિંગ એલિયાસિંગ જોખમ ઘટાડે છે, વધારતું નથી.
વિગતવાર ઉકેલ (મૂળ ભાષામાં)
Statement-wise Verification
- Statement 1 - Correct.
The sampling frequency must be at least twice the maximum frequency component present in the signal.
The sampling theorem requires a band-limited signal to be sampled at at least twice its highest frequency for ideal reconstruction. - Statement 2 - Correct.
Aliasing occurs when the sampling frequency is insufficient to capture the signal bandwidth.
The sampling theorem requires a band-limited signal to be sampled at at least twice its highest frequency for ideal reconstruction. - Statement 3 - Correct.
Reconstruction of a continuous-time signal from its samples requires an ideal low-pass filter.
Anti-aliasing filters limit input bandwidth before sampling, and ideal reconstruction uses a low-pass interpolation filter. - Statement 4 - Incorrect.
Increasing the sampling frequency beyond requirement leads to aliasing effects.
Oversampling increases separation of spectral replicas and reduces aliasing risk
Core Concept
The sampling theorem requires a band-limited signal to be sampled at at least twice its highest frequency for ideal reconstruction. Sampling below this Nyquist rate folds spectral replicas onto one another, causing aliasing that cannot be undone from samples alone. Anti-aliasing filters limit input bandwidth before sampling, and ideal reconstruction uses a low-pass interpolation filter.
Formula / Key Relationship
f_s≥2f_max; Nyquist rate=2f_max.
Why the Other Options Are Wrong
Option A is incorrect because it omits correct statement(s) 3. Option B is incorrect because it includes incorrect statement(s) 4 and omits correct statement(s) 1, 3. Option C is incorrect because it includes incorrect statement(s) 4 and omits correct statement(s) 3.
Exam Shortcut / Approach
Filter before the ADC, not after aliasing has already occurred.
Common Mistake
Oversampling reduces aliasing risk; it does not create aliasing.
Quick Revision
Nyquist Rate Aliasing and Reconstruction is linked with Sampling Theorem, Sampling and Signals and Systems. Filter before the ADC, not after aliasing has already occurred.
More Sampling questions