BISWAJIT ADHIKARI_28101623024_PC - EE - 602.pptx

FBDFXB

1. 1 TOPIC:- Sampling Theorem Department of Electrical Engineering Gargi Memorial Institute of Technology Name of the Student: BISWAJIT ADHIKARI Roll Number: 28101623024 Registration Number: 232810120222 Present Semester: 6 th Course Name: Digital Control System Course Code: PC-EE- 601A

2. 2 INTRODUCTION The "sampling" is one of the most fundamental concepts in digital signal processing that enables the conversion of continuous analog signals to discrete-time digital signals . It establishes the theoretical foundation for the analog-to-digital conversion process which is very important for applications like sound recording, video processing, wireless communication, and more . Since all computer controlled systems operate at discrete times only, it is important to know the condition under which a signal can be retrieved from its values at discrete points. Nyquist explored the key issue and Shannon gave the complete solution which is known as Shannon’s sampling theorem. 

3. MATHEMATICAL MODELING OF SAMPLING PROCESS Sampling operation in sampled data and digital control system is used to model either the sample and hold operation or the fact that the signal is digitally coded. If the sampler is used to represent S/H (Sample and Hold) and A/D (Analog to Digital) operations, it may involve delays, finite sampling duration and quantization errors. On the other hand used to represent digitally coded data the model will be much simpler. Following are two popular sampling operations: 1. Single rate or periodic sampling 2. Multi-rate sampling We would limit our discussions to periodic sampling only. 3

4. MATHEMATICAL MODELING OF SAMPLING PROCESS(continued….) Finite pluse width sampler 4

5. MATHEMATICAL MODELING OF SAMPLING PROCESS ( continued….) The pulse duration is p second and sampling period is T second.Uniform rate sampler is a linear device which satisfies the principle of superposition. 5

6. MATHEMATICAL MODELING OF SAMPLING PROCESS ( continued….) Frequency domain characteristics: Since p(t) is a periodic function, it can be represented by a Fourier series, as 6

7. SAMPLING THEOREM Sampling theorem states that “A band limited signal 𝒙(𝒕) with 𝑿(ɷ) = 𝟎 for |𝑚| ≥ ɷ𝒎 can be represented into and uniquely determined from its samples 𝒙(𝒏𝑻) if the sampling frequency 𝒇𝒔 ≥ 𝟐𝒇𝒎, where 𝒇𝒎 is the frequency. ( i.e ) for signal recovery, the sampling frequency must be at least twice the highest frequency present in the signal. cy component present in it. 7

8. SAMPLING THEOREM(CONTINUED….) 8

9. SAMPLING THEOREM(CONTINUED….) 9

10. SAMPLING THEOREM(CONTINUED….) For 𝝎𝒔 > 2𝝎𝒎 The spectral replicates have a larger separation between them known as guard band which makes process of filtering much easier and effective. Even a non-ideal filter which does not have a sharp cut off can also be used. 10

11. SAMPLING THEOREM(CONTINUED….) For 𝝎𝒔 = 𝟐𝝎𝒎 There is no separation between the spectral replicates so no guard band exists and 𝑋(𝜔) can be obtained from 𝑋𝑠(𝜔) by using only an ideal low pass filter (LPF) with sharp cutoff . For 𝝎𝒔 < 2𝝎𝒎 The low frequency component in 𝑋𝑠 𝜔 overlap on high frequency components of 𝑋 𝜔 so that there is presence of distortion and 𝑋 𝜔 cannot be recovered from 𝑋𝑠 𝜔 by using any filter. This distortion is called aliasing. So we can conclude that the frequency spectrum of 𝑋𝑠(𝜔) is not overlapped for 𝝎𝒔 − 𝝎𝒎 ≥ 𝝎𝒎, therefore the Original signal can be recovered from the sampled signal. For 𝝎𝒔 − 𝝎𝒎 < 𝝎𝒎, the frequency spectrum will overlap and hence the original signal cannot be recovered from the sampled signal. 11

12. SAMPLING THEOREM(CONTINUED….) Aliasing effect (or) fold over effect It is defined as the phenomenon in which a high frequency component in the frequency spectrum of signal takes identity of a lower frequency component in the spectrum of the sampled signal. 12

13. Data Reconstruction or Interpolati on Nyquist Rate It is the theoretical minimum sampling rate at which a signal can be sampled and still be reconstructed from its samples without any distortion Data Reconstruction or Interpolati on The process of obtaining analog signal 𝑥(𝑡) from the sampled signal 𝑥𝑠(𝑡) is called data reconstruction or interpolation. 13

14. Data Reconstruction or Interpolation The reconstruction filter, which is assumed to be linear and time invariant, has unit impulse response h(𝑡). The reconstruction filter, output 𝑦(𝑡) is given by convolu tion of 𝑥𝑠(𝑡) and h(𝑡). 14

15. ………THANK YOU…….. References:- https://www.rcet.org.in/uploads/academics/rohini_65660789483.pdf https://www.gvpcew.ac.in/LN-CSE-IT-22-32/EEE/4-Year/DCS.pdf https://testbook.com/electrical-engineering/sampling-theorem-in-signals-and-systems 15