Overview of sampling

Sampling is a Simple method to convert analog signal into discrete Signal by using any one of its three methods
if the sampling frequency is twice or greater than twice then sampled signal can be convert back into analog signal easily......

1. SAMPLING SAGAR KUMAR 16TC-20 Signal and System 1 TELECOM ENGINEERING

2. C o n t e n t 2 Introduction Sampling Theorem Sampling Methods Ideal sampling Natural sampling Flat top sampling Reconstruction of Sampled Signal Aliasing TELECOM ENGINEERING

3. I n t r o d uction 3 Most of the signals that we use in our daily life are analog in nature ( for eg: speech, weather signals etc). Digital system possess many advantages in comparison to analog system such as they are immune to noise, can be stored, processed with more efficient algorithms, secure, more robust and cost effective etc. Most of the effective signal processor are digital signal processors which needs digital information in order to process it. TELECOM ENGINEERING

4. I n t r o d uction 4 Hence there arises a need to convert our analog signal to discrete time signal in order to process them properly through digital signal processors and then reconvert them back to analog signals so that we can understand them. Sampling is the answer to this need. Samplin g i s a w a y t o c o n v ert a sig n al f r o m continuous time to discrete time. TELECOM ENGINEERING

5. Sampling Theorem 5 A Band-limited continuous time signal can be represented by its samples and can be recovered back when sampling frequency f s is greater than or equal to the twice the highest frequency component of message signal. i. e. f s ≥ 2f m where f m is the maximum frequency component of the continuous time signal. Key words: Band Limited Sampling Frequency TELECOM ENGINEERING

6. Sampling Method f s = 1 / T s i s c al l ed the sam p ling r a t e or sa m p l ing frequency. x ( t ) Analog signal  x s ( t )  x [ nT s ] Disc r e t e signal 6 p ( t ) Analog signal is sampled every T s secs. T s is referred to as the sampling interval. • TELECOM ENGINEERING

7. Sampling Methods There are 3 sampling methods: Ideal Samplin g - a n i m p u lse a t e a c h sam p ling in st a n t w i th a m p l it u d e equals t o signa l a t th a t point of time. Ideal Sampling 7 TELECOM ENGINEERING

8. Sampling Methods Natural Sampling- a pulse of short width with varying amplitude Natural Sampling 8 TELECOM ENGINEERING

9. Sampling Methods Flat-top Sampling– make use of sample and hold circuit almost like natural but with single amplitude value. Flat-top Sampling 9 TELECOM ENGINEERING

10. Impulse Sampling Imp u l s e s ampling can b e per f orme d b y mu l tiplying input signal x(t) with impulse train p(t) of period 'T s '.  p ( t )    ( t  nT s ) n  He r e, the amplit u d e of i m pu ls e chan g es with respect to amplitude of input signal x(t). p ( t ) x ( t ) x s ( t ) 10 10 TELECOM ENGINEERING

11. Impulse Sampling The output of sampler is given by x s ( t )  x ( t )  p ( t )   x ( t )    ( t  nT s ) n  Recall the sifting property of impulse function: x ( t )  ( t  t )  x ( t )  ( t  t ) hence we can write :  x s ( t )  x  ( t )   x ( nT s )  ( t  nT s ) n  (1) 11 TELECOM ENGINEERING

12. Impulse Sampling To take the spectrum of the sampled signal let us take the Fourier Transform of equation (1) as we know multiplication in time domain becomes convolution in frequency domain we have: whe r e hence s  2  X (  )  X (  )  1  X (  )* P (  )  X (  ) and P (  ) are Fourier transform of x(t) and p(t). s 12 T X s (  )  X  (  )   (   n  ) 1  2    X (  )* 2   s     n    TELECOM ENGINEERING

13. Impulse Sampling si n ce Hence the spectrum of sampled signal is given as: • we can not use ideal/impulse sampling because we can not generate the impulse train practically. 1 s T s X s (  )  X  (  )  X (   n  )   n    s s T s 2  n    FT of p ( t )  FT   ( t  nT )    (   n  )   13 TELECOM ENGINEERING

14. Natural Sampling N a tu r al s ampling i s simi l ar t o i m pu ls e samp l i n g , except the impulse train is replaced by pulse train of period T s . n  When we multiply input signal x(t) to pulse train p(t) we get the signal as shown below: The pulse equation is being given as:  p ( t )   p ( t  nT s ) x ( t ) p ( t ) x s ( t )   2  2 14 p ( t ) TELECOM ENGINEERING

15. Natural Sampling is given as: whe r e n  The exponential Fourier series representation of p(t) The output of the sampler is given as: x s ( t )  x ( t )  p ( t )   x ( t )   p ( t  nT s ) (1) n C e n j  s t  p ( t )   n    2 1 1 T n s T s T s P ( n  )  C  p ( t ) e  n j  s t dt  s 2  T s (2) 15 TELECOM ENGINEERING

16. Natural Sampling Now putting the value of C n in equation (2) we have: s 1 P ( n  ) e n j  s t  n  T s p ( t )   s T s P ( n  ) e n j  s t n     1   1 s T s P ( n  ) e n j  s t n    Now putting the value of p(t) in equation (1) we have: x s ( t )  x ( t )  p ( t )  x ( t )    s 16 T s P ( n  ) x ( t ) e n j  s t   1  n    TELECOM ENGINEERING

17. Natural Sampling To get the spectrum of the sampled signal let us take the Fourier Transform of both side: N o w ac c o r din g t o f r eq u e nc y s h i ftin g p r o pe r ty of FT we have: s s s nj  t P ( n  ) x ( t ) e FT  x ( t )   FT  1   T s      n    s s FT x ( t ) e nj  t    X (    )   1 17 s T s n    P ( n  ) FT  x ( t ) e n j  s t       TELECOM ENGINEERING

18. Natural Sampling Hence we can say that Hence the spectrum of sampled signal is given as: s s T s n    P ( n  ) FT  x ( t ) e n j  s t  FT  x ( t )   1     s 1 T s X (  )  P ( n  s ) X (    s )   n     s 18 2  s   s X s (  ) TELECOM ENGINEERING

19. Flat-Top Sampling T h e t op of the samples are fl a t i .e. they have c on s t a n t amplitude. Hence, it i s c al l ed as flat top During transmission, noise is introduced at top of the transmission pulse which can be easily removed if the pulse is in the form of flat top. sampling or practical sampling. x ( t ) 19 p ( t ) x s ( t ) TELECOM ENGINEERING

20. Flat-Top Sampling . x s ( t )  p ( t )* x  ( t )  x s ( t )  p ( t )*[  x ( kT s )  ( t  kT s )] k  Mathematically we can consider the flat top sampled signal is equivalent to the convolved sequence of the pulse p(t) and the ideal sampled signal x δ (t). p ( t ) x  ( t ) s x ( t )   2  2 20 * 20 TELECOM ENGINEERING

21. Flat-Top Sampling . P (  )  Fourier Transform of p ( t ) X  (  )  Fourier Transform of x  ( t ) Now applying Fourier Transform X s (  )  P (  ) X  (  ) s s T s X (  )  P (  ) 1 X (   n  )   n    s s T s X (  )  1 P (  ) X (   n  )   n     s 21 2  s   s X s (  ) where TELECOM ENGINEERING

22.  s 2  s   s  s 2  s   s  s 2  s   s Reconstruction of Sampled Signal Spectrum of a typical Sampled Signal f s > 2f m Oversampling f s = 2f m Perfect sampling f s < 2f m Unde r sa m pling X s (  ) X s (  ) Al i as ing Low-Pass Filter with transfer function H(ω) ω m X s (  ) -ω m ω m -ω m ω s -ω m 22 TELECOM ENGINEERING

23. Al i asing 23 Aliasing refers to the phenomenon of a high frequency component in the spectrum of a signal seemingly taking on the identity of a lower frequency in the spectrum of its sampled version (under- sampled version of the message signal) It is worth to be mention here that a time-limited signal cannot be band-limited. Since all signals are more or less time-limited, they cannot be band- limited. TELECOM ENGINEERING

24. Al i asing 24 Hence we must pass most of signals through low pass filter before sampling in order to make them band-limited. This is called an anti-aliasing filter and are typically built into an analog to digital (A/D) converter. Distortion will occur (If the signal is not band-limited) when the signal is sampled. We refer to this distortion as aliasing. TELECOM ENGINEERING

25. R e f e r ences 25 Forouzan B. A, “Data Communications and Networking”, McGraw-Hill, Fourth Edition T aub H .,Schilling D . L . ,Sah a G. “ T a u b ’ s Principle of Communication Systems”, McGraw-Hill, Third edition Communication Systems, 3Rd Ed Simon Haykin • B. P. Lathi, Modern Digital and Analog Communication Systems, (3rd ed.) Oxford University press, 1998 John G. Proakis and Masoud Salehi, Communication Systems Engineering, Prentice Hall international edition, 1994 TELECOM ENGINEERING

26. 26 TELECOM ENGINEERING