Chapter 7 Sampling Theorem Tín hiệu và hệ thống.pptx

Chapter 7 Sampling Theorem

1. Chapter 7: Sampling 1. Introduction A continuous signal can be represented and reconstructed using its discrete values at specific time points which called samples. To accurately recover the original signal, the sampling theorem must be satisfied The sampling theorem can be considered as a bridge between continuous and discrete signals.

2. 2. Sampling Theorem Let be a band-limited signal with for . Then is uniquely determined by its samples , if where Given these samples, we can reconstruct by generating a periodic impulse train in which successive impulses have amplitudes that are successive sample values. This impulse train is then processed through an ideal lowpass filter with gain and cutoff frequency greater than and less than . The resulting output signal will exactly equal . Notation: : Sampling frequency. : Sampling period. : Maximum frequency of the signal spectrum. Note that : If : No overlap (aliasing) occurs between the shifted replicas of . If : Overlap occurs, leading to information loss and inability to recover the original signal.   when  

3. 3. Representation of Continuous Signals Using Samples 3.1. In the Time Domain The sampling process is performed by multiplying the continuous signal with a periodic unit impulse function : Result:   impulse-train sampling

4. 3. Representation of Continuous Signals Using Samples 3.2. In the Frequency Domain According to the Fourier transform property: Where: Result:   Spectrum of original signal Spectrum of sampling funtion spectrum of sampled signal with  

5. 4. Signal Reconstruction from Samples The signal is reconstructed by filtering the sampled signal through an ideal low-pass filter with a gain of and a cutoff frequency satisfying .   system for sampling and reconstruction representative spectrum for x(t) corresponding spectrum for ideal lowpass filter to recover X(j ) from Xp (j ) spectrum o f  

6. 4. Signal Reconstruction from Samples Interpolation (the fitting of a continuous signal to a set of sample values) is a commonly used procedure for reconstructing a function, either approximately or exactly, from samples. Interpolation formula: ⇒ Interpolation using the impulse response of an ideal lowpass filter as in above is commonly referred to as band-limited interpolation   Ideal band-limited interpolation using the sine function: band-limited signal x(t); impulse train of samples of x(t); ideal band-limited interpolation in which the impulse train is replaced by a superposition of sine functions

7. 5, DISCRETE-TIME PROCESSING OF CONTINUOUS-TIME SIGNALS Continuous-time signals must be converted into discrete-time signals for processing by digital systems such as computers and DSP processors. After processing, the signal can be converted back to continuous-time .   Sampling via multiplication with impulse train Conversion to a sequence Processing via digital filter Reconstruction via lowpass filtering to get  

8. 6, Sampling DT signals Impulse-Train Sampling : with We have: In frequency domain: The frequency domain of the sampled signal is: The Fourier transform of the impulse train: The result:  

9. Signal reconstruction in DT is simmilar to which in CT CONTINUOUS TIME DISCRETE TIME