OKAN UNIVERSITY FACULTY OF ENGINEERING AND ARCHITECTURE. EEE 403 Digital Signal Processing 10 Periodic Sampling

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1 OKAN UNIVERSITY FACULTY OF ENGINEERING AND ARCHITECTURE EEE 403 Digital Signal Processing 10 Periodic Sampling Fall 2013 Yrd. Doç. Dr. Didem Kivanc Tureli 12/20/2013 Lecture 10 1 Periodic Sampling Periodic signal discrete data values In practice, the A/D converter performs sampling. Continuous signal Analog to digital (A/D ) converter Digital values ig question: how fast do we sample a continuous signal so we can preserve its information content? 12/20/2013 Lecture

2 Ambiguity of the sampled waveform 12/20/2013 Lecture 10 3 Aliasing Example: x( t) sin 2 f t Sample x(t) at a rate of f s =1/t s samples/second: f0 ts f0 ts f t th 0 sample sin 2 0 sin 0 0 st 1 sample sin 2 1 nd 2 sample sin s th n sample sin 2 f0 nts 0 s s 0 m x( nts ) sin 2 f0nts sin 2 f0nts 2 m sin 2 f0 nt s nt s nk k sin 2 f0 nts sin 2 f0 nt s nt s t s sin 2 f k f nt 12/20/2013 Lecture

3 Aliasing Defined When sampling at a rate of f s samples/s, if k is any positive or negative integer, we cannot distinguish between the sampled values of a sinewave of f o Hz and a sinewave of (f o +kf s ) Hz. 12/20/2013 Lecture 10 5 Sampling Replication of Spectrum In general sampling a signal at a rate of f s samples/s causes its spectrum to replicate every f s Hz. efore sampling -W W After sampling -3 f s -2 f s -f s -W W f s 2 f s 3 f s f s W f s + W 12/20/2013 Lecture

4 Sampling Replication of Spectrum Replicas Frequency band of interest Replicas -3 f s -2 f s -f s -W W f s 2 f s 3 f s f s W f s + W 12/20/2013 Lecture /20/2013 Lecture

5 Sampling Low Pass Signals The following signal is and Limited and Low Pass -W W Low Pass means its spectrum is centered around frequency 0, and the magnitude of its spectrum becomes 0 after W frequency. (In real life there are NO band limited signals because a signal which is finite in time cannot be band limited in frequency) 12/20/2013 Lecture 10 9 Nyquist Criterion If the signal bandwidth is W and the signal is lowpass, then f s 2W. f s /2 is called the folding frequency 12/20/2013 Lecture

6 Sampling frequency > Nyquist frequency Sampling frequency < Nyquist frequency 12/20/2013 Lecture The noise overlaps with (are on top of) the replicas, and the replicas of the noise overlap with (are on top of) the original signal. 12/20/2013 Lecture

7 Solution: Low pass filter first You should always low pass filter before sampling to fix this problem. 12/20/2013 Lecture andpass Sampling 12/20/2013 Lecture

8 Advantages of bandpass sampling Also called Intermediate Frequency samling (IF sampling) sub-nyquist sampling Undersampling Advantages: Can do with slower A/D converter Need less memory andpass sampling performs digitization and frequency translation in a single process, often called sampling translation. For bandpass signals, Nyquist criterion is fs 2. 12/20/2013 Lecture /20/2013 Lecture

9 - f c f c f c Lecture 10 12/20/ andpass sampling To get m frequency replica without aliasing: 2fc 2fc fs and fs 2 m m1 Example: f c = 20MHz, = 5MHz: m m m m m 2fc 2fc 2 m m MHz 22.5 MHz 10 MHz MHz 15.0 MHz 10 MHz MHz MHz 10 MHz MHz 9.0 MHz 10 MHz 12/20/2013 Lecture

10 12/20/2013 Lecture Selecting the sampling frequency R highest signal frequency component bandwidth fc /2 12/20/2013 Lecture

11 12/20/2013 Lecture R factor Define the ratio R highest signal frequency component bandwidth /2 When we decide on a sampling frequency, we want to stay away from the forbidden zones if we sample at those frequencies we will get aliasing. fc 12/20/2013 Lecture

12 Spectral Inversion When we sample a bandpass signal, we can end up with the spectrum flipped over. Original - f c f c Non-inverted f c Inverted f c 12/20/2013 Lecture To avoid spectral inversion m must be even. Optimum noninverting sampling rate is 2 f c fs 0 m even Choose various values for m even, then find the maximum number such that f s0 still meets the Nyquist criterion. 12/20/2013 Lecture

13 Or not to avoid spectral inversion If spectral inversion is not important, then the question is: "How many replications of the positive and negative images of bandwidth can we squeeze into the frequency range of 2f c + without overlap?" frequency span 2 fc fc / 2 R 2bandwidth 2 Rint R Rint 1 Rint R R int is the largest integer less than R. Then the sampling frequency to use is 2 f c fs min R int 12/20/2013 Lecture Example = 5MHz, fc = 20MHz: If we don t care about spectral inversion: R = 22.5/5=4.5 Rint = 4 fs= (40+5)/4=11.25MHz What if we want to avoid spectral inversion? (2fc-)/2=(40-5)/2=17.5MHz (2fc-)/4=(40-5)/4=8.75MHz does not satisfy Nyquist. So must sample at 17.5MHz. 12/20/2013 Lecture

14 Understanding Digital Signal Processing, Third Edition, Richard Lyons ( ) Pearson Education, /20/2013 Lecture Understanding Digital Signal Processing, Third Edition, Richard Lyons ( ) Pearson Education, /20/2013 Lecture

15 Lecture 10 12/20/ /20/2013 Lecture