1. 8/27/2015 © 2003, JH McClellan & RW Schafer 1 Signal Processing First Lecture 8 Sampling & Aliasing

2. 8/27/2015 © 2003, JH McClellan & RW Schafer 2 READING ASSIGNMENTS  This Lecture:  Chap 4, Sections 4-1 and 4-2  Replaces Ch 4 in DSP First, pp. 83-94  Other Reading:  Recitation: Strobe Demo (Sect 4-3)  Next Lecture: Chap. 4 Sects. 4-4 and 4-5

3. 8/27/2015 © 2003, JH McClellan & RW Schafer 3 LECTURE OBJECTIVES  SAMPLING can cause ALIASING  Nyquist/Shannon Sampling Theorem  Sampling Rate (f s ) > 2f max (Signal bandwidth)  Spectrum for digital signals, x[n]  Normalized Frequency ALIASING

4. 8/27/2015 © 2003, JH McClellan & RW Schafer 4 SYSTEMS Process Signals  PROCESSING GOALS:  We need to change x(t) into y(t) for many engineering applications:  For example, more BASS, image deblurring, denoising, etc SYSTEM x(t)y(t)

5. 8/27/2015 © 2003, JH McClellan & RW Schafer 5 System IMPLEMENTATION  DIGITAL/MICROPROCESSOR  Convert x(t) to numbers stored in memory ELECTRONICS x(t)y(t) COMPUTERD-to-AA-to-D x(t)y(t)y[n]x[n]  ANALOG/ELECTRONIC:  Circuits: resistors, capacitors, op-amps

6. 8/27/2015 © 2003, JH McClellan & RW Schafer 6 SAMPLING x(t)  SAMPLING PROCESS  Convert x(t) to numbers x[n]  “n” is an integer; x[n] is a sequence of values  Think of “n” as the storage address in memory  UNIFORM SAMPLING at t = nT s  IDEAL: x[n] = x(nT s ) A-to-D x(t)x[n]

7. 8/27/2015 © 2003, JH McClellan & RW Schafer 7 SAMPLING RATE, f s  SAMPLING RATE (f s )  f s =1/T s  NUMBER of SAMPLES PER SECOND  T s = 125 microsec  f s = 8000 samples/sec UNITS ARE HERTZ: 8000 Hz  UNIFORM SAMPLING at t = nT s = n/f s  IDEAL: x[n] = x(nT s )=x(n/f s ) A-to-D x(t) x[n]=x(nT s )

8. 8/27/2015 © 2003, JH McClellan & RW Schafer 8

9. 8/27/2015 © 2003, JH McClellan & RW Schafer 9 SAMPLING THEOREM  HOW OFTEN ?  DEPENDS on FREQUENCY of SINUSOID  ANSWERED by NYQUIST/SHANNON Theorem  ALSO DEPENDS on “RECONSTRUCTION”

10. 8/27/2015 © 2003, JH McClellan & RW Schafer 10 Reconstruction? Which One? Given the samples, draw a sinusoid through the values

11. 8/27/2015 © 2003, JH McClellan & RW Schafer 11 STORING DIGITAL SOUND  x[n] is a SAMPLED SINUSOID  A list of numbers stored in memory  EXAMPLE: audio CD  CD rate is 44,100 samples per second  16-bit samples  Stereo uses 2 channels  Number of bytes for 1 minute is  2 X (16/8) X 60 X 44100 = 10.584 Mbytes

12. 8/27/2015 © 2003, JH McClellan & RW Schafer 12 DISCRETE-TIME SINUSOID  Change x(t) into x[n] DERIVATION DEFINE DIGITAL FREQUENCY

13. 8/27/2015 © 2003, JH McClellan & RW Schafer 13 DIGITAL FREQUENCY  VARIES from 0 to 2 , as f varies from 0 to the sampling frequency  UNITS are radians, not rad/sec  DIGITAL FREQUENCY is NORMALIZED

14. 8/27/2015 © 2003, JH McClellan & RW Schafer 14 SPECTRUM (DIGITAL) 2  –2 

15. 8/27/2015 © 2003, JH McClellan & RW Schafer 15 SPECTRUM (DIGITAL) ??? 2  –2  ? x[n] is zero frequency???

16. 8/27/2015 © 2003, JH McClellan & RW Schafer 16 The REST of the STORY  Spectrum of x[n] has more than one line for each complex exponential  Called ALIASING  MANY SPECTRAL LINES  SPECTRUM is PERIODIC with period = 2   Because

17. 8/27/2015 © 2003, JH McClellan & RW Schafer 17 ALIASING DERIVATION  Other Frequencies give the same

18. 8/27/2015 © 2003, JH McClellan & RW Schafer 18 ALIASING DERIVATION–2  Other Frequencies give the same

19. 8/27/2015 © 2003, JH McClellan & RW Schafer 19 ALIASING CONCLUSIONS  ADDING f s or 2f s or –f s to the FREQ of x(t) gives exactly the same x[n]  The samples, x[n] = x(n/ f s ) are EXACTLY THE SAME VALUES  GIVEN x[n], WE CAN’T DISTINGUISH f o FROM (f o + f s ) or (f o + 2f s )

20. 8/27/2015 © 2003, JH McClellan & RW Schafer 20 NORMALIZED FREQUENCY  DIGITAL FREQUENCY

21. 8/27/2015 © 2003, JH McClellan & RW Schafer 21 SPECTRUM for x[n]  PLOT versus NORMALIZED FREQUENCY  INCLUDE ALL SPECTRUM LINES  ALIASES  ADD MULTIPLES of 2   SUBTRACT MULTIPLES of 2   FOLDED ALIASES  (to be discussed later)  ALIASES of NEGATIVE FREQS

22. 8/27/2015 © 2003, JH McClellan & RW Schafer 22 SPECTRUM (MORE LINES) 2  –0.2  1.8  –1.8 

23. 8/27/2015 © 2003, JH McClellan & RW Schafer 23 SPECTRUM (ALIASING CASE) –0.5  –1.5  0.5  2.5  –2.5  1.5 

24. 8/27/2015 © 2003, JH McClellan & RW Schafer 24 SAMPLING GUI (con2dis)

25. 8/27/2015 © 2003, JH McClellan & RW Schafer 25 SPECTRUM (FOLDING CASE) 0.4  –0.4  1.6  –1.6 

26. 8/27/2015 © 2003, JH McClellan & RW Schafer 26 SAMPLING DEMO (Chap. 4)

27. 8/27/2015 © 2003, JH McClellan & RW Schafer 27 FOLDING (a type of ALIASING)  MANY x(t) give IDENTICAL x[n]  CAN’T TELL f o FROM (f s -f o )  Or, (2f s -f o ) or, (3f s -f o )  EXAMPLE:  y(t) has 1000 Hz component  SAMPLING FREQ = 1500 Hz  WHAT is the “FOLDED” ALIAS ?

28. 8/27/2015 © 2003, JH McClellan & RW Schafer 28 DIGITAL FREQ AGAIN FOLDED ALIAS ALIASING

29. 8/27/2015 © 2003, JH McClellan & RW Schafer 29 FOLDING DIAGRAM