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Digital Signal Processing

Digital Signal Processing. Prof. Nizamettin AYDIN naydin @ yildiz .edu.tr http:// www . yildiz .edu.tr/~naydin. Digital Signal Processing. Lecture 8 Sampling & Aliasing. READING ASSIGNMENTS. This Lecture: Chap 4, Sections 4-1 and 4-2 Replaces Ch 4 in DSP First, pp. 83-94

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Digital Signal Processing

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  1. Digital Signal Processing Prof. Nizamettin AYDIN naydin@yildiz.edu.tr http://www.yildiz.edu.tr/~naydin

  2. Digital Signal Processing Lecture 8 Sampling & Aliasing

  3. 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

  4. LECTURE OBJECTIVES ALIASING • SAMPLING can cause ALIASING • Sampling Theorem • Sampling Rate > 2(Highest Frequency) • Spectrum for digital signals, x[n] • Normalized Frequency

  5. SYSTEMS Process Signals x(t) SYSTEM y(t) • PROCESSING GOALS: • Change x(t) into y(t) • For example, more BASS • Improve x(t), e.g., image deblurring • Extract Information from x(t)

  6. System IMPLEMENTATION x(t) ELECTRONICS y(t) x(t) A-to-D x[n] COMPUTER y[n] D-to-A y(t) • DIGITAL/MICROPROCESSOR • Convert x(t) to numbers stored in memory • ANALOG/ELECTRONIC: • Circuits: resistors, capacitors, op-amps

  7. SAMPLING x(t) x(t) C-to-D x[n] • SAMPLING PROCESS • Convert x(t) to numbersx[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 = nTs • IDEAL: x[n] = x(nTs)

  8. SAMPLING RATE, fs • SAMPLING RATE (fs) • fs =1/Ts • NUMBER of SAMPLES PER SECOND • Ts= 125 microsec fs = 8000 samples/sec • UNITS ARE HERTZ: 8000 Hz • UNIFORM SAMPLING at t = nTs= n/fs • IDEAL: x[n] = x(nTs)=x(n/fs) x[n]=x(nTs) x(t) C-to-D

  9. SAMPLING THEOREM • HOW OFTEN ? • DEPENDS on FREQUENCY of SINUSOID • ANSWERED by SHANNON/NYQUIST Theorem • ALSO DEPENDS on “RECONSTRUCTION”

  10. Reconstruction? Which One? Given the samples, draw a sinusoid through the values

  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 × (16/8) × 60 × 44100 = 10.584 Mbytes

  12. DISCRETE-TIME SINUSOID • Change x(t) into x[n] DERIVATION DEFINE DIGITAL FREQUENCY

  13. DIGITAL FREQUENCY • VARIES from 0 to 2p, as f varies from 0 to the sampling frequency • UNITS are radians, not rad/sec • DIGITAL FREQUENCY is NORMALIZED

  14. SPECTRUM (DIGITAL) –0.2p 2p(0.1)

  15. SPECTRUM (DIGITAL) ??? ? –2p 2p(1) x[n] is zero frequency???

  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 = 2p • Because

  17. ALIASING DERIVATION • Other Frequencies give the same

  18. ALIASING DERIVATION–2 • Other Frequencies give the same

  19. ALIASING CONCLUSIONS • ADDING fsor 2fsor –fs to the FREQ of x(t) gives exactly the same x[n] • The samples, x[n] = x(n/ fs) are EXACTLY THE SAME VALUES • GIVEN x[n], WE CAN’T DISTINGUISH fo FROM (fo +fs) or (fo + 2fs)

  20. NORMALIZED FREQUENCY • DIGITAL FREQUENCY

  21. SPECTRUM for x[n] • PLOT versus NORMALIZED FREQUENCY • INCLUDE ALL SPECTRUM LINES • ALIASES • ADD MULTIPLES of 2p • SUBTRACT MULTIPLES of 2p • FOLDED ALIASES • (to be discussed later) • ALIASES of NEGATIVE FREQS

  22. SPECTRUM (MORE LINES) –1.8p –0.2p 2p(0.1) 1.8p

  23. SPECTRUM (ALIASING CASE) –1.5p 0.5p –0.5p 1.5p –2.5p 2.5p

  24. SAMPLING GUI (con2dis)

  25. SPECTRUM (FOLDING CASE) –0.4p 0.4p –1.6p 1.6p

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