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Basics of Signal Processing

Basics of Signal Processing. RECEIVER. SOURCE. SIGNAL. ACTION. describe waves in terms of their significant features. understand the way the waves originate. effect of the waves . will the people in the boat notice ?. frequency = 1/T. sine wave period (frequency) amplitude phase.

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Basics of Signal Processing

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  1. Basics of Signal Processing

  2. RECEIVER SOURCE SIGNAL ACTION describe waves in terms of their significant features understand the way the waves originate effect of the waves will the people in the boat notice ?

  3. frequency = 1/T • sine wave • period (frequency) • amplitude • phase l = speed of sound × T, where T is a period

  4. sine cosine Phase F

  5. Sinusoidal grating of image

  6. TO • Fourier idea • describe the signal by a sum of other well defined signals

  7. Fourier Series A periodic function as an infinite weighted sum of simpler periodic functions!

  8. A good simple function Ti=T0 / i

  9. e.t.c. ad infinitum

  10. T=1/f T=1/f e.t.c…… e.t.c……

  11. period T period T Fourier’s Idea • Describe complicated function as a weighted sum of simpler functions! • simpler functions are known • weights can be found

  12. + + 0 0 - - x x + + + 0 0 - - - = = + + 0 - - + + 0 area is positive (T/2) area is zero

  13. 0 0 b1T/2 0 0 …………… = area=b1T/2 area=b2T/2

  14. area = DC area = b1T/2 area = b2T/2 f(t) f(t) sin(4πt) f(t) sin(2πt) T=2p + + + - - - + + + + + + - - - - + + + +

  15. Aperiodic signal Discrete spectrum becomes continuous (Fourier integral) Phase spectrum Magnitude spectrum 0 0 1/T0 1/T0 2/T0 2/T0 frequency frequency Spacing of spectral components is f0 =1/T0

  16. sampling ts=1/fs 22 samples per 4.2 ms  0.19 ms per sample  5.26 kHz

  17. Sampling T = 10 ms (f = 1/T=100 Hz) • > 2 samples per period, • fs > 2 f Sinusoid is characterized by three parameters Amplitude Frequency Phase We need at least three samples per the period

  18. Undersampling T = 10 ms (f = 1/T=100 Hz) T’ = 40 ms (f’= 25 Hz) ts = 7.5 ms (fs=133 Hz < 2f )

  19. Sampling of more complex signals highest frequency component period period Sampling must be at the frequency which is higher than the twice the highest frequency component in the signal !!! fs > 2 fmax

  20. Sampling • Make sure you know what is the highest frequency in the signal spectrum fMAX • Chose sampling frequency fs > 2 fMAX NO NEED TO SAMPLE ANY FASTER !

  21. T Magnitude spectrum 0 1/T 2/T frequency Periodicity in one domain implies discrete representation in the dual domain

  22. Sampling in time implies periodicity in frequency ! T F=1/ts fs = 1/T ts time frequency DISCRETE FOURIER TRANSFORM Discrete and periodic in both domains (time and frequency)

  23. Recovery of analog signal Digital-to-analog converter (“sample-and-hold”) Low-pass filtering

  24. 0.000000000000000 0.309016742003550 0.587784822932543 0.809016526452407 0.951056188292881 1.000000000000000 0.951057008296553 0.809018086192214 0.587786969730540 0.309019265716544 0.000000000000000 -0.309014218288380 -0.587782676130406 -0.809014966706903 -0.951055368282511 -1.000000000000000 -0.951057828293529 -0.809019645926324 -0.587789116524398 -0.309021789427363 -0.000000000000000

  25. Quantization 11 levels 21 levels 111 levels

  26. a part of vowel /a/ 16 levels (4 bits) 32 levels (5 bits) 4096 levels (12 bits)

  27. Quantization • Quantization error = difference between the real value of the analog signal at sampling instants and the value we preserve • Less error  less “quantization distortion”

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