1. Basics of Signal Processing

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

4. sinecosine Phase 

5. Sinusoidal grating of image

6. Fourier idea –describe the signal by a sum of other well defined signals TOTO

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

8. A good simple function T i =T 0 / i

10. e.t.c. ad infinitum

11. T=1/f e.t.c…… T=1/f e.t.c……

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

13. Orthogonality

14. 0 + - 0 + - 0 + - x 0 + + -- x 0 ++ = 0 ++ - - = area is positive (T/2)area is zero

15. f(t) = 1.5 + 3sin(2πt/T) + 4sin(4πt/T) = + + 0 T 0 T 1.5 = + + 0 T 030 One example of a function

16. = 00b 1 T/200 …………… area=b 1 T/2 area=b 2 T/2

17. T=2  + - ++ ++++ + + + + ++ -- ---- f(t) f(t) sin(2πt) f(t) sin(4πt) area = DCarea = b 1 T/2area = b 2 T/2

18. Spacing of spectral components is 1/T Periodicity in one domain (here time) implies discrete representation in the dual domain (here frequency) 01/T2/T frequency 01/T2/T frequency Phase spectrum Magnitude spectrum

19. Aperiodic signal Discrete spectrum becomes continuous (Fourier integral) 01/T 0 2/T 0 frequency 01/T 0 2/T 0 frequency Phase spectrum Magnitude spectrum Spacing of spectral components is f 0 =1/T 0

20. Magnitude spectrum of voiced speech signal frequency log | S(  ) | F1F1 F2F2 F3F3 F4F4 f0f0

21. 1/f 0 1/F 1 1/F 2 Waveform Logarithmic power spectrum f0f0 F1F1 F2F2 F3F3

22. Going digital

23. sampling 22 samples per 4.2 ms  0.19 ms per sample  5.26 kHz t s =1/f s

24. Sampling > 2 samples per period, f s > 2 f T = 10 ms (f = 1/T=100 Hz) Sinusoid is characterized by three parameters 1.Amplitude 2.Frequency 3.Phase We need at least three samples per the period

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

26. Sampling at the Nyquist frequency 2 samples per period, f s = 2 f Nyquist rate t s = 5 ms (f s =200 Hz) ? ?? f s > 2 f

27. Sampling of more complex signals period highest frequency component Sampling must be at the frequency which is higher than the twice the highest frequency component in the signal !!! f s > 2 f max

28. Sampling 1.Make sure you know what is the highest frequency in the signal spectrum f MAX 2.Chose sampling frequency f s > 2 f MAX NO NEED TO SAMPLE ANY FASTER !

29. Periodicity in one domain implies discrete representation in the dual domain 01/T2/T frequency Magnitude spectrum T

30. frequency F =1/t s f s = 1/T time tsts T Sampling in time implies periodicity in frequency ! Discrete and periodic in both domains (time and frequency) DISCRETE FOURIER TRANSFORM

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

32. 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 -0.951057828293529 -0.809019645926324 -0.587789116524398 -0.309021789427363 -0.000000000000000

33. Quantization 11 levels 21 levels 111 levels

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

35. 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”

36. Homework 2 Create for your fun (and education) various periodic functions using Fourier principle (and listen to the created signals)

37. Homework 3 What happens if the signal is not periodic and why?

38. Homework 4 How would you transmit the signal below and why? t0t0 t0t0

39. Homework 5 A signal generator produced a triangular wave which was sampled as indicated below. Was the signal samples correctly? If not, what went wrong and how would you fix it? time