1. Basics of Signal Processing

2. SIGNALSOURCE RECEIVER 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 ? ACTION

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

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. PROCESSING One way of “signal processing”

23. Going digital

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

25. 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

26. 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)

27. 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

28. 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

29. 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 !

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

31. 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

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

33. 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

34. Quantization 11 levels 21 levels 111 levels

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

36. 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”