Advanced Digital Signal Processing

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

2. Introduction to the Fourier Transform

3. Review • Frequency Response • Fourier Series • Definition of Fourier transform Relation to Fourier Series • Examples of Fourier transform pairs

4. Everything = Sum of Sinusoids • One Square Pulse = Sum of Sinusoids • ??????????? • Finite Length • Not Periodic • Limit of Square Wave as Period  infinity • Intuitive Argument

5. Fourier Series: Periodic x(t) Fourier Synthesis Fourier Analysis

6. Square Wave Signal

7. Spectrum from Fourier Series

8. What if x(t) is not periodic? • Sum of Sinusoids? • Non-harmonically related sinusoids • Would not be periodic, but would probably be non-zero for all t. • Fourier transform • gives a “sum” (actually an integral) that involves ALL frequencies • can represent signals that are identically zero for negative t. !!!!!!!!!

9. Limiting Behavior of FS T0=2T T0=4T T0=8T

10. Limiting Behavior of Spectrum T0=2T T0=4T T0=8T

11. FS in the LIMIT (long period) Fourier Synthesis Fourier Analysis

12. Fourier Transform Defined • For non-periodic signals Fourier Synthesis Fourier Analysis

13. Example 1:

14. Frequency Response • Fourier Transform of h(t) is the Frequency Response

15. Magnitude and Phase Plots

16. Example 2:

18. Example 3:

20. Example 4: Shifting Property of the Impulse

22. Example 5:

24. Table of Fourier Transforms

25. Fourier Transform Properties

26. The Fourier transform • More examples of Fourier transform pairs • Basic properties of Fourier transforms • Convolution property • Multiplication property

27. Fourier Transform Fourier Synthesis (Inverse Transform) Fourier Analysis (Forward Transform)

28. WHY use the Fourier transform? • Manipulate the “Frequency Spectrum” • Analog Communication Systems • AM: Amplitude Modulation; FM • What are the “Building Blocks” ? • Abstract Layer, not implementation • Ideal Filters: mostly BPFs • Frequency Shifters • aka Modulators, Mixers or Multipliers: x(t)p(t)

29. Frequency Response • Fourier Transform of h(t) is the Frequency Response

33. Table of Fourier Transforms

35. Fourier Transform of a General Periodic Signal • If x(t) is periodic with period T0 ,

36. Square Wave Signal

37. Square Wave Fourier Transform

38. Table of Easy FT Properties Linearity Property Delay Property Frequency Shifting Scaling

39. Scaling Property

40. Scaling Property

41. Uncertainty Principle • Try to make x(t) shorter • Then X(jw) will get wider • Narrow pulses have wide bandwidth • Try to make X(jw) narrower • Then x(t) will have longer duration • Cannot simultaneously reduce time duration and bandwidth

42. Significant FT Properties Differentiation Property

43. Convolution Property • Convolution in the time-domain corresponds to MULTIPLICATION in the frequency-domain

44. Convolution Example • Bandlimited Input Signal • “sinc” function • Ideal LPF (Lowpass Filter) • h(t) is a “sinc” • Output is Bandlimited • Convolve “sincs”

45. Ideally Bandlimited Signal

46. Convolution Example

47. Cosine Input to LTI System

48. Ideal Lowpass Filter

49. Ideal Lowpass Filter

50. Signal Multiplier (Modulator) • Multiplication in the time-domain corresponds to convolution in the frequency-domain.