1. Advanced Digital Signal Processing
Prof. Nizamettin AYDIN

2. Introduction to the Fourier Transform

3. Definition of Fourier transform
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. More examples of Fourier transform pairs
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.

51. Frequency Shifting Property

53. Differentiation Property
Multiply by jw