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- 1. Communication System Ass. Prof. Ibrar Ullah BSc (Electrical Engineering) UET Peshawar MSc (Communication & Electronics Engineering) UET Peshawar PhD (In Progress) Electronics Engineering (Specialization in Wireless Communication) MAJU Islamabad E-Mail: ibrar@cecos.edu.pk Ph: 03339051548 (0830 to 1300 hrs) 1
- 2. Chapter-3 • • • • • • • • • Aperiodic signal representation by Fourier integral (Fourier Transform) Transforms of some useful functions Some properties of the Fourier transform Signal transmission through a linear system Ideal and practical filters Signal; distortion over a communication channel Signal energy and energy spectral density Signal power and power spectral density Numerical computation of Fourier transform 2
- 3. Fourier Transform Motivation • The motivation for the Fourier transform comes from the study of Fourier series. • In Fourier series complicated periodic functions are written as the sum of simple waves mathematically represented by sines and cosines. • Due to the properties of sine and cosine it is possible to recover the amount of each wave in the sum by an integral • In many cases it is desirable to use Euler's formula, which states that e2πiθ = cos 2πθ + i sin 2πθ, to write Fourier series in terms of the basic waves e2πiθ. • From sines and cosines to complex exponentials makes it necessary for the Fourier coefficients to be complex valued. complex number gives both the amplitude (or size) of the wave present in the function and the phase (or the initial angle) of the wave. 3
- 4. Fourier Transform • The Fourier series can only be used for periodic signals. • We may use Fourier series to motivate the Fourier transform. • How can the results be extended for Aperiodic signals such as g(t) of limited length T ? 4
- 5. Fourier Transform Toois made long enough T is made long enough to avoid overlapping to avoid overlapping between the repeating between the repeating pulses pulses The pulses in the periodic signal repeat after an infinite interval 5
- 6. Fourier Transform Observe the nature of the spectrum changes as To increases. Let define G(w) a continuous function of w Fourier coefficients Dnnare Fourier coefficients D are 1/Tootimes the samples of 1/T times the samples of G(w) uniformly spaced at G(w) uniformly spaced at woorad/sec w rad/sec 6
- 7. Fourier Transform is the envelope for the coefficients Dn Let To → ∞ by doubling To repeatedly Doubling Toohalves the Doubling T halves the fundamental frequency fundamental frequency wooand twice samples in w and twice samples in the spectrum the spectrum 7
- 8. Fourier Transform If we continue doubling To repeatedly, the spectrum becomes denser while its magnitude becomes smaller, but the relative shape of the envelope will remain the same. To → ∞ wo → 0 Dn → 0 Spectral components are spaced at Spectral components are spaced at zero (infinitesimal) interval zero (infinitesimal) interval Then Fourier series can be expressed as: ⇒ 8
- 9. Fourier Transform As 9
- 10. Fourier Transform ⇒ 10
- 11. Fourier Transform 11
- 12. Fourier Transform 12
- 13. Fourier Transform 13
- 14. Example 3.1 Solution: G ( w) = ∞ g ( t ) e − jwt dt ∫ −∞ ⇒ 14
- 15. Example 3.1 Fourier spectrum 15
- 16. Compact Notation for some useful Functions 16
- 17. Compact Notation for some useful Functions 2) Unit triangle function: 17
- 18. Compact Notation for some useful Functions 3) Interpolation function sinc(x): The function sin x “sine over argument” is called sinc x function given by sinc function plays an sinc function plays an important role in signal important role in signal processing processing 18
- 19. Fourier Fourier series: and Fourier transform: and 19
- 20. Some useful Functions 20
- 21. Some useful Functions 2) Unit triangle function: 21
- 22. Some useful Functions 3) Interpolation function sinc(x): The function sin x “sine over argument” is called sinc x function given by sinc function plays an sinc function plays an important role in signal important role in signal processing processing 22
- 23. Example 3.2 Consider ⇒ Fourier transform Fourier transform 23
- 24. Example 3.2 Therefore 24
- 25. Example 3.2 Spectrum: 25
- 26. Example 3.3 26
- 27. Example 3.4 Spectrum of aaconstant signal g(t) =1 is an Spectrum of constant signal g(t) =1 is an impulse impulse 2πδ ( w ) Fourier transform of g(t) is spectral representation of everlasting exponentials Fourier transform of g(t) is spectral representation of everlasting exponentials components of of the form e jwt . .Here we need single exponential e jwt components of of the form Here we need single exponential component with w = 0, results in a single spectrum at a single frequency component with w = 0, results in a single spectrum at a single frequency w=0 w=0 27
- 28. Example 3.5 Spectrum of the everlasting exponential Spectrum of the everlasting exponential e jw o t is aasingle impulse at w = w o is single impulse at Similarly we can represent: 28
- 29. Example 3.6 According to Euler formula: As and and 29
- 30. Example 3.6 Spectrum: 30
- 31. Some properties of Fourier transform 31
- 32. Some properties of Fourier transform 32
- 33. Some properties of Fourier transform 33
- 34. Example 3.5 Spectrum of the everlasting exponential Spectrum of the everlasting exponential e jw o t is aasingle impulse at w = w o is single impulse at Similarly we can represent: 34
- 35. Example 3.6 According to Euler formula: As and and 35
- 36. Example 3.6 Spectrum: 36
- 37. Some properties of Fourier transform 37
- 38. Some properties of Fourier transform 38
- 39. Some properties of Fourier transform Symmetry of Direct and Inverse Transform Operations— 1- Time frequency duality: •g(t) and G(w) are remarkable similar. •Two minor changes, 2π and opposite signs in the exponentials 39
- 40. Some properties of Fourier transform 2- Symmetry property 40
- 41. Some properties of Fourier transform Symmetry property on pair of signals: 41
- 42. Some properties of Fourier transform 42
- 43. Some properties of Fourier transform 3- Scaling property: 43
- 44. Some properties of Fourier transform The function g(at) represents the function g(t) compressed in time by a factor a The scaling property states that: Time compression → spectral expansion Time expansion → spectral compression 44
- 45. Some properties of Fourier transform Reciprocity of the Signal Duration and its Bandwidth As g(t) is wider, its spectrum is narrower and vice versa. Doubling the signal duration halves its bandwidth. Bandwidth of a signal is inversely proportional to the signal duration or width. 45
- 46. Some properties of Fourier transform 4- Time-Shifting Property Delaying aasignal by Delaying signal by its spectrum. its spectrum. to does not change does not change Phase spectrum is changed by Phase spectrum is changed by − wt o 46
- 47. Some properties of Fourier transform Physical explanation of time shifting property: Time delay in a signal causes linear phase shift in its spectrum 47
- 48. Some properties of Fourier transform 5- Frequency-Shifting Property: Multiplication of aasignal by aafactor of Multiplication of signal by factor of e jwo t shifts its spectrum by shifts its spectrum by w = wo 48
- 49. Some properties of Fourier transform e jwot is not a real function that can be generated In practice frequency shift is achieved by multiplying g(t) by a sinusoid as: Multiplying g(t) by aasinusoid of frequency Multiplying g(t) by sinusoid of frequency shift the spectrum G(w) by ± wo shift the spectrum G(w) by Multiplication of sinusoid by g(t) amounts to modulating the sinusoid amplitude. This type of modulation is called amplitude modulation. 49 wo

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