Advanced Digital Signal Processing

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

2. Amplitude Modulation

3. Review of FT properties • Convolution <--> multiplication • Frequency shifting • Sinewave Amplitude Modulation • AM radio • Frequency-division multiplexing • FDM

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

5. Table of FT Properties Differentiation Property

6. Frequency Shifting Property

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

8. Cosine Input to LTI System

9. Ideal Lowpass Filter

10. Ideal LPF: Fourier Series

11. The way communication systems work How do we share bandwidth ?

12. Table of FT Properties Differentiation Property

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

15. Amplitude Modulator • x(t)modulates the amplitude of the cosine wave. The result in the frequency-domain is two shifted copies of X(jw).

18. DSBAM Modulator • If X(jw)=0 for |w|>wb and wc >wb,the result in the frequency-domain is two shifted and scaled exactcopies of X(jw).

19. DSBAM Waveform • In the time-domain, the “envelope” of sine-wave peaks follows |x(t)|

20. Double Sideband AM (DSBAM) Upper sideband Lower sideband “Typical” bandlimited input signal Frequency-shifted copies

21. DSBAM DEmodulator

22. DSBAM Demodulation

23. Frequency-Division Multiplexing (FDM) • Shifting spectrum of signal to higher frequency: • Permits transmission of low-frequency signals with high-frequency EM waves • By allocating a frequency band to each signal multiple bandlimited signals can share the same channel • AM radio: 530-1620 kHz (10 kHz bands) • FM radio: 88.1-107.9 MHz (200 kHz bands)

24. FDM Block Diagram (Xmitter) Spectrum of inputs must be bandlimited

25. Frequency-Division De-Mux

26. Bandpass Filters for De-Mux

27. Pop Quiz: FT thru LPF

28. Sampling and Reconstruction (Fourier View)

29. Sampling Theorem Revisited • GENERAL: in the FREQUENCY DOMAIN • Fourier transform of sampled signal • Reconstruction from samples • Review of FT properties • Convolution  multiplication • Frequency shifting • Review of AM

30. Table of FT Properties Delay Property Frequency Shifting Scaling

31. Amplitude Modulator Phase • x(t)modulates the amplitude of the cosine wave. The result in the frequency-domain is two SHIFTED copies of X(jw).

32. DSBAM: Frequency-Domain Upper sideband Lower sideband “Typical” bandlimited input signal Frequency-shifted copies

33. DSBAM Demod Phase Synch

34. Quadrature Modulator TWO signals on ONE channel: “out of phase” Can you “separate” them in the demodulator ?

35. Demod: Quadrature System

36. Quadrature Modulation: 4 sigs 8700 Hz 3600 Hz

37. Ideal C-to-D Converter FOURIER TRANSFORM of xs(t) ??? • Mathematical Model for A-to-D

38. Periodic Impulse Train Fourier Series

39. FT of Impulse Train

40. Impulse Train Sampling

41. Illustration of Sampling

42. Sampling: Freq. Domain EXPECT FREQUENCY SHIFTING !!!

43. Frequency-Domain Analysis

44. Frequency-Domain Representation of Sampling “Typical” bandlimited signal

45. Aliasing Distortion • If ws< 2wb, the copies of X(jw) overlap, and we have aliasing distortion. “Typical” bandlimited signal

46. Reconstruction of x(t)

47. Reconstruction: Frequency-Domain

48. Ideal Reconstruction Filter

49. Signal Reconstruction Ideal bandlimited interpolation formula

50. Shannon Sampling Theorem • “SINC” Interpolation is the ideal • PERFECT RECONSTRUCTION • of BANDLIMITED SIGNALS