1. Digital Signal Processing
Prof. Nizamettin AYDIN

2. Digital Signal Processing
Lecture 22
Sampling and Reconstruction
(Fourier View)

3. LECTURE OBJECTIVES Sampling Theorem Revisited
GENERAL: in the FREQUENCY DOMAIN
Fourier transform of sampled signal
Reconstruction from samples
Reading: Chap 12, Section 12-3
Review of FT properties
Convolution  multiplication
Frequency shifting
Review of AM

4. Table of FT Properties
Delay Property
Frequency Shifting
Scaling

5. 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).

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

7. DSBAM Demod Phase Synch

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

9. Demod: Quadrature System

10. Quadrature Modulation: 4 sigs
8700 Hz
3600 Hz

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

12. Periodic Impulse Train
Fourier Series

13. FT of Impulse Train

14. Impulse Train Sampling

15. Illustration of Sampling

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

17. Frequency-Domain Analysis

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

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

20. Reconstruction of x(t)

21. Reconstruction: Frequency-Domain

22. Ideal Reconstruction Filter

23. Signal Reconstruction
Ideal bandlimited interpolation formula

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

25. Reconstruction in Time-Domain

26. Ideal C-to-D and D-to-C
Ideal Sampler
Ideal bandlimited interpolator