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Digital Signal Processing

Digital Signal Processing

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Digital Signal Processing

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

  2. Digital Signal Processing Lecture 20 Fourier Transform Properties

  3. READING ASSIGNMENTS • This Lecture: • Chapter 11, Sects. 11-5 to 11-9 • Tables in Section 11-9 • Other Reading: • Recitation: Chapter 11, Sects. 11-1 to 11-9 • Next Lectures: Chapter 12 (Applications)

  4. LECTURE OBJECTIVES • The Fourier transform • More examples of Fourier transform pairs • Basic properties of Fourier transforms • Convolution property • Multiplication property

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

  6. 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)

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

  8. Table of Fourier Transforms

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

  10. Square Wave Signal

  11. Square Wave Fourier Transform

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

  13. Scaling Property

  14. Scaling Property

  15. 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

  16. Significant FT Properties Differentiation Property

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

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

  19. Ideally Bandlimited Signal

  20. Convolution Example

  21. Cosine Input to LTI System

  22. Ideal Lowpass Filter

  23. Ideal Lowpass Filter

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

  25. Frequency Shifting Property

  26. Differentiation Property Multiply by jw