Fourier_Transform_Properties_Full_Presentation.pptx

Fuzzy logic controller with MATLAB

1. Properties of the Fourier Transform Analog and Digital Signals & Systems Assistant Professor: Dr. Adel Rawea Department of Mechatronics Engineering

2. Introduction Fourier Transform is a vital tool for analyzing signals. Its properties simplify signal/system analysis. Applications: communication, robotics, control.

3. Linearity Property F{αx1+βx2} = αX1+βX2 Superposition applies to FT Example: cos+sin → sum in frequency

4. Time Shifting F{x(t-t0)} = X(jω)e^{-jωt0} Time delay ↔ phase shift Magnitude unchanged

5. Frequency Shifting F{x(t)e^{jω0t}} = X(j(ω-ω0)) Time modulation ↔ frequency translation

6. Time Scaling F{x(at)} = 1/|a| X(jω/a) Compression in time ↔ expansion in frequency

7. Convolution in Time F{x1*x2} = X1·X2 Convolution in time ↔ multiplication in frequency

8. Modulation Property F{x(t)cos(ω0t)} = ½[X(ω-ω0)+X(ω+ω0)] Spectrum shifted around ±ω0

9. Conclusion FT properties map time ↔ frequency operations Simplify signal/system analysis Essential in mechatronics and communications