Fourier_Transform_Properties_All_Diagrams.pptx

Fourier Transform Properties

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

2. Introduction Fourier Transform links time and frequency domains Properties simplify complex signal operations Applications in communications, robotics, control

3. Linearity Property F{αx1+βx2} = αX1+βX2 Superposition principle Example: cos+sin

4. Conjugation Property F{x*(t)} = X*(-ω) Conjugation ↔ conjugate and flip

5. Area Under x(t) ∫x(t)dt = X(0) Time area = DC spectrum value

6. Area Under X(ω) ∫X(ω)dω = 2πx(0) Frequency area = value at t=0

7. Time Reversal F{x(-t)} = X(-ω) Flip in time ↔ flip in frequency

8. Time Scaling F{x(at)} = 1/|a| X(ω/a) Compression ↔ Expansion

9. Time Shifting F{x(t-t0)} = X(ω)e^{-jωt0} Delay ↔ phase shift

10. Frequency Shifting F{x(t)e^{jω0t}} = X(ω-ω0) Translation in frequency

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

12. Differentiation in Time F{dx/dt} = jωX(ω) Differentiation ↔ multiply by jω

13. Integration in Time F{∫x} = (1/jω)X(ω)+πX(0)δ(ω) Integration ↔ divide by jω

14. Differentiation in Frequency F{t x(t)} = j dX/dω Multiply by t ↔ differentiate X

15. Modulation Property F{x(t)cos(ω0t)} = ½[X(ω-ω0)+X(ω+ω0)] Spectrum replication at ±ω0

16. Conclusion Fourier properties connect domains Simplify system analysis Essential in engineering applications