1. I. Previously on IET

2. Complex Exponential Function
Im-Axis ω
Re-Axis

3. Complex Exponential Function
Im-Axis ω
Re-Axis

4. Complex Exponential Function
Im-Axis ω
Re-Axis

5. Complex Exponential Function
Im-Axis ω
Re-Axis

6. Complex Exponential Function
Im-Axis ω
Re-Axis

7. Complex Exponential Function
Im-Axis ω
Re-Axis

8. Complex Exponential Function
Im-Axis ω
Re-Axis

9. Complex Exponential Function
Im-Axis ω
Re-Axis

10. Complex Exponential Function
Im-Axis ω
Re-Axis

11. Complex Exponential Function
Im-Axis ω
Re-Axis

12. The Fourier Transform
Representing functions in terms of complex exponentials with different frequencies

13. The Fourier Transform (Cosine Function)
Im-Axis
Im-Axis + -ω
Re-Axis
Re-Axis ω +

14. The Fourier Transform (Cosine Function)
Im-Axis
Im-Axis + -ω
Re-Axis
Re-Axis ω +

15. The Fourier Transform (Cosine Function)
Im-Axis
Im-Axis + -ω
Re-Axis
Re-Axis ω +

16. The Fourier Transform (Cosine Function)
Im-Axis
Im-Axis + -ω
Re-Axis
Re-Axis ω +

17. The Fourier Transform (Cosine Function)
Im-Axis
Im-Axis + -ω
Re-Axis
Re-Axis ω +

18. The Fourier Transform (Sine Function)
Im-Axis
Im-Axis - -ω
Re-Axis
Re-Axis ω -

19. The Fourier Transform (Sine Function)
Im-Axis
Im-Axis + -ω
Re-Axis
Re-Axis ω +

20. The Fourier Transform (Sine Function)
Im-Axis
Im-Axis -
Re-Axis
Re-Axis -ω ω -

21. The Fourier Transform (Sine Function)
Im-Axis -ω
Im-Axis +
Re-Axis
Re-Axis ω +

22. Fourier Transform of Sinusoids
1/2
1/2
j(1/2) -ω ω -ω ω
-j(1/2)
Notes
A real value for the coefficients in the frequency domain means that the starting point for rotation is on the real axis
An Imaginary value for the coefficients in the frequency domain means that the starting point for rotation is on the imaginary axis

23. Fourier Transform of Real Valued Functions
Im-Axis
Im-Axis
Im-Axis ωn ω1
Re-Axis
Re-Axis
Re-Axis ω2
Im-Axis
Im-Axis
Im-Axis
-ω2
Re-Axis
Re-Axis
Re-Axis
-ω1
-ωn
A real-valued function in time implies that
G(-f) = G*(f)