1. Sinusoids: continuous time
Amplitude
Frequency Hz
Phase radians
delay
amplitude
period

2. Example of a Sinusoid
delay
period
amplitude
Then we can compute:

3. Sampled Sinusoids
sampling interval

4. Analog and Digital Frequencies
Analog Frequency in Hz (1/sec)
Digital Frequency in radians (no dimensions)

5. Example Analog Frequency: Sampling Frequency: Digital Frequency:
Given:
Analog Frequency:
Sampling Frequency:
compute:
Digital Frequency:

6. Complex Numbers where A complex number is defined as Real Part
Imaginary Part
where
It can be expressed as a vector in the Complex Plane:

7. Complex Numbers: Magnitude and Phase
You can represent the same complex number in terms of magnitude and phase:
Then:

8. Complex Numbers
Recall:
where
Then:

9. Example
Let:
then

10. Example
Let:
then

11. Euler Formulas
Since:
Then:

12. Complex Exponentials
Using Euler’s Formulas we can express a sinusoidal signal in terms of complex exponentials:
This can be written as:

13. Complex Exponentials
Same for Discrete Time:
which can be written as:

14. Why this is Important?
It is much easier to deal with complex exponentials than with sinusoids.
In fact:
differentiation, integration, time delay are just multiplications and division, for complex exponentials only.