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

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

Sampled Sinusoids sampling interval

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

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

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:

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

Complex Numbers Recall: where Then:

Example Let: then

Example Let: then

Euler Formulas Since: Then:

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

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

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.