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Lecture 2: Sinusoids

Reading assignments

• Ch. 2: Sinusoids

Topics Today

• Standard form
• Two equivalences
  • Formula ←→ plot
  • Sinusoid delay ←→ phase shift
• Complex amplitudes
• Phasor addition theorem

Quick brush-up on sinusoids

• Fixed radius/magnitude of the vector
• As angle increases, the triangle changes shape

• shown in red, shown in blue
• Blue sinusoid has 4x frequency of red

A peek into later chapters: Any signal can be written as a sum of sinusoids

Representing frequency

• Note the sinusoid is periodic. After a fundamental period , its values repeat

Units of frequency

• = period [s, s/cycle]
• = frequency (hertzian) [Hz, s^-1, cycle/s]
• = frequency (radian) [rad/s]

Defining a sinusoid

Most general form of a sinusoid

Three parameters that define a sinusoid

• amplitude
• frequency
• phase (can be derived from time delay -- see 05.18 Lecture - ECE2026#Converting Equivalences)

Standard Form

“Standard form” constraints make parameters in a sinusoid unique

Standard Form -- Exercise

Quick refresher on Trig

Tuning Fork Example

[r, fs] = audioread('tuningfork440.mp3'); % audio clip attached to slides
t = (1:length(r)) / fs;
plot(t,r);

soundsc(r,fs);
soundsc(cos(2*pi*440*t), fs);

ginput(2) % measuring wavelength
T0 = diff(ans(:,1))
f0 = 1 / T0

• Looks sinusoidal, as you zoom in in MATLAB
• 440 Hz?
  • Verify by measuring wavelength/period
  • Built-in tool in MATLAB when in plot
• Compare sound to pure case:

Converting Equivalences

Two skills

1. From formula to plot
2. From plot to formula

Both based on equivalence of delay and phase shift:

• Therefore, ⭐⭐⭐
• is also written as t_p (time of peak) and t_p (time delay)

Skill 1: Given formula/equation, draw plot

Solution

Steps

• Locate the first peak, which can be found by setting cos() arguments to 0
• Locate another peak by setting cos() arguments to
• Location of peak at t=10s
• Determine the phase

Note: You can arrive at the next peak by incrementing/decrementing by T_0=1/f_0. Additionally, when using the formula, the time for any peak can be chosen. The simplest choice would be the closest peak in this example

Skill 2: Given plot, write formula

Solution

1. Measure time between peaks
2. Measure time delay . In this case, you can use (wait i have some words here)
3. Measure amplitude

Last step: Multiply everything out

Representing Sinusoids via Complex Exponentials

Euler

Graphically, plotted on the unit circle

• Fixed value of theta
• Results in an triangle with length values able to be determined using trig functions

Complex Amplitude

To get a general sinusoid, change from a set magnitude of 1 to a general complex amplitude (which represents the starting point in a rotating phasor)

• Circle radius A
• Starting angle phi
• Introduces phase and amplitude to our “pure sinusoid”

Going back to Euler’s

Theta is a function of time

Therefore…

Note real component of z(t) looks familiar!

Important to note

• Starting vector * frequency of rotation

Complex Amplitude Represents

Note how more compact it is to represent sinusoids in polar form and complex amplitudes, as we know the frequency.

Sum of Sinusoids

Motivating question: What is the result of the following expression?

While not as obvious, the sum is also a sinusoid, of the same frequency!

Key Result

The sum of 2 sinusoids having the same frequency is a single sinusoid of that frequency, regardless of amplitudes and phases!

Why? Phasors Make it Clear

• Essentially, thinking geometrically again, we are adding the starting position of the two sinusoids, which are represented in the complex amplitude
• Essentially performing vector addition

Phasor addition rule

Here it is, restated once more

We know:

• Vector addition

Any number of sinusoids can be added this way

• Important condition: All sinusoids must have the same frequency

Exercise

• Would be solved using a calculator
• A represents the combined magnitude
• phi represents the combined phase

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Next lecture: 05.23 Lecture - ECE2026