Frequency & Phase
Speed and starting point
In the first lesson, you saw that sine and cosine come from a point moving around a circle. You even played with the speed slider and watched the wave compress and stretch.
That speed has a name: frequency. And there's a second property, phase, that tells you where on the circle the point starts. Together, they completely describe any pure tone or oscillation.
Frequency: How Fast It Spins
Frequency measures how many complete circles the point makes per second. One full trip around = one cycle.
- f = frequency in Hertz (Hz) = cycles per second
- ω = angular frequency in radians per second = 2πf
A 1 Hz signal goes around the circle once per second. A 440 Hz signal (the note A above middle C) goes around 440 times per second. The math is the same, just faster.
The ω = 2πf Connection
You already know from the Radians lesson that one full circle = 2π radians. So if the point completes f circles per second, it covers:
ω = 2πf radians per second
This is angular frequency. In DSP, you'll see ω far more often than f, because it plugs directly into the sine function:
sin(ωt) = sin(2πft)
Phase: Where It Starts
So far, our point always starts at the 3 o'clock position on the circle (angle = 0). But what if it starts somewhere else?
The phase (φ, pronounced "phi") is the starting angle. Change the phase, and the entire wave shifts left or right:
The Complete Sinusoid
Every pure sinusoidal signal in DSP can be written as:
x(t) = A · sin(ωt + φ)
Three parameters, three meanings:
| Symbol | Name | What it controls |
|---|---|---|
| A | Amplitude | How tall the wave is (radius of the circle) |
| ω | Angular frequency | How fast the point spins (radians/sec) |
| φ | Phase | Where the point starts on the circle |
Why Phase Matters in DSP
You might think: if the wave looks the same, who cares where it starts? Here's why:
- Interference: Two signals at the same frequency can amplify or cancel each other depending on their phase difference. In-phase = double the amplitude. 180° out of phase = total cancellation.
- Filters: Every filter changes the phase of different frequencies. Understanding this is essential for filter design.
- Communication: Phase-shift keying (PSK) encodes data by changing the phase of a carrier wave, and it's how WiFi and cellular signals carry information.
Frequently Asked Questions
What is the frequency of a signal?
Frequency is how many complete cycles a signal makes per second, measured in Hertz (Hz). A 1000 Hz signal oscillates 1000 times per second. Angular frequency ω = 2πf gives the same information in radians per second, which is more natural for the math of signal processing.
What is phase in signal processing?
Phase is the starting position of a wave within its cycle, measured in radians or degrees. A phase of 0 means the wave starts at zero crossing upward. A phase of π/2 means it starts at its peak. Phase differences between signals determine whether they reinforce or cancel when combined.
Quick Check
Test your understanding of the key concepts from this lesson.
