Explanation of sinusoidal function: $f(x,y) = A\cos(2\pi(ux + vy) + \phi)$

I never got to take a signals and systems course and this has come up in the math of my image processing review.

Can this be explained? The equation for a sinusoidal signal is $f(x,y) = A\cos(2\pi(ux + vy) + \phi)$ What do these pieces mean and what is a good way of visualizing it?

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Have you tried plotting it? Would that help? –  Patrick Da Silva Jan 17 '12 at 3:33
A start: The absolute value of $A$ is the amplitude. If $ux+vy\ne 0$, then $1/(|ux+vy|)$ is the period. –  André Nicolas Jan 17 '12 at 5:37
More: So if say $u$ and $v$ are constants, the signal has period $1/|k|$ om the line $ux+vy=k$. The constant $\phi$ represents a phase shift from the standard cosine function. –  André Nicolas Jan 17 '12 at 6:44

It think it would be best to see first what happens in one dimension.

I have an example for the function $y=A\cos(\omega\theta+\phi)$.

• For the graph of $$y=A\cos(\omega\theta+\phi)$$
• $A$ is the amplitude. The crest of the wave has height $A$.
• $\omega=2\pi f$ where $f$ is the frequency. One complete wave is generated over an interval of length $T={1/ f}={2\pi\over\omega}$.
• $\phi/\omega$ is the phase shift. The wave "starts'' at $(-\phi/\omega,0)$.

If you like, you may download an interactive version of the above diagram here. You may use the sliders on the top, left, and bottom, (by clicking and dragging the "circle") to change the parameters $A$, $\phi$, and $\omega$.

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