How do you relate imaginary numbers with phase shift? How to imagine this?

Question

So I found this answer on one of the questions. I wanted to ask how the complex part can be related with the phase shift. I wanted to ask this on the same question but it didn't allow me. Could anyone please help me understand this concept. Even if I imagine this as a spiral with the value of the imaginary part changing, the starting time of the wave remains the same doesn't it? So how is it related to phase shift?

"Negative frequency doesn't make much sense for sinusoids, but the Fourier transform doesn't break up a signal into sinusoids, it breaks it up into complex exponentials:

$F(\omega)=\int_{-\infty}^{\infty} f(t)e^{−jωt}dt$

These are actually spirals, spinning around in the complex plane:

complex exponential showing time and real and imaginary axes

Spirals can be either left-handed or right-handed (rotating clockwise or counterclockwise), which is where the concept of negative frequency comes from. You can also think of it as the phase angle going forward or backward in time.

In the case of real signals, there are always two equal-amplitude complex exponentials, rotating in opposite directions, so that their real parts combine and imaginary parts cancel out, leaving only a real sinusoid as the result. This is why the spectrum of a sine wave always has 2 spikes, one positive frequency and one negative. Depending on the phase of the two spirals, they could cancel out leaving a purely real sine wave or a real cosine wave or a purely imaginary sine wave, etc."

Answer 1

Paint a red dot on the blue complex spiral at t=0, where the value is 1.0. Now roll the spiral upside-down until the red dot is at -1.0, still at t=0. This roll would be equivalent to a phase shift of Pi. Note that the real part no longer corresponds to a cosine wave that starts at t=0.

But the real part of the rolled spiral does correspond to a cosine wave that starts (with value 1.0) at +Pi or -Pi, which is a different starting time, contradicting one of your assumptions above.

Thus phase is related to starting time.

Answer 2

The way I understand it is with a equation like

y=sin(x)*cos(n)+cos(x)*sin(n)

plotting y vs x, if you change the value of n from [0, π/2], you will see a sine wave move in phase from [0,-π/2] as cos(n) goes from [1, 0] and sin(n) goes the opposite way from [0, 1], so you are varying the amount to sine component vs cosine component to generate a phase shifting sine wave. This comes from the trig identities

sin(x+y) = sin(x)*cos(y)+cos(x)*sin(y)

The Fourier transform performs a series of correlations with sine wave and cosine wave giving you an amount of sine correlation vs cosine correlation for each frequency. The sine and cosine represent orthogonal components of the signal and so to determine the amplitude you need to sum the two vectors that are at right angles to each other, the angle of the resulting vector is the phase and the amplitude of the resulting vector is the amplitude.

Answer 3

I think the 3D plot is making this concept more difficult than it really is.

If your signal is f(t) = cos(2πft), then the phase of f(t) is simply 2πft. The real part of the vector is cos(2πft) and the imaginary part is sin(2πft).

I am not sure why someone would use a 3D plot to describe this. These concepts remain on the complex plane, and the angle of the vector (its phase) is simply 2πft.

The 3D plot you show is how we explain the phase of a propagating plane wave. Then the distance between each loop of the spiral is called the wavelength and is equal to c*T where c is the speed of light and T is the period of the signal. These concepts are important to RF engineers for example, who need to understand the phase of the voltage and current at each point in space as the signal propagates.

I think you will have less difficulty if you stay on the complex plane.