Quadrature phase: two signals that are shifted 90°/2π radians from each other

Centre frequency: the frequency at which the unmodulated signal (the carrier wave) is transmitted

Proof:

• You may be familiar with the angle sum identity:

$$ sin(a+b)=sin(a)cos(b)+cos(a)sin(b) $$

• Knowing that $cos(a)=sin(a+\frac{\pi}{2})$, if we shift the signal by -90 degrees, we get the following:

  $$ sin(a+\frac{\pi}{2}+b)=sin(a+\frac{\pi}{2})cos(b)+cos(a+\frac{\pi}{2})sin(b) $$

  $$ cos(a+b)=cos(a)cos(b)+cos(a+\frac{\pi}{2})sin(b) $$

• If we let the carrier frequency of a signal be $f$, its time-varying amplitude be $A(t)$, and its time-varying phase shift $\varphi(t)$

• We can represent any arbitrary amplitude and phase modulated signal with $A(t)cos(2\pi f t+\varphi(t))$. Note that FM signals are essentially created by shifting phase as well. You can try this out yourself in Desmos:

• (Interesting thing to note: MSK signals, which are FSK signals with continuous phase, have continuous and differentiable functions for g(t))
• We can represent the signal using two varying amplitude cosines, 90 degrees out of phase with one another, but with the same carrier frequency:

• LHS is called the Amplitude-phase form, while RHS is called the quadrature-carrier or IQ form
• Technical detail about orthogonality I don’t fully understand: these two functions are not actually orthogonal, but are assumed to be because $A(t)$, $\varphi(t)$ are considered to be slowly varying compared to $2\pi ft$. If someone ends up understanding why that’s important, please add notes here.