How to build a phase shifter with arbitrary phase shift
Nice question! It uses one of my favorite trig identities (which can also be used to show that quadrature modulation is actually simultaneous amplitude and phase modulation).


   The Hilbert transform of $\sin(2\pi f_0t)$ is $-\cos(2\pi f_0t)$. Also, $$\sin(2\pi f_0t+\theta)=a\sin(2\pi f_0t)+b\cos(2\pi f_0t)$$ (constrained to $a^2+b^2=1$), with $\theta=\text{atan2}(b,a)$. This suggest one possible approach. Say Fred needs $\theta=2.1$ radians. He calculates $\tan(2.1)\approx-1.71$. Then, he needs to find $a$ and $b$ such that $a^2+b^2=1$ and $b/a=-1.71$, with $a&lt;0$ and $b&gt;0$, which is a simple algebra problem. Set $a_0=-1$, $b_0=1.71$, $n=\sqrt{a_0^2+b_0^2}$, $a=a_0/n$, and $b=b_0/n$. Then, Fred can easily generate a sine with the desired phase by using a Hilbert transformer, two multipliers, two DC sources (one set to $a$ volts and the other to $-b$ volts, to take care of the sign of the cosine), and one adder.


The impulse response of the system described above is given by:


   $a\delta(t)+\frac{b}{\pi t}$


Block diagram: