Frequency response with an odd real factor breaks the |H| cos shortcut

The rule A cos(w0 n + t) -> |H(w0)| A cos(w0 n + t + angle H(w0)) requires H(-w) = H*(w), meaning a real impulse response. Filters like H(w) = w e^(j pi cos w) fail this because the factor w is odd while the exponential is even. Test the symmetry with one number, then split the sinusoid into e^(jt) and e^(-jt), multiply them by H(+w0) and H(-w0) separately, and recombine. The output can be complex even for a real input, such as sin(t) becoming -j(pi/2)cos(t) when H(+w0) = pi/2 and H(-w0) = -pi/2.