4.3 Complex lms

LMS for complex data and coefficients (such as quadrature communication systems) takes the form $$y_k=W_k^H{X}_k$$ $$e_k=d_k-y_k$$ $$W_{k+1}=W_k+2e_k^{*}{X}_k$$ It is derived in exactly the same way as LMS, using the following complex vector differentiation formulas $$\frac{d P^{H}W}{d W}=0$$ $$\frac{d W^{H}P}{d W}=2P$$ $$\frac{d W^{H}RW}{d W}=2RW$$ or by differentiating with respect to the real and imaginary parts separately and recombining the results.