(a) Show that the kernels K() in Eq. (6.47) produce the indicated outputs w(t). Deduce the ratio S w (f)/S y (f ) =|K(f)| 2

(a) Show that the kernels K(τ) in Eq. (6.47) produce the indicated outputs w(t). Deduce the ratio S_w(f)/S_y(f ) =|K̃(f)|² in two ways: 

(i) By Fourier transforming each K(τ); 

(ii) By setting y = e^i2πft , deducing the corresponding filtered output w directly from the expression for w in terms of y, and then squaring to get|K̃ (f)|².

(b) Derive Eqs. (6.52b) and (6.52c) for the kernel (6.52a).

Data from Eq. (6.47)

K(T) = 0 for t < 0, K(T) = 2 T for t > 0; (6.52a)