(a) Show that the kernels K() in Eq. (6.47) produce the indicated outputs w(t). Deduce the ratio...

(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)

Transcribed Image Text:

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