2.8

(a) Consider a signal f(t) with unit energy defined over the time interval [0, 7] that is zero outside that time interval. For example,

f(t) = √T/Tpr (1). Consider two signals of duration NT:

so(t) =

N-1

are orthogonal.

Σsouf(t-iT)

i-0

N-1

si(t) = Σsuf(t-iT).

132

Determine (so(1), $1(1)) in terms of the sequence Sq, i = 0, 1,...,N-1 and $₁,,i = 0, 1,..., N-1.

1-0

(b) Consider the two signals of duration 27 generated from f(t) and the two vectors so = (+1, +1) and s₁ = (+1,-1). We will

form a matrix of vectors. In this case, N = 2:

H₂ =

50

51

+1

(39)

where the first row can be used to generate a first signal and the second row used to generate a second signal. Using part (a), show that

the two signals

so(t) = 50,0 f(t)pr (1) + 50,1f(t-T)pr(t-T)

$₁(t) = $1,0 f(t)pr(t) + $1,1ƒ(t – T)pr(t – T)