Question

4) When the correlation is done with a perfectly aligned symbol the result is t100 What is the value of the correlation with the symbol is off by 45° ? 05 05 20 40 100 130 +

Answer 1

The correlation between two signals which is displaced by any angle is the cosine of that angle. Proof: the correlation between two signals is given as let g_1 (t) and g_2 (t) are the two signals correlation = integral g_1 (t) g_2 (t) dt/squareroot integral [g_1 (t)]^2 dt squareroot integral [g_2 (t)]^2 dt according to question: g_1 (t) = sin t g_2 (t) = sin (t + 45) numerator integral sin t sin (t + 45) dt over a period of 2 pi = integral^2 pi_0 sin t sin (t + 45) dt = 1/2 integral_0^2 pi [cos (t - (t + 45)) - cos (t + t + 45)] dt = 1/2 integral^2 pi_0 [cos 45 - cos (2t + 45)] dt = pi cos 45 \r\ denominator = squareroot integral^2 pi_0 (sin t)^2 dt squareroot integral^2 pi_0 (sin (t) - 45)^2 dt = squareroot 1/2 2 pi squareroot 1/2 2 pi